The thermal conductivity of magnesium silicon nitride, MgSiN2, ceramics and related materials
Citation for published version (APA): Bruls, R. J. (2000). The thermal conductivity of magnesium silicon nitride, MgSiN2, ceramics and related materials. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR535906
DOI: 10.6100/IR535906
Document status and date: Published: 01/01/2000
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Download date: 04. Oct. 2021 The Thermal Conductivity of Magnesium Silicon Nitride,
MgSiN2, Ceramics and Related Materials
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. M. Rem, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 5 oktober 2000 om 16.00 uur
door
Richard Joseph Bruls
geboren te Sittard Dit proefschrift is goedgekeurd door de promotoren: prof.dr. R. Metselaar en prof.dr. K. Itatani
Copromotor: dr. H.T. Hintzen
Druk: Universiteitsdrukkerij, Technische Universiteit Eindhoven
CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN
Bruls, Richard J.
The Thermal Conductivity of Magnesium Silicon Nitride, MgSiN2, Ceramics and Related Materials / by Richard J. Bruls. - Eindhoven: Technische Universiteit Eindhoven, 2000. - Proefschrift. - ISBN 90-386-3011-5 NUGI 813
Trefwoorden: keramische materialen; nitriden / warmtegeleidbaarheid / phononen Subject headings: ceramic materials; nitrides / thermal conductivity / phonons
Kaft: temperatuur-tijd afhankelijkheid van een thermische diffusiviteitsmeting met een atomic force microscoop thermisch beeld als achtergrond. Aan mijn ouders en grootouders
Aan Marianne
Table of contents
Chapter 1. Introduction 11
1. General introduction 11 2. Substrate materials 13 2.1. Requirements 13 2.2. Relation between heat conduction and material characteristics 14 2.3. AlN as a promising substrate material 16
2.4. The new ceramic material MgSiN2 18 3. Objective and outline 20 References 22
Chapter 2. Preparation and characterisation of MgSiN2 powders 29
1. Introduction 30 2. Experimental section 30 2.1. Starting materials 30 2.2. Preparation 31 2.3. Characterisation 32 3. Results and discussion 33 3.1. Starting powder characteristics 33
3.1.1. Mg3N2 33
3.1.2. Si3N4 35
3.2. Phase formation of MgSiN2 36
3.3. Oxygen content of the MgSiN2 powders 41
3.4. X-ray diffraction data of MgSiN2 44 3.5. Powder characteristics 49
3.6. Oxidation behaviour of MgSiN2 powders 52 4. Conclusions 53 References 55
5 Table of contents
Chapter 3. Preparation, characterisation and properties of MgSiN2 ceramics 59
1. Introduction 59 2. Experimental 61 2.1. Preparation 61 2.2. Characterisation 64 2.3. Properties 66 3. Results and discussion 67 3.1. Characterisation 67 3.1.1. Phase formation and lattice parameters of MgSiN2 67 3.1.2. Density 71 3.1.3. Chemical composition 72 3.1.4. Microstructure 74 3.1.5. TEM/EDS 77 3.2. Properties 80 3.2.1. Oxidation resistance 80 3.2.2. Hardness 81 3.2.3. Young's modulus 82 3.2.4. Thermal expansion 82 3.2.5. Thermal diffusivity/conductivity 83 4. Theoretical considerations 86 4.1. Secondary phases 86 4.2. Grain size 87 4.3. Defects 87 4.4. Maximum influence of secondary phases, grain size and defects 88 5. Conclusions 89 References 89
Chapter 4. Anisotropic thermal expansion of MgSiN2 97
1. Introduction 97 2. Experimental procedure 99 3. Results and discussion 101
6 Table of contents
3.1. Neutron diffraction data refinement 101 3.2. Thermal expansion 107 4. Conclusions 112 References 112
Chapter 5. The heat capacity of MgSiN2 117
1. Introduction 117 2. Experimental 119 2.1. Adiabatic calorimeter measurements 119 2.2. Differential scanning calorimeter measurement 120 3. Results and discussion 121 o 3.1. Cp of MgSiN2 121
3.2. Debye temperature of MgSiN2 124 o o o 3.3. Thermodynamic functions ST , (HT - H0 ) and o o (GT - H0 ) of MgSiN2 127 o 3.4. H0 of MgSiN2 130 4. Conclusions 133 References 133
Chapter 6. The Young's modulus of MgSiN2, AlN and Si3N4 137
1. Introduction 137 2. Experimental section 138 3. Results and discussion 140 3.1. Evaluation of the measurements 140 3.2. Interpretation of the fitting parameters 144
3.2.1. E0 144
3.2.2. B and T0 145 4. Conclusions 146 References 147
Chapter 7. The Grüneisen parameters of MgSiN2, AlN and β-Si3N4 153
1. Introduction 153 2. Evaluation of the input parameters 156
7 Table of contents
2.1. Lattice linear thermal expansion coefficient α lat 156
2.2. Molar volume Vm 159
2.3. Adiabatic compressibility βS 159
2.4. Heat capacity at constant pressure Cp 160 3. Evaluation of the Grüneisen parameter γ 162 4. Discussion 166 4.1. The temperature dependence of the Grüneisen parameter 166 4.2. The absolute value of the Grüneisen parameter at the Debye temperature 168 5. Conclusions 170 References 170
Chapter 8. Theoretical thermal conductivity of MgSiN2, AlN and
β-Si3N4 using Slack's equation 177
1. Introduction 177 2. The Slack equation 179 3. Influence of input parameters 181 4. The modification of the Slack equation 185 5. Applicability, reliability and limitations of Slack modified 193 6. Conclusions 195 References 196
Chapter 9. A new method for estimation of the intrinsic thermal conductivity 203
1. Introduction 203 2. The temperature dependence of the thermal diffusivity and conductivity 204 3. Experimental 208
4. Results for MgSiN2, AlN and β-Si3N4 209 4.1. The temperature dependence of the thermal diffusivity a 209 4.2. Inverse thermal diffusivity a -1 versus temperature T plots 212
8 Table of contents
5. Discussion 220 5.1. Interpretation of the fitting parameters 220
5.2. Thermal conductivity estimates for MgSiN2, AlN and β-Si3N4 221 5.3. Comparison with other estimates 225 5.4. Limitations, accuracy and reliability 227 6. Conclusions 228 References 229
Chapter 10. Conclusions 237
List of symbols 241
Lower-case symbols 241 Upper-case symbols 242 Greek symbols 243
Summary 245
Samenvatting 247
Nawoord 251
Curriculum Vitae 254
List of publications 255
9
10 Chapter 1. Introduction
1. General introduction
"So the Lord God banished him from the Garden of Eden to work the ground" [1]. Since then people try to improve their existence by making life more comfortable. They used their intellect, knowledge and inventiveness to increase the standard of living. It started with stone tools, the ability of making fire and the production of food by farming and is now (after making a large step in history in only few seconds of writing [2]) continuing in the age of the computer information and automation.
1.E+08
Pentium® II 1.E+07 Processor Pentium® Pentium® Pro Processor Processor 1.E+06 80486DX 80486SX
80386DX 80386SX 80286 1.E+05
8086 8088
Number of transistors 1.E+04 8080 8008 4004 1.E+03 1970 1975 1980 1985 1990 1995 2000 2005 t [year]
Fig. 1-1: Number of transistors per chip versus time (t) (Data supplied by Intel Corp.).
More and more processes are computer controlled and/or guided. Due to the increasing number and complexity of tasks in e.g. the industry, and in order to
11 Chapter 1.
reduce the human intervention the complexity, speed and calculating power of these machines is still increasing. E.g. the last decades the number of transistors per chip and the processing power have increased tremendously (see Figs. 1-1 and 1-2).
1.E+09 Pentium® Pro Processor Pentium® 1.E+08 Processor
1.E+07 80386DX
1.E+06 8008
1.E+05 Instructions per second Instructions per 4004
1.E+04 1970 1975 1980 1985 1990 1995 2000 2005 t [year]
Fig. 1-2: Processing power in instructions per second versus time (t) (Data supplied by Intel Corp.).
Related to this development there is a tendency to increase the processing power per unit volume by miniaturisation. E.g. the computing power of the house size first computer in 1945 (ENIAC) containing 17468 vacuum tubes is nowadays easily surpassed by a microprocessor with a size much smaller than a match box containing 10000000 transistors [3, 4]. This resulted in the use of more and more chip controlled electronic devices during the last decades. One of the best examples is the introduction of the personal computer (PC) with a high computing power. But also a (mobile) telephone, audio equipment, bank/credit card and microwave oven contain one or more chips. So microelectronics is playing an essential role in nowadays life. For several applications in microelectronics the (bare) chip is directly attached to a substrate [5] which is for unencapsulated chip design one of the most important parts. Besides for substrates, also for an enormous amount of other applications the thermal properties of a material are of crucial importance. For one type of
12 Introduction
application the heat flow should be minimised like insulation material for a furnace, heat shield of the space shuttle, etc. whereas for another application like heat exchangers, lamp envelope materials, substrates, etc. the heat flow should be high.
2. Substrate materials
2.1. Requirements
A substrate in electronic integrated circuits has two main functions. The first is obvious viz. carry the chip attached to it giving it mechanical stability. Second, the substrate is used as a heat sink in order to avoid over-heating and eventually damage of the electronic circuit attached to it. For that purpose, in general a substrate should fulfil the following requirements [6 - 10]: - high mechanical strength - high thermal conductivity - high electrical resistance - low dielectric constant - low dielectric loss - thermal expansion coefficient similar to silicon - good thermal shock resistance - good metallisation properties - chemical and electrical stability - non-toxic - smooth surface - easy to produce, economic viability, cheap and convenient processing
Recently, the requirements for substrates are becoming more strict [5, 10, 11] due to the tendency of miniaturising the electronic circuit attached to the substrate, resulting in a higher heat dissipation per surface area. This implies that the thermal
13 Chapter 1.
conductivity of the substrate should be typically above 100 W m-1 K-1 at room temperature resulting in a sufficient heat transport and a good thermal shock -1 -1 resistance. As a consequence the traditional substrate material Al2O3 (38 W m K at room temperature [10]) does no longer fulfil the requirements. So the selection of a substrate material that complies with all the needs is of crucial importance for the design and development of better and new microelectronics. Because substrates should have a high electrical resistivity combined with a high thermal conductivity only a limited number of materials come into account.
2.2. Relation between heat conduction and material characteristics
Since substrate materials have to be electrical insulators with a high thermal conductivity, only materials showing a high phonon conduction (viz. heat is transported by lattice vibrations, the so-called phonons) are suitable candidates. Materials with good phonon conductivity should fulfil the following requirements [6, 7, 12, 13]: - simple structure - low atomic mass - strong covalent bonding - low anharmonicity - high purity
Especially materials with the simple adamantine type (diamond related) crystal structure show a high phonon thermal conductivity [12] in combination with a high electrical resistivity, which makes this class of materials potentially interesting as substrate materials. In Table 1-1 some values for the thermal conductivity (κ ) [W m-1 K-1] of some commonly used adamantine type materials are presented. Considering the above mentioned requirements for obtaining a high lattice thermal conductivity, it is not supprising that carbon (C) with its simple diamond structure, low atomic mass and strong covalent bonding has the highest thermal
14 Introduction
conductivity. For the other group IV elements Si and Ge with the diamond structure the thermal conductivity decreases due to the increase in atomic mass and the decrease of the strength of the covalent bonding as compared to diamond. Also the binary compounds (IV - IV, III - V, II - VI) having an adamantine crystal structure show a relatively high thermal conductivity although lower than diamond due to the increasing complexity of the crystal structure, decrease of covalent character of the bonds and in most cases also increase of average mass.
Table 1-1: The thermal conductivity (κ ) of some unary and binary adamantine type materials at room temperature (300 K).
unary material κ binary Material κ [W m-1 K-1] [W m-1 K-1] IV C (diamond) 2000 [13] IV - IV SiC 490 [13, 15] Si 160 [14] III - V BN (cubic) 1300 [13] Ge 60 [14] BP 350 [13] AlN 285 [16] AlP 130 [6] GaN 130 [17] GaP 100 [18] II - VI BeO 370 [19]
Each of the materials mentioned in Table 1-1 has its specific disadvantages [6, 7, 20]. C and cubic BN are difficult to produce (high pressures and temperatures) and therefore expensive. SiC, Si and Ge are semiconductors. GaN, GaP and AlP have a large mismatch in thermal expansion coefficient with Si and BP is thermally unstable. BeO has the disadvantage of its toxicity especially of its dust. The compound that is most intensively studied during the last two decades, viz. AlN ceramics, was considered expensive due to the necessity of using pure raw materials, the relatively high processing temperature (around 1700 °C) and poor metallisation properties as compared to the traditional substrate material
Al2O3. However, nowadays AlN ceramics can be processed at normal pressure, and
15 Chapter 1.
recently production costs are minimised by optimisation of processing routes and metallisation process.
2.3. AlN as a promising substrate material
Not only is AlN ceramics used as a substrate material, it has also several other interesting applications due to the potential superior thermal and mechanical properties and chemical stability as compared to more traditional materials like
Al2O3 and stainless steel. For example crucibles, tube envelopes, heater plates for chemical vapour deposition (CVD) applications and nozzles for extrusion (Fig. 1-3).
Fig. 1-3: Several examples of commercially available AlN products showing a heater plate, nozzles and several substrates. To obtain an impression of the size a 3.5 inch diskette is included.
The most important problem that had to be solved for obtaining AlN ceramics with a high thermal conductivity were the oxygen impurities in the starting powder and the resulting ceramics. Oxygen dissolved in the AlN lattice results in the formation of Al vacancies [8, 13] that are very effective in scattering
16 Introduction
the phonons [21, 22] resulting in a low thermal conductivity. Moreover, oxygen impurities present as secondary phases normally hamper the heat transport between the AlN grains due to the formation of a thermally insulating layer [23 - 25]. Furthermore, the grain size itself can be of importance because phonon - grain boundary scattering also reduces the thermal conductivity [26 - 28]. So for obtaining a high thermal conductivity the intrinsic properties (lattice defects) as well as the extrinsic properties (microstructure) have to be controlled [29] (Fig. 1-4).
Unfavorable microstructure Favorable microstructure
Fig. 1-4: A schematic drawing of a microstructure showing left a situation, with defects in the lattice and a grain boundary phase, which is not beneficial for obtaining a high thermal conductivity, and right a more favourable situation in which the defect concentration within the grains is reduced and the secondary phases are located at the triple points, resulting in less phonon-defect scattering and a good thermal contact between the grains.
It is suggested that AlN has an intrinsic thermal conductivity of 320 W m-1 K-1 at 300 K [16, 31] (for comparison Cu 400 W m-1 K-1 at room temperature [30]). Around 1975 it was possible to synthesise on lab-scale single
17 Chapter 1.
crystals with a high thermal conductivity (> 200 W m-1 K-1, see Fig. 1-5) [31, 32]. However, it took several decades to improve and optimise the thermal conductivity of AlN polycrystalline ceramics [6, 7, 29, 31, 33 - 45] until the above mentioned problems considering the oxygen impurities were solved and samples approaching the theoretical value could be synthesised (see Fig. 1-5). Around 1990 it was possible to synthesise commercial polycrystalline ceramic samples (Fig. 1-3) with an excellent thermal conductivity (> 200 W m-1 K-1) and the research subsequently concentrated on reducing the processing costs by lowering the processing temperature and using other (cheaper) additives [46 - 49], and improvement of the metallisation properties [50]. Nowadays, commercial AlN substrates are available with a thermal conductivity of about 200 W m-1 K-1.
350 AlN theoretical 300 AlN single crystal AlN polycrystalline 250 ] -1
K 200 -1
150 [Wm κ 100
50
0 1950 1960 1970 1980 1990 2000 t [year]
Fig. 1-5: Development of the room temperature thermal conductivity (κ ) of polycrystalline and single crystalline AlN as compared to the theoretically predicted value versus the time (t) (Data from Ref. 6, 7, 29, 31 and 33 - 45.). The lines are drawn as a guide to the eye.
2.4. The new ceramic material MgSiN2 Already in 1973 Slack [13] noted that, besides unary and binary, several ternary compounds having an adamantine crystal structure might have a high thermal
18 Introduction
conductivity (> 100 W m-1 K-1). Although this value is lower than that of the corresponding binary compounds (having the same average atomic mass) due to the increasing complexity of the crystal structure, nevertheless these compounds might be interesting. So, instead of the full optimisation of already known materials for substrate application there is also a drive for finding ternary and even quaternary adamantine compounds which can be suitable new substrate materials.
II-IV-V2 V N
MgSiN2
VI O IV V VI III2 -VI-IV C Si Ge N POAs II-VI II Be Al2OC BeO Mg III-V B BN BP III Al AlN AlP GaP Ga IV IV-IV GaN GaAs C C SiC IV Si Si Ge Ge
Fig. 1-6: Diamond and diamond-structure related unary, binary and ternary materials.
Recently, several materials derived from AlN (see Fig. 1-6, 2 Al substituted by 1 Mg + 1 Si → MgSiN2 or by 1 Ca + 1 Si → CaSiN2, and 2 N substituted by
1 O + 1 C → Al2OC) and other adamantine type materials (AlCON (Al28O21C6N6)) [20, 51 - 55] were suggested as being new potentially interesting substrate materials. In view of the requirements for good phonon conduction it can be concluded that MgSiN2 is the most promising material as CaSiN2 has a higher average mass, Al2OC is only stable as a solid solution [55 - 57] and AlCON has a rather complex crystal structure (large crystallographic unit cell) [54, 55]. MgSiN2 is a covalent electrical insulator with a rather simple structure, comparable with that of AlN [51]. Before starting this Ph. D. work the thermal conductivity value -1 -1 measured for MgSiN2 was 17 W m K at room temperature [51], whereas a
19 Chapter 1.
maximum value was predicted of approximately 120 W m-1 K-1 [53] (in -1 -1 comparison AlN (single crystal) 320 W m K , Al2O3 (single crystal (sapphire)) 60 W m-1 K-1). So at that time considerable improvement of the thermal conductivity of MgSiN2 was expected, if the processing could be optimised and better starting materials were used.
3. Objective and outline
The objective of this work was to optimise the thermal conductivity of MgSiN2 ceramics. In order to obtain a high thermal conductivity the impurity content and especially the oxygen content in the MgSiN2 lattice was considered to be of crucial importance analogous to the situation with AlN. Therefore this work first concentrated on the optimisation of the synthesis of pure MgSiN2 powder and ceramics by suitable processing. Although, originally a high thermal conductivity was expected for optimised MgSiN2, this value could by far not be confirmed experimentally notwithstanding improvement of the processing resulting in pure
MgSiN2 powders and ceramics. Therefore, the available theoretical method to predict the maximum achievable thermal conductivity by phonon conduction (Slack's theory) was reconsidered. This resulted in an improved theory of Slack and moreover the development of a new prediction method based on extrapolation of temperature dependent thermal diffusivity measurements allowing discrimination between phonon-phonon and phonon-defect scattering processes. This procedure is generally applicable for ceramic materials showing heat conduction by phonons. So, this procedure can be used for identifying the potential thermal conductivity of (new) non-optimised materials and for guiding the process optimisation resulting in a decrease of the time and effort needed to optimise the thermal properties. The improved prediction methods were also applied to the commercially interesting materials AlN and β-Si3N4 in order to check the general validity. So, an experimental as well as a theoretical approach is described in this thesis. The results of this work are presented in the different parts of this thesis as follows:
20 Introduction
This part (Chapter 1) provides a short overview why MgSiN2 was considered a potential interesting material. Furthermore it becomes clear that good estimates for the intrinsic thermal conductivity are very important to choose the most promising materials. Furthermore, it is important to minimise the effort put in material optimisation. Usually the main problem in achieving a high thermal conductivity in phonon conductors is phonon scattering due to defects. For nitrides these defects are mainly caused by oxygen impurities in the nitride starting powders. Therefore it is considered important to synthesise pure MgSiN2 powder which is discussed in Chapter 2.
Chapter 3 deals with the processing of MgSiN2 ceramics by hot uni-axial pressing and the resulting properties. By suitable processing it should be possible to identify and eliminate the mechanism that is limiting the thermal conductivity of
MgSiN2. This was done by changing the processing conditions viz. temperature, time and/or using an additive during processing. The experimental determination and modelling of the thermal expansion, heat capacity and Young's modulus of MgSiN2 are discussed in Chapters 4, 5 and 6 respectively. These properties and, in particular, their temperature dependence are needed for calculating the Grüneisen parameter and Debye temperature that are required for the theoretical estimation of the maximum achievable thermal conductivity with the Slack equation. Both the specific heat as well as the Young's modulus is used to evaluate the Debye temperature.
The evaluation of the Grüneisen parameter of MgSiN2 is discussed in Chapter 7. The Grüneisen parameter is related to the complexity of the crystal structure and the characteristics of the bonding between the atoms. The Grüneisen parameter of MgSiN2 is compared with that of AlN, which has a similar wurtzite-like crystal structure, and β-Si3N4, with a phenakite (Be2SiO4) structure. Besides the Grüneisen parameter and the Debye temperature, also the number of atoms per primitive unit cell is an important parameter for estimating the maximum achievable thermal conductivity. The comparison with β-Si3N4 (having a relatively high thermal conductivity) is made because it has about the same number of atoms
21 Chapter 1.
per primitive unit cell as MgSiN2 (with a relatively low thermal conductivity) whereas it has much more atoms per as per primitive unit cell as AlN (having a high thermal conductivity). Furthermore, recently β-Si3N4 substrates are commercially applied triggering a detailed comparison with MgSiN2 and AlN. In Chapter 8 the theory of Slack is discussed. This theory describes a relatively simple method to predict the maximum achievable thermal conductivity of non-metallic materials from the crystal structure, Debye temperature, Grüneisen parameter and number of atoms per primitive unit cell. The assumptions made in this theory are briefly discussed and some improvements are presented, resulting in a modified Slack theory. The applicability of this adapted theory is discussed by calculating the intrinsic thermal conductivity values at room temperature for
MgSiN2, AlN and β-Si3N4 and comparing them with experimental data for validation. Another new method for predicting the intrinsic thermal conductivity is presented in Chapter 9. This method is based on extrapolation of thermal diffusivity measurements as a function of the temperature. The general applicability, validity and limitations of this method are discussed using MgSiN2,
AlN and β-Si3N4 as model compounds. Therefore the results were compared with those obtained in Chapter 8 and experimental values. The final conclusions of this thesis are summarised in Chapter 10.
References
1. The Holy Bible, The fall of Man, Genesis 3:23. 2. After N.A. Armstrong (the first man on the moon), "one small step for a man - one giant leap for mankind", at 10:56 p.m. Eastern Daylight Time July 20, 1969. 3. D.J.W. Sjobbema, Geschiedenis van de elektronica; Van voltacel naar digitale televisie, first edition (Kluwer BedrijfsInformatie b.v., Deventer, The Netherlands, 1998).
22 Introduction
4. E. Braun and S. MacDonald, Revolution in miniature, second edition (Cambridge University Press, Cambridge, UK, 1982). 5. Microelectronics Packaging Handbook; Part II: Semiconductor Packaging, edited by R.R. Tummala, E.J. Rymaszewski and A.G. Klopfenstein, second edition (Chapman and Hall, London, 1997). 6. W. Werdecker and F. Aldinger, Aluminum Nitride - An Alternative Ceramic Substrate for High Power Applications in Microcircuits, IEEE Trans. Compon., Hybrids, Manuf. Technol. CHMT-7 (1984) 399. 7. Y. Kurokawa, K. Utsumi, H. Takamizawa, T. Kamata and S. Noguchi, AlN Substrates with High Thermal Conductivity, IEEE Trans. Compon., Hybrids, Manuf. Technol. CHMT-8 (1985) 247. 8. A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and Kinetic Effects of Oxygen Removal on the Thermal Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 72 (1989) 2031. 9. G.W. Prohaska and G.R. Miller, Aluminum Nitride: A Review of the Knowledge Base for Physical Property Development, Mat. Res. Soc. Symp. Proc. 167, Advanced Electronic Packaging Materials, Boston, Massachusetts, USA, November 27 - 29 1989, edited by A.T. Barfknecht, J.P. Patridge, C.J. Chen and C.-Y. Li, (Materials Research Society, Pittsburg, 1990) 215. 10. A. Roosen, Modern Substrate Concepts for the Microelectronic Industry, Electroceramics IV 2, Aachen, Germany, September 5 - 7 1994, edited by R. Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann, (Augustinus Buchhandlung, 1994) 1089. 11. H. Treichel, E. Eckstein and W. Kern, New Dielectric Materials and Insulators for Microelectronic Applications, Ceramics International 22 (1996) 435. 12. D.P. Spitzer, Lattice Thermal Conductivity of Semi-Conductors: a Chemical Bond Approach, J. Phys. Chem. Solids 31 (1970) 19. 13. G.A. Slack, Nonmetallic Crystals with High Thermal Conductivity, J. Phys. Chem. Solids 34 (1973) 321.
23 Chapter 1.
14. C.J. Glassbrenner and G.A. Slack, Thermal Conductivity of Silicon and Germanium from 3 °K to the Melting Point, Phys. Rev. 134 (1964) A1058. 15. G.A. Slack, Thermal Conductivity of Pure and Impure Silicon, Silicon Carbide, and Diamond, J. Appl. Phys. 35 (1964) 3460. 16. G.A. Slack, R.A. Tanzilli, R.O. Pohl and J.W. Vandersande, The Intrinsic Thermal Conductivity of AlN, J. Phys. Chem. Solids 48 (1987) 641. 17. E.K. Sichel, and J.I. Pankove, Thermal Conductivity of GaN, 25 - 360 K, J. Phys. Chem. Solids 38 (1977) 330. 18. H. Wagini, Die Wärmeleitfähigkeit von GaP und AlSb, Z. Naturforschg. 21 a (1966) 2096. 19. G.A. Slack and S.B. Austerman, Thermal Conductivity of BeO Single Crystals, J. Appl. Phys. 42 (1971) 4713. 20. W.A. Groen, P.F. van Hal, M.J. Kraan, N. Sweegers and G. de With,
Preparation and Properties of ALCON (Al28C6O21N6), J. Mater. Sci. 30 (1995) 4775. 21. V. Ambegaokar, Thermal Resistance due to Isotopes at High Temperature, Phys. Rev. 114 (1959) 488. 22. M.G. Holland, Phonon Scattering in Semiconductors From Thermal Conductivity Studies, Phys. Rev. 134 (1964) A471. 23. A. Hafidi, M. Billy and J.P. Lecompte, Influence of Microstructural Parameters on Thermal Diffusivity of Aluminium Nitride-Based Ceramics, J. Mater. Sci. 27 (1992) 3405. 24. C.F. Chen, M.E. Perisse, A.F. Ramirez, N.P. Padture and H.M. Chan, Effect of Grain Boundary Phase on the Thermal Conductivity of Aluminium Nitride Ceramics, J. Mater. Sci. 29 (1994) 1595. 25. P.S. de Baranda, A.K. Knudsen, and E. Ruh, Effect of CaO on the Thermal Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 76 (1993) 1751. 26. A.K. Collins, M.A. Pickering and R.L. Taylor, Grain size dependence of the thermal conductivity of polycrystalline chemical vapor deposited β-SiC at low temperatures, J. Appl. Phys. 68 (1990) 6510.
24 Introduction
27. K. Watari, K. Ishizaki and T. Fujikawa, Thermal Conduction Mechanism of Aluminium Nitride Ceramics, J. Mater. Sci. 27 (1992) 2627. 28. K. Watari, K. Hirao, M. Toriyama and K. Ishizaki, Effect of Grain Size on the
Thermal Conductivity of Si3N4, J. Am. Ceram. Soc. 82 (1999) 777. 29. Y. Kurokawa, K. Utsumi and H. Takamizawa, Development and Microstructural Characterization of High-Thermal-Conductivity Aluminum Nitride Ceramics, J. Am. Ceram. Soc. 71 (1988) 588. 30. CRC Materials Science and Engineering Handbook, second edition, edited by J.F. Shackelford, W. Alexander and J.S. Park (CRC Press, Boca Raton, Florida, USA., 1994). 31. M.P. Borom, G.A. Slack and J.W. Szymaszek, Thermal Conductivity of Commercial Aluminum Nitride, Bull. Am. Ceram. Soc. 51 (1972) 852. 32. G.A. Slack and T.F. McNelly, Growth of High Purity AlN Crystals, J. Cryst. Growth 42 (1977) 560. 33. K.M. Taylor and C. Lenie, Some Properties of Aluminum Nitride, J. Electrochem. Soc. 107 (1960) 308. 34. N. Kuramoto and H. Taniguchi, Transparent AlN Ceramics, J. Mater. Sci. Lett. 3 (1984) 471 35. I.C.Huseby and R.F. Bobik, U.S. Pat. No. 4547471 (High Thermal Conductivity Aluminum Nitride Ceramic Body), Oct. 15, 1985, Nos. 4578232 - 4578234 (Pressureless Sintering Process to Produce High Thermal Conductivity Ceramic Body of Aluminum Nitride), 4578384 and 4578365 (High Thermal Conductivity Ceramic Body of Aluminum Nitride), Mar. 25, 1986, see A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and Kinetic Effects of Oxygen Removal on the Thermal Conductivity of Aluminum Nitride, J. Am. Cer. Soc. 72 (1989) 2031. 36. A. Horiguchi, F. Ueno, M. Kasori, K. Shinozaki and A. Tsuge, 25th Ceramic Basic Seminar Proceedings Abstract (1987) 155 (see also M. Okamoto, H. Arakawa, M. Oohasi and S. Ogihara, Effect of Microstructure on Thermal
25 Chapter 1.
Conductivity of AlN Ceramics, J. Ceram. Soc. Jpn. Inter. Ed. 97 (1998) 1486). 37. R.R. Tummala, Ceramics in Microelectronic Packaging, Am. Ceram. Soc. Bull. 67 (1988) 752. 38. N. Kuramoto, H. Taniguchi and I. Aso, Development of Translucent Aluminum Nitride, Am. Ceram. Soc. Bull. 68 (1989) 883. 39. M. Okamoto, H. Arakawa, M. Oohashi and S. Ogihara, Effect of Microstructure on Thermal Conductivity of AlN Ceramics, J. Ceram. Soc. Jpn. Inter. Ed. 97 (1989) 1486. 40. F. Ueno and A. Horiguchi, Grain Boundary Phase Elimination and Microstructure of Aluminium Nitride, Proceedings of the 1st European Ceramic Society Conference (EcerS'89) 1, Processing of Ceramics, Maastricht, The Netherlands, 18 - 23 June 1989, edited by G. de With, R.A. Terpstra and R. Metselaar (Elsevier Applied Science, 1989) 383. 41. M. Hirano and N. Yamauchi, Development of As-Fired Aluminium Nitride Substrates with Smooth Surface and High Thermal Conductivity, J. Mater. Sci. 28 (1993) 5737. 42. K. Watari, K. Ishazaki and F. Tsuchiya, Phonon Scattering and Thermal Conduction Mechanisms of Sintered Aluminium Nitride Ceramics, J. Mater. Sci. 28 (1993) 3709. 43. J. Jarrige, P.J. Lecompte, J. Mullot and G. Müller, Effect of Oxygen on the Thermal Conductivity of Aluminium Nitride Ceramics, J. Eur. Ceram. Soc. 17 (1997) 1891. 44. T.B. Jackson, A.V. Virkar, K.L. More, R.B. Dinwiddle, Jr., R.A. Cutler, High-Thermal-Conductivity Aluminum Nitride Ceramics: The Effect of Thermodynamic, Kinetic, and Microstructural Factors, J. Am. Ceram. Soc. 80 (1997) 1421. 45. A. Witek, M. Bockowski, A. Presz, M. Wróblewski, S. Krukowski, W. Wlosinski and K. Jablonski, Synthesis of Oxygen-free Aluminium Nitride Ceramics, J. Mater. Sci. 33 (1998) 3321.
26 Introduction
46. J. Jarrige, K. Bouzouita, C. Doradoux and M. Billy, A New Method for Fabrication of Dense Aluminium Nitride Bodies at Temperatures as Low as 1600 °C, J. Eur. Ceram. Soc. 12 (1993) 279. 47. K. Watari, M.C. Valecillos, M.E. Brito, M. Toriyama and S. Kanzaki,
Densification and Thermal Conductivity of AlN Doped with Y2O3, CaO and
Li2O, J. Am. Ceram. Soc. 79 (1996) 3103. 48. G.M. Gross, H.J. Seifert and F. Aldinger, Thermodynamic Assessment and Experimental Check of Fluoride Sintering Aids for AlN, J. Eur. Ceram. Soc. 18 (1998) 871. 49. K. Watari, M.E. Brito, T. Nagaoka, M. Toriyama and S. Kanzaki, Additives for Low-Temperature Sintering of AlN Ceramics with High Thermal Conductivity and High Strength, Key Engineering Materials 159-160, Novel Synthesis and Processing of Ceramics, (Trans Tech Publications, Switzerland, 1999) 205. 50. A. Adlaßnig, J.C. Schuster, R. Reicher and W. Smetana, Development of Glass Frit Free Metallization Systems for AlN, J. Mater. Sci. 33 (1998) 4887. 51. W.A. Groen, M.J. Kraan and G. de With, Preparation, Microstructure and
Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413. 52. W.A. Groen, M.J. Kraan and G. de With, New Ternary Nitride Ceramics:
CaSiN2, J. Mater. Sci. 29 (1994) 3161. 53. W.A. Groen, M.J. Kraan, G. de With, and M.G.A. Viegers, New covalent
ceramics: MgSiN2, Mat. Res. Soc. Symp. 237, Covalent Ceramics II: Non-oxides, Boston, Massachusetts, U.S.A., November 29 - December 2 1993, edited by Barron, A.R., Fischman, G.S., Furry, M.A. and Hepp, A.F. (Materials Research Society, 1994) 239. 54. W.A. Groen, M.J. Kraan, P.F. van Hal and A.E.M. De Veirman, A New
Diamond - Related Compound in the System Al2O3-Al4C3-AlN, J. Sol. State Chem. 120 (1995) 211. 55. W.A. Groen, P.F. van Hal, M.J. Kraan and G. de With, New High Thermal Conductivity Ceramics, Fourth Euro Ceramics 3, Basic Science -
27 Chapter 1.
Optimisation of Properties and Performance by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo Editoriale Faenza Editrice S.p.A., Faenza, Italy, 1995) 343. 56. H. Yokokawa, M. Dokiya, M. Fujishige, T. Kameyama, S. Ujiie and K. Fukuda, X-Ray Powder Diffraction Data for Two Hexagonal Aluminum Monoxycarbide Phases, J. Am. Ceram. Soc. 65 (1982) C-40. 57. J.M. Lihrmann, T. Zambetakis and M. Daire, High-Temperature Behavior of
the Aluminum Oxycarbide Al2OC in the System Al2O3-Al4C3 and with Additions of Aluminum Nitride, J. Am. Ceram. Soc. 72 (1998) 1704.
28 Chapter 2.
Preparation and characterisation of MgSiN2 powders
Abstract
The powder preparation of MgSiN2 was studied using several starting mixtures
(Mg3N2/Si3N4, Mg/Si3N4 and Mg/Si) in the temperature range 800 - 1500 °C in N2 or N2/H2 atmospheres. The phase formation was followed with thermo gravimetric analysis and differential thermal analysis (TGA/DTA) and powder X-ray diffraction (XRD). At 1250 °C Mg/Si mixtures did not yield single phase MgSiN2 whereas for Mg/Si3N4 and Mg3N2/Si3N4 mixtures nearly single-phase powders were obtained. The Mg/Si3N4 mixtures yielded MgSiN2 at the lowest processing temperature but the Mg3N2/Si3N4 mixtures yielded the most pure MgSiN2 powder with respect to secondary phases. The main secondary phase detectable with XRD was MgO when starting from Mg3N2/Si3N4 or MgO and metallic Si when starting from Mg/Si3N4 mixtures. When the processing starting from Mg3N2/Si3N4 mixtures was optimised MgSiN2 powders containing only 0.1 wt. % oxygen could be prepared. Using XRD the solubility of oxygen in the MgSiN2 lattice was estimated to be at maximum 0.5 wt. %. The MgSiN2 powder was oxidation resistant in air till 830 °C. The morphology and particle size were studied with the scanning electron microscope (SEM) and the sedimentation method. Two different kinds of morphology were observed, determined by the morphology of the Si3N4 starting material.
29 Chapter 2.
1. Introduction
As a consequence of the ever increasing miniaturisation of integrated circuits combined with a high energy dissipation, in recent years there is a strong need for substrate materials with improved thermal conductivity [1]. Because the electrical conductivity must be low, only non-metallic materials showing phonon conduction are suitable.
The traditional material Al2O3 does not longer fulfil the recent requirements. Several binary alternatives deduced from diamond, which has a high thermal conductivity and electrical resistivity, were considered [2], each material having its own disadvantages: SiC is electrically conducting, BeO is toxic, and the compound which is most intensively studied during the last years, viz. AlN, is considered to be expensive. Also the ternary compounds deduced from AlN were proposed e.g. 3+ 2+ 4+ MgSiN2 (by replacing two Al ions by a combination of Mg and Si ) and Al2OC (by replacing two N3- ions by a combination of O2- and C4-) [3]. Recently, for
MgSiN2 ceramics a fairly high thermal conductivity was reported [4, 5]. For optimum thermal conductivity it is expected that the oxygen content of the MgSiN2 ceramics should be low, similar to that in AlN [6]. Therefore, for achieving MgSiN2 ceramics with a high thermal conductivity the oxygen concentration of the starting material preferably should be low. In this chapter the preparation, phase formation and characterisation of
MgSiN2 powders with a low oxygen content is reported. Preliminary results have already been published [7]. The present situation concerning the preparation of ceramic samples and thermal conductivity is described elsewhere [8, 9].
2. Experimental section
2.1. Starting materials
MgSiN2 powders were prepared from Mg3N2/Si3N4, Mg/Si3N4 or Mg/Si powder mixtures. The influence of the composition and impurity content (quality) of various starting materials on the characteristics of the resulting MgSiN2 powders
30 Preparation and characterisation of MgSiN2 powders
was investigated for Mg3N2 (Table 2-1) and Si3N4 (Table 2-2 and Table 2-3). Mg powder of Merck (5815) and Si powder of Riedel de Haen AG were used.
Table 2-1: Characteristics of the Mg3N2 starting materials used (data from the supplier). Manufacturer Code [N] [N] + [Mg] [wt. %] [wt. %] Alfa 932825 27.4 99.5 Cerac M1014 26.0 99.5 theoretical — 27.8 100
Table 2-2: Characteristics of the Si3N4 starting materials used (data from this work; a: measured with Kjeldahl method, b: measured with LECO O/N gas analyzer, and c: data given by supplier).
a b Manufacturer Code [N]spec [N] meas [O]spec [O] meas [wt. %] [wt. %] SKW Trostberg Silzot HQ > 38.5 (38.74c) 38.7 ± 0.3b < 1.0 (0.34c) 0.7 ± 0.1 Cerac S1177 > 38.0 38.4 ± 0.4 — 0.7 ± 0.1 HCST LC12N > 38.5 39.2 ± 0.1 1.4 - 1.7 1.4 ± 0.1 Kema Nord — — 38.4 ± 0.3 — 2.4 ± 0.1 Sylvania — — 29.5 ± 0.3 — 4.1 ± 0.2 Tosoh TS10 — 39.3 ± 0.5 — 1.6 ± 0.1 Ube SNE10 > 38.0 37.7 ± 0.6 < 2.0 1.2 ± 0.1 theoretical 39.9 0
2.2. Preparation
The starting materials were mixed using a porcelain mortar and pestle in stoichiometric amounts in a glove-box to prevent oxidation and hydrolysis of the starting materials, especially Mg3N2. Subsequently, the mixed powders were put in a closed stainless steel (AISI 304) tube. When further purification of the resulting powders became necessary, molybdenum (Plansee, regular grade) tubes were used. The tubes had a small gas inlet/outlet to prevent pressure built up. The starting
31 Chapter 2.
mixtures were normally fired at 1250 °C during 16 hours in a horizontal tube furnace in a flowing N2 (99.95 % pure) or 85 vol. % N2 (99.95 % pure) / 15 vol. %
H2 (99.95 % pure) atmosphere. The firing temperature of 1250 °C was taken from two earlier studies on the preparation of MgSiN2 [4, 10]. Also other firing temperatures in the range of 900 °C - 1500 °C were used.
2.3. Characterisation
The starting powders and the powders resulting after firing the starting materials were characterised with powder X-ray diffraction (XRD, Philips PW 1050/25,
Cu-Kα). The Mg3N2 starting materials were characterised in the range of 10 - 100 °
2θ (scan rate 1 °/min.) and the Si3N4 starting materials in the range of 10 - 80 ° 2θ (scan rate 2 °/min.). The phase formation of the fired materials was investigated with powder X-ray diffraction. They were investigated in the range of 10 - 100 ° 2θ using standard continuous scans (1 °/min. or 2 °/min.) as well as step scans (0.1 °/min.).
The lattice parameters of MgSiN2 were calculated with the computer program Refcel [11] using the fact that MgSiN2 has an orthorhombic cell (space group Pna21 [12]). At least ten reflections (200, 002, 121, 201, 122, 202, 040, 320, 123, 203, 042, 241, 322, 401, 242 and 243) including a zero point correction were used for calculating the lattice parameters.
The nitrogen content ([N] [wt. %]) of the Si3N4 starting materials was determined by the Kjeldahl method or a O/N gas analyser (Leco TC 436). For the former method the sample (0.1 g powder) was decomposed in molten LiOH. The released ammonia was binded in a saturated boric acid solution. The amount of ammonia was determined by titration with 0.1 M hydrochloric acid using bromophenolblue as indicator. For the latter method the nitrogen present in the sample was thermally converted at high temperatures to N2 which was measured with a catharometer.
The oxygen content ([O] [wt. %]) for the Si3N4 starting materials and the
MgSiN2 powders was measured using a O/N gas analyser (Leco TC 436). The
32 Preparation and characterisation of MgSiN2 powders
powder sample was mixed with carbon whereafter the oxygen present in the sample was carbothermally converted at high temperatures in an inert atmosphere into CO, which after further oxidation to CO2 was measured with infra-red (IR) absorption spectroscopy. Thermo gravimetric analysis and differential thermal analysis (TGA/DTA) was performed with a Netzsch STA 409 thermobalance to investigate the phase formation and oxidation of MgSiN2. The phase formation was studied in flowing
N2 atmosphere using Al2O3 sample holders applying a constant heating rate of
10 °C/min. The oxidation study was performed in flowing air using Al2O3 sample holders applying a constant heating rate of 5 °C/min. Also tube furnace oxidation experiments in air were performed in combination with XRD and mass measurements to determine the (intermediate) reaction products and to study the oxidation kinetics. Scanning electron microscopy (SEM, JEOL 840A) was used to study the particle size and morphology of some of the prepared powders, and energy dispersive spectrometry (EDS) to determine the chemical composition of the powders, especially the presence of contamination(s). The particle size distribution was measured with a Sedigraph 5100 Micromeritics using a 60 wt. % ethylene glycol / 40 wt. % water mixture. Before measuring the particle size distribution the dispersed powder mixture was ultrasonic treated for 20 min. to break up powder agglomerates.
3. Results and discussion
3.1. Starting powder characteristics
3.1.1. Mg3N2
At maximum 0.5 wt. % of impurities are present in the Mg3N2 starting materials (Table 2-1). The major impurity is oxygen which is present as MgO as observed with XRD. The significant difference between the nitrogen concentration given by
33 Chapter 2.
the supplier for Mg3N2 from Cerac (26.0 wt. %) and the expected value (27.8 wt. %) gives evidence for the presence (besides MgO) of free Mg metal. During the reaction of Mg metal with Si metal in a flowing nitrogen atmosphere an
Mg3N2 ceramic disk (∅ ≈ 20 mm × 0.4 mm) was formed (see 3.2. Phase formation of MgSiN2). This disk was also investigated with XRD using the same conditions as for the investigated Mg3N2 powders. Almost no MgO could be detected in this sample with XRD.
10.06 Cerac 10.04 Alfa 10.02 JCPDS 35-778 Ceramic 10.00 [Å] a 9.98
9.96
9.94
9.92 024681012 f(θ ) [-]
Fig. 2-1: Lattice parameter a of Mg3N2 powder of Cerac (+), Alfa
(◊), JCPDS 35-778 (□), and Mg3N2 ceramic disk (∆) as a function of f(θ ).
Because Mg3N2 has a cubic lattice it is possible to calculate the true lattice parameter by plotting the lattice parameter a calculated for each reflection versus the function f(θ ), which is given by:
cos2 (θ ) cos 2 (θ ) f (θ ) = + (1) sin(θ ) θ and extrapolating to f(θ ) = 0 (see Ref. 13). In Fig. 2-1 the lattice parameter, a, for each reflection of the Mg3N2 starting powders and the ceramic Mg3N2 disk is plotted versus f(θ). For comparison data of the JCPDS card 35-778 for Mg3N2 powder (Cerac) are also included. From this figure it can be deduced that, although marked differences occur for the lattice parameters calculated from the individual
34 Preparation and characterisation of MgSiN2 powders
reflections, the extrapolated lattice parameter for all samples is the same viz. 9.963 ± 0.002 Å, which is comparable with the lattice parameter mentioned in JCPDS card 35-778 (9.9657 Å). Because the lattice parameter was the same for all investigated Mg3N2 samples and the impurity content of the powder samples was at maximum 0.5 wt. % it can be concluded that the Mg3N2 lattice is saturated with oxygen and that the solubility of oxygen in the Mg3N2 lattice is very low.
3.1.2. Si3N4
In Table 2-2 and Table 2-3 the measured powder characteristics of the used Si3N4 powders are presented. The nitrogen content measured for all investigated Si3N4 powders is in good agreement with the specification of the suppliers (see
Table 2-2). The oxygen concentration in the Si3N4 starting materials ranges from
0.7 to 4.1 wt. %. For the SKW Trostberg Si3N4 powder the measured oxygen content (0.7 wt. %) is well within the specifications (< 1.0 wt. %) but considerably higher than the content given by the supplier (0.34 wt. %). It can be seen that for materials with a nitrogen concentration close to the theoretical value (> 39 wt. %), the oxygen concentration is low. A considerable deviation of the nitrogen concentration from the theoretical value combined with a low oxygen content was measured for the Cerac, Ube and SKW Trostberg Si3N4 powders. This indicates that some free silicon or silicon containing compound like SiC may be present.
According to the supplier, for the SKW Trostberg Si3N4 the free Si metal content is smaller than 0.5 wt. % and some SiC (0.4 wt. %) is present.
The crystallographic modification of the Si3N4 powders, viz. amorphous, α (JCPDS card 41-360), β (JCPDS card 33-1160) or tetragonal phase (JCPDS card 40-1129), was determined with XRD (Table 2-3). Only the Sylvania powder appeared to be amorphous. Most powders mainly consist of the α-modification, except for Si3N4 of Cerac which contained predominantly β. For three powders also the presence of the tetragonal modification could be demonstrated. For the crystalline powders the α/(α+β ) ratio was calculated (Table 2-3) using the methods described in Refs. 14 - 18. The calculated α/(α+β ) ratio agrees quite well
35 Chapter 2.
with the specification of the suppliers. Only the measured α/(α+β ) ratio of SKW
Trostberg Si3N4 deviates about 15 % from the specified ratio.
Table 2-3: Characteristics of the used Si3N4 starting materials (data from this work; **: also some
Si3N4 with the tetragonal modification present). α α Manufacturer Code α + β α + β (d50)spec (d50)meas (d90)meas spec meas [-] [µm] SKW Trostberg Silzot HQ > 0.80 0.66 1.7 2.2 4.9 Cerac S1177 ± 0.1 0.08 < 2.0 1.2 2.6 HCST LC12N 0.94 0.89** 0.6 0.6 3.0 Kema Nord — — 0.91** — 2.3 9.0 Sylvania — amorphous amorphous — — — Tosoh TS10 — 0.93 — 1.1 7.5 Ube SNE10 > 0.95 1.00** 0.6 0.7 1.4
For all investigated Si3N4 powders the median particle size, d50, was less than 2.5 µm (Table 2-3) and some are submicrometer size (Ube and HCST). The
Si3N4 powders of HCST, Kema Nord and Tosoh have a broad particle size distribution (3d50 < d90) which indicates that the primary particles are most probably agglomerated, even after ultrasonic treatment.
3.2. Phase formation of MgSiN2
The TGA/DTA experiments show that when starting with an Mg3N2/Si3N4 mixture the temperature should surpass about 1100 - 1150 °C to get fast formation of
MgSiN2, in agreement with literature data [10]. In the DTA signal two endothermic peaks are present. Which peak or whether both peaks can be ascribed to the formation of MgSiN2 is not clear because both peaks are less than 50 °C separated from each other. No attempts were made to discriminate between them because the
36 Preparation and characterisation of MgSiN2 powders
used standard synthesis temperature of 1250 °C is sufficiently high to obtain a fast reaction and a fully reacted product.
For the Mg/Si3N4 starting mixture, the reaction mechanism is much more complicated than for the previous case. Several exothermic DTA peaks are present (Fig. 2-2), the strongest at 612 °C, and some smaller ones at 897 °C, 920 °C, and (not visible in Fig. 2-2) 1061 °C. At about 612 °C nitridation of Mg takes place accompanied by a mass gain of about 9.5 wt. %. The total mass gain at 1000 °C is about 12.5 wt. % which is comparable with the expected mass gain of 13.1 wt. % for the nitridation of the Mg present in the Mg/Si3N4 starting mixture. XRD showed that a Mg/Si3N4 mixture fired at 700 °C in an N2 atmosphere resulted in a mixture of Mg3N2 and Si3N4 whereas a mixture fired at 900 °C resulted in MgSiN2 giving further evidence that the DTA peaks at 897 °C and 920 °C are related with the formation of MgSiN2.
800 21 DTA 700 18 TGA 600 15 500 12 V]
µ 400 [ 9 × 100 [%] 0 T 300 ∆ m 6 / 200 m ∆ 100 3 0 0 -100 -3 0 200 400 600 800 1000 1200 T [°C]
Fig. 2-2: TGA/DTA plot of an Mg/Si3N4 mixture in a nitrogen atmosphere showing the temperature difference (∆T ) and relative mass
difference (∆m/m0) as function of the temperature (T ).
Also the nitridation of metallic Mg powder was studied with TGA/DTA (Fig. 2-3). At 648 °C an endothermic peak is observed which can be ascribed to the melting of Mg metal. Two exothermic nitridation peaks were observed at 660 °C
37 Chapter 2.
and 690 °C. The last one is related to the rapid nitridation of Mg. The observed results are in good agreement with earlier published data [19] on the nitridation of
Mg. When these results are compared with those obtained for the Mg/Si3N4 mixtures, a lowering by about 50 °C of the nitridation temperature of Mg and no melting peak of Mg are observed when using the Mg/Si3N4 mixtures. A possible explanation might be a different reactivity of Mg in the presence of Si3N4.
500 50 DTA 400 TGA 40
300 30 V] µ × 100 [%]
[ 200 20 0 T m ∆ /
100 10 m ∆
0 0
-100 -10 600 650 700 750 800 850 900 950 T [ºC]
Fig. 2-3: TGA/DTA plot of Mg powder in a nitrogen atmosphere showing the
temperature difference (∆T ) and relative mass difference (∆m/m0) as function of the temperature (T ).
When comparing the phase formation of MgSiN2 starting with Mg3N2/Si3N4 and Mg/Si3N4 mixtures it can be concluded that when starting with an Mg/Si3N4 mixture nearly single-phase MgSiN2 can already be obtained at a temperature of about 900 °C, which is much lower than the minimal temperature of about 1150 °C necessary for an Mg3N2/Si3N4 mixture. This difference in phase formation temperature might be related to the fact that during nitridation of Mg an Mg3N2 phase is formed different from the room temperature modification [20] with a higher reactivity. Also gas phase reactions may play an important role in the observed difference in temperature. When Mg(g) condenses on the Si3N4 particles the reactivity of the starting mixture might be increased due to the small particle size of the condensed Mg resulting in a lower reaction temperature.
38 Preparation and characterisation of MgSiN2 powders
In order to study the observed differences between the Mg3N2/Si3N4 and
Mg/Si3N4 starting mixtures in more detail the phase formation of MgSiN2 for several Mg3N2/Si3N4 fired at 1250 °C and for Mg/Si3N4 starting mixtures fired at 900 - 1250 °C using different starting materials was studied with XRD. For completeness also the phase formation for a Mg/Si starting mixture at 1250 °C was studied.
Nearly single-phase grey-brown coloured MgSiN2 materials were obtained when starting with Mg3N2/Si3N4 mixtures fired at 1250 °C or Mg/Si3N4 mixtures fired at 900 - 1250 °C in an N2 atmosphere. For all Si3N4 starting materials (amorphous, α- or β-modification, irrespective of the presence of tetragonal phase or free Si), MgSiN2 is readily formed. In all cases some MgO (periclase, JCPDS card 4-829) could be detected with XRD as a secondary phase. Sometimes white powder was observed at the outside of the reaction tube. This powder was also MgO, as observed with XRD, indicating that the oxygen in the starting materials or the gas atmosphere reacts with Mg or Mg3N2 to MgO. The MgO contamination is caused by oxygen impurities in the starting material and oxygen pickup during the processing (mixing) and the synthesis (oxygen impurities in the N2 atmosphere / reaction with oxides from the stainless steel tubes). The relative MgO content
(I/I0)MgO in the MgSiN2 powders was determined by dividing the intensity of the strongest reflection of MgO (hkl = 200) by the intensity of the strongest reflection of MgSiN2 (hkl = 121) multiplied by 100 %. As expected the observed MgO content decreases for the purer Si3N4 starting materials. Almost no MgO could be detected ((I/I0)MgO = 3) for the Mg3N2/Si3N4 and Mg/Si3N4 mixtures using oxygen poor Si3N4 starting powders of SKW Trostberg and Cerac, respectively. In general for the same Si3N4 starting material the least amount of MgO was observed when using Mg instead of Mg3N2 indicating that the purity of the resulting MgSiN2 might be improved by using a Mg/Si3N4 instead of a Mg3N2/Si3N4 mixtures. Another advantage of using Mg/Si3N4 mixture is that, due to the lower firing temperature necessary, a less non-stoichiometric product, caused by possible evaporation of magnesium [4], will be formed. Moreover, a lower firing temperature yields less contamination of the prepared materials with metals from the stainless steel (or
39 Chapter 2.
molybdenum) tubes, and a smaller particle size which will improve the sinterability of the resulting powders. So using Mg/Si3N4 instead of Mg3N2/Si3N4 starting mixtures might be beneficial for preparing a pure MgSiN2 powder because the reaction temperature can be lowered. However, in the powders synthesised from
Mg/Si3N4 mixtures always some free Si metal (JCPDS 27-1402) was detected with
XRD. So, Mg can not only react with the N2 atmosphere to form Mg3N2 but also with the Si3N4 powder to form Mg3N2 and metallic Si [12]. Because the nitridation of metallic Si is kinetically hampered even at the standard processing temperature of 1250 °C [21] removing of this secondary phase is a problem. So, the advantages of the lower reaction temperature when using Mg/Si3N4 mixtures are cancelled by the reaction of Mg with Si3N4 forming metallic Si which cannot be removed at low reaction temperatures. Starting with an Mg/Si mixture in a stainless steel reaction tube fired in a flowing N2 atmosphere at 1250 °C MgSiN2 was formed. However, in this case no single-phase MgSiN2 was obtained. The black coloured reaction product consisted mainly of MgSiN2 and several not identified secondary phases. In the coldest part of the reaction tube a light brown-orange coloured ceramic disk was formed. This disk (∅ ≈ 20 mm × 0.4 mm) was investigated with XRD. It was concluded that Mg condensed in the coldest part of the reaction tube as Mg3N2 ceramic. Considering those difficulties, no further attempts were made to obtain single-phase MgSiN2 powder using a Mg/Si starting mixture.
In summary the phase formation study at 1250 °C using Mg3N2/Si3N4,
Mg/Si3N4 and Mg/Si starting mixtures showed that only the first two starting mixtures resulted in nearly single phase MgSiN2. Although the TGA/DTA and furnace experiments indicate that MgSiN2 can be synthesised at 900 °C using
Mg/Si3N4 starting mixtures, the use of a Mg3N2/Si3N4 starting mixture at 1250 °C is preferred because the resulting MgSiN2 powder contains less Si impurities. In general when a molybdenum tube was used instead of a stainless steel tube the resulting MgSiN2 powder had a much more homogeneous and lighter colour indicating that the powder contained less metallic contaminations (Fe, Cr and Ni as detected with SEM/EDS). Based on the MgO found at the outside of the reaction
40 Preparation and characterisation of MgSiN2 powders
tube it can be assumed that MgO(g) can evaporate from the starting mixture. Using these results it was tried to synthesise an oxygen poor MgSiN2 powder.
For this the Mg3N2/Si3N4 starting mixture with the lowest oxygen content was used (Mg3N2 (Alfa)/Si3N4 (SKW Trostberg)). If the oxygen of the MgO at the outside of the reaction tube originates from the starting mixture then it is possible to purify the resulting MgSiN2 powder by adding an excess amount of Mg or
Mg3N2 to the starting mixture. The starting powder, with a small excess of Mg3N2
(± 1 wt. %) intentionally added, was fired in a 50 ml/h N2 (99.995 % pure)/5 ml/h
H2 (99.9999 % pure) atmosphere for 3 h at 1250 °C and subsequently 1 h at 1500 °C in a molybdenum tube using a heating and cooling rate of 3 °C/min. The excess of Mg3N2 is used for maintaining the stoichiometry in the resulting MgSiN2 powder. The heat treatment at 1500 °C was performed to ensure that the starting materials had fully reacted, to nitridate possible metallic Mg and Si impurities in the starting powders, to evaporate the MgO present in the reaction mixture and to remove the Mg3N2 excess present in the reaction product by decomposition into
Mg(g) and N2 [22]. So the stoichiometry of the reaction product is maintained because MgSiN2 is stable at 1500 °C [22] and the added excess of Mg3N2 which did not react to MgO is also removed from the reaction mixture.
Using this procedure single phase white MgSiN2 powder was formed. With
XRD using a scan rate of 0.033 °/min. only a small trace of MgO ((I/I0)MgO = 0.4) could be detected. This indicates that the excess Mg3N2 does not increase the MgO content in the resulting MgSiN2 powder under the given reaction conditions.
Because also no Mg3N2 could be detected this indicates that during the reaction
Mg3N2 or Mg3N2 and MgO evaporates from the reaction mixture.
3.3. Oxygen content of the MgSiN2 powders
The oxygen content of the MgSiN2 powders synthesised at 1250 °C is presented in Table 2-4. The influence of the reaction temperature, in the range of
1000 - 1250 °C, on the oxygen content of the resulting MgSiN2 powders was negligible. As expected, the overall oxygen content becomes lower when using
41 Chapter 2.
purer Si3N4 starting materials. Also, using Mg3N2 from Alfa (with the highest nitrogen content, Table 2-1) instead of Mg3N2 from Cerac decreases the oxygen content of the synthesised MgSiN2 powder. Use of Mg instead of Mg3N2 as a starting material results in an even somewhat lower oxygen concentration in the
MgSiN2 powder. However, if the oxygen content in the used Si3N4 starting material is low the difference in the oxygen content of MgSiN2 starting from Mg3N2/Si3N4 or Mg/Si3N4 mixtures appears to be negligible. So the oxygen content of the Si3N4 starting material is the dominating factor. For the standard synthesis temperature of 1250 °C the lowest oxygen content of about 0.9 - 1.0 wt. % is obtained for the purest Si3N4 starting materials (Cerac S1068 and SKW Trostberg Silzot HQ). It is significantly below the value of about 4 wt. % obtained in a previous study [4].
Table 2-4: Overall oxygen content of the MgSiN2 powders prepared from different starting materials at 1250 °C.
Starting Materials Si3N4 Mg3N2 Mg-metal [O] Cerac Alfa Merck [wt. %] SKW Trostberg 0.3 - 0.7 — 0.9 — Cerac 0.7 1.4 1.0 1.0 HCST 1.2 — 1.6 —
Si3N4 Kema Nord 1.4 — 1.6 1.3 Sylvania 1.6 2.0 1.7 1.5 Tosoh 2.4 3.0 2.2 2.8 Ube 4.1 5.3 6.1 3.9
Fig. 2-4 shows the relative MgO content of the MgSiN2 powders synthesised at 1250 °C as a function of the overall oxygen content. When the oxygen content is high (> 2 wt. %), no strong correlation between the relative MgO content and the oxygen content is observed because the oxygen can be present in several secondary phases. Whereas, in case the oxygen content is low (≤ 2 wt. %), a correlation is observed. Assuming that MgO is the only oxygen containing component at overall oxygen concentrations ≤ 2 wt. %, a crude estimation of the maximum solubility of
42 Preparation and characterisation of MgSiN2 powders
oxygen in the MgSiN2 lattice was made by extrapolation to a relative MgO content equalling 0, yielding a maximum oxygen concentration of about 0.5 ± 0.2 wt. %. Above this solubility limit oxygen MgO is formed as a secondary phase, whereas below this limit oxygen is assumed to incorporate in the MgSiN2 lattice. The maximum solubility of 0.5 wt. % oxygen in the MgSiN2 lattice corresponds to 0.5 1021 O/cm3 at maximum, as compared with about 6 1021 O/cm3 reported for
AlN [3]. So the solubility of oxygen in the MgSiN2 lattice is much lower than in AlN.
30
25
20 [%] 15 MgO ) 0 I / I ( 10
5
0 012345 overall oxygen content [wt. %]
Fig. 2-4: The relative MgO content ((I/I0)MgO) of several MgSiN2
powders synthesised at 1250 °C from Mg3N2/Si3N4 (+)
and Mg/Si3N4 (⊕) mixtures as a function of the overall oxygen content (as determined with the O/N gas analyser (Leco TC 436)).
The MgSiN2 powder synthesised using the purest starting materials by firing first at 1250 °C and subsequently at 1500 °C contained only 0.1 ± 0.1 wt. % O as determined with the O/N gas analyser. This value is considerably lower than that measured for the MgSiN2 powder synthesised at 1250 °C using the same starting materials (0.9 wt. % O). This value is even lower than the value expected from the oxygen content of the used starting mixture (∼ 0.6 wt. % O) indicating that during the synthesis the oxygen content in the reaction mixture decreases. The unexpected
43 Chapter 2.
low oxygen content might be caused, as discussed before, by a (partial) reaction of the weighed-out Mg3N2 with the oxygen present in the starting mixture to MgO which evaporates from the reaction mixture. An additional effect might be the carbothermal nitridation reaction occurring at the higher firing temperature between the trace SiC, present in the SKW Trostberg Si3N4 starting powder, and the oxygen containing compounds present in the starting mixture. Also the use of a purer gas atmosphere might be beneficial for the obtained oxygen content. The low oxygen content in combination with the fact that still some MgO was detectable with XRD ((I/I0)MgO = 0.4 %) indicates that the maximum solubility of oxygen in the MgSiN2 lattice is most probably well below the estimated 0.5 wt. % based on Fig. 2-4. So the estimation of the maximum solubility of oxygen in the
MgSiN2 lattice might be conservative.
The MgSiN2 powder sample with the low oxygen content of 0.1 wt. % contained 34.2 ± 1.7 wt. % N which is considerably higher than the value obtained in a previous study (30.7 wt. % [4]), and only somewhat lower than theoretical value (34.8 wt. %). This is in agreement with the fact that due to the presence of some residual oxygen and possibly other contaminations, the nitrogen content should be somewhat lower than the theoretical value.
3.4. X-ray diffraction data of MgSiN2
In order to calculate reliable lattice parameters for MgSiN2 powders the reflections of MgSiN2 should be correctly indexed. From the present XRD study of
MgSiN2 powders and another study of MgSiN2 ceramics [9] it is known that the indexing of the MgSiN2 reflections given in JCPDS card 25-530 is not completely correct. The data of ceramic samples were used to revise the published XRD data of MgSiN2 powders because the ceramic samples gave a better signal-noise ratio than the powder samples. The revised data (Table 2-5) were obtained from MgSiN2 ceramic samples (Ref. 9) with an average grain size of 0.25 - 1.5 µm in which no preferential orientation was detectable with XRD using a cylindrical camera. The data were used to identify the powder samples. The d-values, d obs , presented are
44 Preparation and characterisation of MgSiN2 powders
Table 2-5: List of d-values and relative intensities of pure MgSiN2 evaluated from ceramic samples.
hkl value d obs dobs I / I 0 obs d-value according I/I0 according to to JCPDS 25-530 JCPDS 25-530 [Å] [Å] [%] [Å] [%] 110 4.09 4.08 9 4.1 12 011 3.949 3.945 10 3.96 14 111 3.160 3.158 1 3.14 8 120 2.758 2.756 88 2.76 85 200 2.6349 2.6332 45 2.642 55 002 2.4922 2.4907 80 2.496 100 210 2.4405 2.4384 3 121 2.4133 2.4113 100 2.415 95 201 2.3294 2.3283 23 2.336 30 211 2.1919 2.1909 1 112 2.1278 2.1258 1 220 2.0434 2.0426 < 1 130 1.9969 1.9952 < 1 031 1.9803 1.9792 3 1.983 3 221 1.8907 1.8909 < 1 122 1.8491 1.8483 28 1.850 30 202 1.8106 1.8098 12 1.811 18 212 1.7437 1.7429 2 310 1.6953 1.6947 2 040 1.6184 1.6185 25 1.621 20 013 1.6093 1.6089 1 231 1.5830 1.5826 1 132 1.5584 1.5578 1 140 1.5471 1.549 45 320 1.5439 1.5436 36 141 1.4775 1.482 3 321 1.4748 1.4745 1 123 1.4232 1.4228 34 1.425 40 203 1.4054 1.4051 11 1.409 18 240 1.3790 1.3786 7 1.381 10 213 1.3734 1.3740 1 042 1.3573 1.3569 15 1.359 16 241 1.3291 1.3288 11 1.328 12 400/033 1.3175/1.3164 1.3173 1 322 1.3125 1.3122 22 1.314 30 401 1.2737 1.2735 5 1.275 8 150 1.2573 1.2569 1 051 1.2531 1.2526 1 004 1.2461 1.2458 2 1.248 5 420 1.2202 1.2201 < 1 151 1.2191 1.2186 < 1 242 1.2066 1.2063 5 1.208 7 332 1.1954 1.1950 < 1 114 1.1919 1.1918 1 313/421 1.1866/1.1852 1.1868 < 1 233 1.1776 1.1773 1 402 1.1647 1.1643 1 124 1.1356 1.1353 2 1.133 3 251 1.1317 1.1309 1 204 1.1265 1.1262 1 1.129 2 calculated using the average observed lattice parameters (orthorhombic lattice a = 5.2698 ± 0.0013 Å, b = 6.4736 ± 0.0014 Å and c = 4.9843 ± 0.0010 Å)
45 Chapter 2.
determined for several ceramic samples. The relative intensities, I / I 0 obs , are the average measured relative intensities for those ceramic samples. For determining the lattice parameters the computer program Refcel with zero point correction was used taking into account at least ten reflections. For comparison an observed d-value list (dobs) of a ceramic sample is included in Table 2-5 and also the data of
JCPDS card 25-530 of MgSiN2, which refers to the results of David [23], are presented.
As can be seen from Table 2-5 the d-value list of MgSiN2 was revised by adding some low intensity peaks which are not mentioned in JCPDS card no. 25-530. We especially mention the 210 (d = 2.4405 Å), 212 (d = 1.7437 Å) and 310 (d = 1.6953 Å) reflections because they have a relative strong intensity
(I/I0 ≈ 2 - 3) as compared to the other low intensity peaks (I/I0 ≤ 1) which were added. Another difference is that in JCPDS card no. 25-530 the d-values 1.549 Å and 1.482 Å are indexed with hkl = 140 and hkl = 141 whereas in the present study these reflections were indexed as 320 and 321, respectively. Furthermore, some differences in observed intensity I/I0 are noticed, especially for the 111 reflection for which I/I0 = 8 according to JCPDS 25-530 whereas the observed value is much lower (I/I0 = 1).
However, David et al. [12] calculated a theoretical intensity of I/I0 = 0.7, which is in excellent agreement with the intensity of I/I0 = 1 observed in the present study. This mismatch in calculated and measured intensity by David et al. is tentatively ascribed by the present author to the presence of free Si (JCPDS card
27-1402) in their MgSiN2 powder, which increased the intensity measured for the
111 reflection. The 111 reflections of Si and MgSiN2 have similar d-values of 3.136 Å and 3.160 Å, respectively. The indexation of the 140 and 141 reflections was changed because if a synthetic pattern (d-value list of all possible reflections) was generated using calculated lattice parameters, the d-value of the 320 and 321 reflection matched much better the experimentally found d-values than the calculated d-value of the
140 and 141 reflection. As an example the d-values observed for a ceramic MgSiN2 sample, dobs, can be compared with the calculated d-values using the average lattice
46 Preparation and characterisation of MgSiN2 powders
parameters, d obs , determined from the ceramic samples (Table 2-5). Using the atomic positions for MgSiN2 taken from Ref. 24 and a computer program for calculating X-ray diffraction intensities (Powder Cell [25]) we also concluded that the 140 and 141 reflection should indeed be replaced by the 320 and 321 reflection, respectively. The intensity calculated for the 140 and 141 reflection is << 1 % whereas the calculated intensity of the 320 and 321 reflection were in good agreement with the measured intensity. Furthermore David et al. [12] calculated a much higher intensity for the 320 than for the 140 reflection (49.7 versus 0.1 respectively) whereas the calculated intensity of the 141 matched better than the one calculated for the 321 reflection (3.4 versus 0.4 respectively). Wild et al. [26] also used the 320 reflection instead of the 140 reflection. In Fig. 2-5 the lattice parameters and in Fig. 2-6 the cell volume measured for the MgSiN2 powders processed at 1250 °C are presented as a function of the measured overall oxygen content. The error bar indicated in Fig. 2-5 and Fig. 2-6 equals 3 times the standard deviation of the Refcel calculation. From the figures we can conclude that the powder samples processed at 1250 °C have the same lattice parameters (aaverage = 5.275 ± 0.007 Å, baverage = 6.472 ± 0.009 Å and caverage = 4.987 3 ± 0.011 Å) and cell volume (Vaverage = 170.25 ± 0.70 Å ). Within the limits of accuracy the results are in agreement with the lattice parameters used in Table 2-5 for the ceramic samples. Because the lattice parameters for all samples synthesised at 1250 °C are the same irrespective of the overall oxygen content ranging from 0.9 - 6.1 wt. % it is concluded that the maximum solubility of oxygen in the
MgSiN2 lattice is surpassed. This is in accordance with the estimated maximum oxygen solubility of 0.5 wt. % in the MgSiN2 lattice; therefore no influence of the overall oxygen concentration on the lattice parameters is expected above 0.5 wt. % oxygen.
For the MgSiN2 powder with an oxygen content of about 0.1 wt. %, first fired at 1250 °C and subsequently 1500 °C, the lattice parameters are a = 5.276 ± 0.006 Å, b = 6.477 ± 0.006 Å and c = 4.990 ± 0.005 Å. This is comparable with the calculated average lattice parameters observed for the ceramic
47 Chapter 2.
and powder samples indicating that the solubility of oxygen is most probably even less than 0.1 wt. % oxygen.
5.40 6.70
5.30 6.60 a
5.20 6.50 b [Å] c [Å] , b a 5.10 6.40
5.00 c 6.30
4.90 6.20 01234567 overall oxygen content [wt. %]
Fig. 2-5: The calculated lattice parameters a, b and c determined for several
MgSiN2 powders versus the measured overall oxygen content.
176
174 ]
3 172 [Å V 170
168
166 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 overall oxygen content [wt. %]
Fig. 2-6: The calculated cell volume (V ) determined for several
MgSiN2 powders versus the measured overall oxygen content.
48 Preparation and characterisation of MgSiN2 powders
3.5. Powder characteristics
The morphology and particle size as observed with SEM is similar for MgSiN2 powders prepared from Mg3N2/Si3N4 and Mg/Si3N4 mixtures when the same Si3N4 starting material is used (Fig. 2-7 and Fig. 2-8). This was confirmed by sedigraph measurements of MgSiN2 powders prepared from Mg3N2/Si3N4 and Mg/Si3N4 mixtures. The MgSiN2 powder prepared from Mg has a broader particle size distribution and a somewhat larger median particle size. In Fig. 2-9 the mass cumulative particle size distribution of the Si3N4 powder (SKW Trostberg), and the
MgSiN2 powders synthesised thereof with Mg3N2 (Alfa) and Mg (Merck) at
1250 °C are presented. As can be seen in Fig. 2-9 the starting Si3N4 powder has a narrow particle size distribution and a median particle size of 2.2 µm. The MgSiN2 powder synthesised from the Mg3N2/Si3N4 starting mixture has also a narrow particle size distribution but the powder is coarser. The median particle size equals
3.2 µm. The MgSiN2 powder prepared from the Mg/Si3N4 starting mixture has a broad particle size distribution but the median particle size, viz. 3.8 µm, is only slightly larger than the powder synthesised from the Mg3N2/Si3N4 starting mixture.
Fig. 2-7: SEM picture of an MgSiN2 powder synthesised at 1250 °C from a
Si3N4 (SKW Trostberg)/ Mg3N2 (Alfa) starting mixture.
49 Chapter 2.
Fig. 2-8: SEM picture of an MgSiN2 powder synthesised at 1250 °C from a
Si3N4 (SKW Trostberg)/Mg (Merck) starting mixture.
100 90 80 70 60 50 40 30 20 cumulative mass [wt. %] [wt. mass cumulative 10 0 012345678910 particle size [µm]
Fig. 2-9: The particle size distribution as determined by Sedigraph
measurements of Si3N4 (SKW Trostberg) starting material
(+) and MgSiN2 powder prepared thereof with Mg3N2 (Alfa) (∆) or Mg (Merck) (○).
The powder consisted partially of hard agglomerates that could not be removed by ultrasonic treatment and are probably related to the formation of free Si metal during the synthesis.
50 Preparation and characterisation of MgSiN2 powders
Fig. 2-10: SEM picture of an MgSiN2 powder synthesised at 1250 °C from a
Si3N4 (HCST)/Mg (Merck) starting mixture.
For the MgSiN2 powders two different morphologies were observed (Fig. 2-10 and Fig. 2-11). The first one showed equi-axed grains with primary particle sizes smaller than 3 µm and agglomerates smaller than 10 µm. The second one consisted of large porous sponge like agglomerates (~ 100 µm). The observed type of morphology was independent of the α/β ratio of the starting Si3N4 powder used and whether a Mg/Si3N4 or Mg3N2/Si3N4 starting mixture was used. If the
Si3N4 starting powders of Tosoh or Ube were used then the sponge like MgSiN2 particles were observed. If the Si3N4 starting powders of HCST and SKW
Trostberg were used then small equi-axed MgSiN2 particles were observed. For comparison the morphology of the Si3N4 starting material of HCST and Tosoh was investigated with the SEM. The Si3N4 powder of HCST consisted of small grains whereas the Tosoh powder consisted of large agglomerates of small grains. This indicates that the morphology of the Si3N4 starting material determines the morphology of the resulting MgSiN2 powder.
51 Chapter 2.
Fig. 2-11: SEM picture of a MgSiN2 powder synthesised at 1250 °C from a
Si3N4 (Tosoh) / Mg3N2 (Alfa) starting mixture.
3.6. Oxidation behaviour of MgSiN2 powders TGA/DTA measurements and furnace experiments in combination with XRD were used to study the oxidation behaviour of MgSiN2. TGA/DTA experiments show that MgSiN2 powders are oxidation resistant in air up to 830 °C. At higher temperatures 4 reaction peaks are observed (Fig. 2-12); 3 exothermic peaks at 904 °C, 1082 °C and 1362 °C, and 1 endothermic peak at 1459 °C. The total weight gain for the first 2 DTA peaks is about 18 wt. %. This mass gain can be represented by the following overall reaction:
8 MgSiN2 + 9 O2 → 4 Mg2SiO4 + 2 Si2N2O + 6 N2 (+ 18.6 wt. %) The total weight gain after the third DTA peak is about 25 wt. %. This can be represented by the following overall reaction:
2 MgSiN2 + 3 O2 → Mg2SiO4 + SiO2 + 2 N2 (+ 24.8 wt. %)
So after the third DTA peak the MgSiN2 powder is totally oxidised. This peak is related to the fast oxidation of Si2N2O to SiO2. The reaction temperature of about 1362 °C is in favourable agreement with the temperature mentioned in the
52 Preparation and characterisation of MgSiN2 powders
literature [27] for the fast oxidation of Si2N2O powder at about 1330 °C. So the oxidation of MgSiN2 is a two step process as shown by the TGA/DTA measurements. The fourth DTA peak at about 1459 °C is an endothermic one and is related to the phase transformation of SiO2 from tridimite to cristobalite (1477 °C as deduced from Fig. 5 of Ref. 28).
The isothermal oxidation behaviour of MgSiN2 powder was studied in air, just above the oxidation temperature, at 850 °C. From the isothermal oxidation study it was clear that MgSiN2 can be totally oxidised at 850 °C indicating that the intermediate reaction products are not stable. No parabolic oxidation behaviour [27] was observed probably due to a superposition of the two above mentioned oxidation reactions.
70 28
DTA 60 TGA 21 50
40 14 V] µ 30 [ × [%] 100 0 T ∆ m 20 7 / m ∆ 10 0 0
-10 -7 0 300 600 900 1200 1500 T [°C]
Fig. 2-12: TGA/DTA plot of the oxidation behaviour of MgSiN2 powder in air showing the temperature difference (∆T ) and relative mass
difference (∆m/m0) as function of the temperature (T ).
4. Conclusions
The phase formation study of MgSiN2 showed that nearly single phase MgSiN2 powders can be obtained from Mg3N2/Si3N4 or Mg/Si3N4 mixtures. However, the reaction paths are different as shown with TGA/DTA. Oxygen poor MgSiN2
53 Chapter 2.
powders can be prepared, not only by the conventional synthesis route starting with
Si3N4 and Mg3N2, but also by starting with Si3N4 and Mg in a flowing N2/(H2) atmosphere. This alternative synthesis route has the benefit of a lower reaction temperature and the disadvantage of a more critical processing due to the formation of free Si metal during the synthesis. When using the standard synthesis temperature of 1250 °C, the overall oxygen content obtained for the MgSiN2 powders varied between 0.9 - 6.1 wt. %
O, mainly determined by the oxygen content of the Si3N4 starting material. The lattice parameters of these powders do not depend on the overall oxygen concentration indicating that the maximum solubility of oxygen in the lattice is surpassed in accordance with the observed presence of some residual MgO. Its concentration in these powders suggest that the maximum solubility of oxygen in the MgSiN2 lattice does not exceed 0.5 ± 0.2 wt. %. By using improved processing conditions it is possible to synthesise powders with an oxygen content of only 0.1 wt. % O. However, even in these powders containing only 0.1 wt. % O some MgO could be detected with XRD indicating that the maximum solubility of oxygen in the MgSiN2 lattice is probably even much lower than 0.5 wt. %.
The study of the MgSiN2 powders with SEM and the sedimentation method showed that the morphology of the MgSiN2 powders is most probably determined by the morphology of the used Si3N4 starting material. If the starting Si3N4 powder was agglomerated, large sponge like MgSiN2 particles were observed.
Oxidation experiments showed that MgSiN2 powder is oxidation resistant in air up to 830 °C as determined by TGA/DTA. At two different temperatures (1082 °C and 1362 °C) fast oxidation takes place indicating that the oxidation of
MgSiN2 is at least a two step process. An isothermal oxidation experiment at
850 °C showed that MgSiN2 could be fully oxidised indicating that the intermediate oxidation products are not stable.
Finally it can be concluded that it is possible to synthesise MgSiN2 powders with a very low oxygen content which are very suitable for further processing to ceramics with optimum thermal properties.
54 Preparation and characterisation of MgSiN2 powders
References
1. A. Roosen, Modern Substrate Concepts for the Microelectronic Industry, Electroceramics IV 2, Aachen, Germany, September 5 - 7 1994, edited by R. Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann, (Augustinus Buchhandlung, 1994) 1089. 2. C.-F. Chen, M.E. Perisse, A.F. Ramirez, N.P. Padture, H.M. Chan, Effect of Grain Boundary Phase on the Thermal Conductivity of Aluminium Nitride Ceramics, J. Mater. Sci. 29 (1994) 1595. 3. G.A. Slack, Nonmetallic Crystals with High Thermal Conductivity, J. Phys. Chem. Solids 34 (1973) 321. 4. W.A. Groen, M.J. Kraan, and G. de With, Preparation, Microstructure and
Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413. 5. H.T. Hintzen, W.A. Groen, P. Swaanen, M.J. Kraan and R. Metselaar,
Hot-pressing of MgSiN2 Ceramics, J. Mater. Sci. Lett. 13 (1994) 1314. 6. G.A. Slack, R.A. Tanzilli, R.O. Pohl and J.W. Vandersande, The Intrinsic Thermal Conductivity of AlN, J. Phys. Chem. Solids 48 (1987) 641. 7. H.T. Hintzen, R.J. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Densification of MgSiN2, Ceram. Trans. 51, Int. Conf. Ceramic Processing Science Technology, Friedrichshafen (Germany), September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995) 585.
8. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics 2, Basic Science - Developments in Processing of Advanced Ceramics - Part II, Faenza (Italy), October 1995, edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, Italy, 1995) 289. 9. Chapter 3; R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar,
Preparation, Characterisation and Properties of MgSiN2 ceramics, to be published.
55 Chapter 2.
10. J. David et J. Lang, Sur un nitrure double de magnésium et de silicium, C. R. Acad. Sc. Paris 261 (1965) 1005. 11. Refcel, Calculation of cell constants and calculation of all possible lines in a powder diagram by H.M. Rietveld, October 1972.
12. J. David, Y. Laurent et J. Lang, Structure de MgSiN2 et MgGeN2, Bull. Soc. Fr. Minéral. Cristallogr. 93 (1970) 153. 13. B.D. Cullity, Elements of X-Ray Diffraction, second edition, (Addison- Wesley Publishing Company, Inc., 1978).
14. C.P. Gazzara and D.R. Messier, Determination of Phase Content of Si3N4 by X-ray Diffraction Analysis, Am. Ceram. Soc. Bull. 56 (1977) 777.
15. G. Petzow and R. Sersale, Characterization of Si3N4 Powders, Pure and Appl. Chem. 59 (1987) 1673. 16. N. Matter, A. Riedel and A. Wassermann, Quantitative Phase Analysis of
Si3N4 Ceramics using the Powder Diffraction Standard Data Base, Mat. Sci. Forum 133-136, EPDIC 2 (Trans Tech Publications, Switzerland, 1993) 39. 17. W. Pfeiffer and M. Schulze, A Method for the Determination of Weight Factors for Quantitative Phase Analysis using Dual Phase Starting Powders with Application to α/β-Silicon Nitride, Mat. Sci. Forum 133-136, EPDIC 2 (Trans Tech Publications, Switzerland, 1993) 39. 18. D.Y. Li, B.H. O'Conner, Q.T. Chen and M.G. Zadnik, Quantitative Powder X-ray Diffractometry Phase Analysis of Silicon Nitride Materials by a Multiline, Mean-Normalized-Intensity Method, J. Am. Ceram. Soc. 77 (1994) 2195. 19. T. Murata, K. Itatani, F.S. Howell, A. Kishioka and M. Kinoshita, Preparation of Magnesium Nitride Powder by Low-Pressure Chemical Vapor Deposition, J. Am. Ceram. Soc. 76 (1993) 2909. 20. I.S. Gladkaya, G.N. Kremkova and N.A. Bendeliani, Phase diagram of magnesium nitride at high pressures and temperatures, J. Mater. Sci. Lett. 12 (1993) 1547.
56 Preparation and characterisation of MgSiN2 powders
21. M. Barsoum, P. Kangutkar and M.J. Koczak, Nitridation Kinetics and Thermodynamics of Silicon Powder Compacts, J. Am. Ceram. Soc. 74 (1991) 1248. 22. R. Muller, Konstitutionsuntersuchungen und thermodynamischen Berechnungen im system Mg, Si/N, O, Ph. D. Thesis, University of Stuttgart, Stuttgart, Germany (1981) p. 107.
23. J. David, Étude sur Mg3N2 et quelques-unes de ses combinaisons, Rev. Chim. Miner. 9 (1972) 717. 24. M. Wintenberger, F. Tcheou, J. David and J. Lang, Verfeinerung der Struktur
des Nitrids MgSiN2 - eine Neutronenbeugungsuntersuchung, Z. Naturforsch. 35b (1980) 604. 25. Powder Cell, Programm zur Manipulation von Kristallstrukturen und Berechnung der Röntgenpulverdiffraktogramme. Werner Kraus, Dr. Gert Nolze, Bundesanstalt für Materialforschung und -prüfung 12205 Berlin Unter den Eichen 87 (1995). 26. S. Wild, P. Grieveson and K.H. Jack, The Crystal Chemistry of New Metal- Silicon-Nitrogen Ceramic Phases, Spec. Ceram. 5 (1972) 289. 27. J. Persson, P.O. Käll and M. Nygren, Interpretation of the Parabolic and Nonparabolic Oxidation Behaviour of Silicon Oxynitride, J. Am. Ceram. Soc. 5 (1992) 3377. 28. M. Hillert, S. Jonssen and B. Sundman, Thermodynamic Calculation of the Si-N-O System, Z. Metallkd. 83 (1992) 648.
57 Chapter 2.
58 Chapter 3. Preparation, characterisation and properties of
MgSiN2 ceramics
Abstract
MgSiN2 ceramics with and without sintering additives were prepared by hot uni-axial pressing. For the sintered samples the lattice parameters, density, oxygen and nitrogen content, microstructure, oxidation resistance, hardness, elastic constants, linear thermal expansion coefficient and thermal diffusivity were determined. By suitable processing fully dense MgSiN2 ceramics with an oxygen content < 1.0 wt. % could be obtained. The size of the MgSiN2 grains increased with increasing hot-pressing temperature and time. Transmission electron microscopy (TEM) showed that no grain boundary phases were present on the
MgSiN2 grains and that secondary phases are present as separate grains in the
MgSiN2 matrix. Atomic force microscopy (AFM) thermal imaging revealed a thermal barrier at the grain boundaries. However, the influence of the grain size / microstructure on the thermal diffusivity was limited. Furthermore, the influence of the oxygen content and defect chemistry on the thermal diffusivity was limited.
From these data it was concluded that the thermal conductivity of the MgSiN2 ceramic samples, which did not exceed 25 W m-1 K-1 at 300 K, is determined by intrinsic phonon-phonon scattering.
1. Introduction
Non-electrically conducting materials with a high thermal conductivity are potentially interesting as a heat sink material in integrated circuits. Due to the
59 Chapter 3.
miniaturisation of these integrated circuits there is a strong need for replacing the traditional heat sink material Al2O3 (alumina) by new and better ones with a higher -1 -1 thermal conductivity than Al2O3 (viz. 17 - 38 W m K [1]). Several binary adamantine type compounds like SiC, BeO and AlN have been investigated for this purpose. Some years ago, also ternary and quaternary adamantine type oxy-carbo- nitride, (oxy-)nitride, and (oxy-)carbide compounds have been suggested as potential substrate materials [2 - 4]. Especially MgSiN2, which can be deduced from AlN by replacing two Al3+ ions by a combination of Mg2+ and Si4+, was considered to be very promising [2, 5]. Previous studies showed that the mechanical and electrical properties of non-optimised MgSiN2 ceramics are comparable with those of AlN and Al2O3 [2, 6]. A first estimate using the theory of Slack [7] of the maximum achievable thermal conductivity at 300 K resulted in a value of 75 W m-1 K-1 [8]. In a more recent study [9] we suggest that the maximum -1 -1 achievable thermal conductivity of MgSiN2 does not exceed 50 W m K and more probably is limited to a value of about 30 W m-1 K-1, which is comparable with the highest reported experimental value of 25 W m-1 K-1 [10]. However, in the literature no systematic research on the influence of the processing conditions on the thermal conductivity of MgSiN2 ceramics has been reported which might give a better insight in the thermal conductivity limiting mechanism in this material. If this mechanism can be identified, possibly a better, more reliable estimate of the maximum achievable thermal conductivity can be made. Moreover the thermal conductivity of MgSiN2 can be optimised more effectively. The thermal resistivity of phonon conductors is in general determined by the sum of the occurring resistivities, that means the inverse phonon mean free paths of all individual independent phonon scattering processes [11]. Therefore, for phonon conductors like MgSiN2 phonon-phonon scattering, phonon-defect scattering and phonon-grain boundary scattering are considered to influence the thermal conductivity [12, 13]. Furthermore, the heat transport between the grains can be hampered by (oxygen containing) secondary phases. So phonon scattering processes in the MgSiN2 lattice itself (phonon-phonon, phonon-defect), or due to
60 Preparation, characterisation and properties of MgSiN2 ceramics
the microstructure of the ceramics ((phonon scattering at the grain boundaries depending on the grain size), (secondary phases at the grain boundaries)) influence the heat transport in MgSiN2 ceramics.
In this chapter the characterisation and properties of MgSiN2 ceramics processed in different ways is described. The problem of obtaining not fully dense
MgSiN2 ceramics encountered during pressureless-sintering can be solved by using the hot uni-axial pressing technique [14]. This technique was reported to give fully dense ceramic samples necessary for obtaining a high thermal conductivity. Special attention will be paid to the influence of the processing parameters and sintering additives on the (thermal) properties in order to identify the mechanism that determines the thermal conductivity of MgSiN2 ceramics. The synthesis of MgSiN2 powders with a low oxygen content, which is considered to be necessary for obtaining a high thermal conductivity [2, 14], has been reported previously [15 - 17]. Preliminary results concerning the ceramics have already been published elsewhere [10, 15].
2. Experimental
2.1. Preparation
Dense MgSiN2 ceramic samples were prepared using the (reaction) hot uni-axial pressing ((R)HUP) technique. The hot-press (HP 20, Thermal Technology Ind.) was equipped with an Astro furnace (model 100-4560-FP) fitted with graphite heating elements. The interior of the hot-press including the die and the ram is made from graphite. During hot-pressing the temperature, force and displacement were monitored.
Although HUP is a suitable method for obtaining fully dense MgSiN2 ceramics without sintering additives [14], during the present work additives were still sometimes used. The additive was not only used to promote further densification and/or grain growth but especially in an attempt to purify the MgSiN2 bulk by formation of a secondary phase. The intention is to minimise phonon-
61 Chapter 3.
defect scattering by (oxygen) impurities in the bulk causing vacancies, which is reported to have a detrimental effect on the thermal conductivity of AlN [18, 19].
Y2O3 was used as an additive because it is known to be a suitable additive for AlN and because it forms low melting compounds with MgO and SiO2 [20 - 22]. Also
CaO and 2Al2O3.Si3N4 were used because they form in combination with MgO and
SiO2 the low melting compounds CaMgSiO4 (1485 °C [23]) and Mg2SiAlO4N
(1600 °C [24]) or MgSi4Al2O6N4 (1450 - 1650 °C [24]), respectively. Also Si3N4 and Mg3N2 were used as an additive in order to influence the defect chemistry of the MgSiN2 bulk [2]. Finally, the addition of Mg3N2 might be beneficial for obtaining MgSiN2 ceramics with a low oxygen content as has been reported previously for the synthesis of MgSiN2 powders [17] and ceramics [25] with a low oxygen content. The ceramic pellets were prepared with HUP starting from as prepared
MgSiN2 powder or RHUP starting from Mg3N2 (Alfa 932825)/Si3N4 (SKW Trostberg Silzot HQ or Cerac S1177) powder mixtures with or without additives
(see Table 3-1). The MgSiN2 powders were prepared starting from either Mg
(Merck 5815)/Si3N4 (HCST LC12N) or Mg3N2 (Alfa 932825 or Cerac
M1014)/Si3N4 (Tosoh TS10, Cerac S1177 or Ube SNE10) mixtures. The characteristics of the starting materials used and the experimental details for the preparation of the MgSiN2 powders are described elsewhere [17]. To prevent oxidation and hydrolysis of Mg3N2, the Mg3N2/Si3N4 starting mixture was handled and put into the hot-press die in a glove-box purged with nitrogen. All parts of the hot-press that were in contact with the starting powder during hot-pressing were coated with boron nitride or protected with boron nitride coated graphite foil. The following processing parameters (see Table 3-1) were varied: hot-pressing temperature (1450 - 1750 °C), hot-pressing pressure (20 - 75 MPa) and time applying the maximum pressure (0.5 - 5 h). After filling of the die (∅ = 34 mm) with about 10 g of starting powder, the powder was pre-pressed at 10 MPa. Subsequently the hot-press was closed and purged with nitrogen for at least 0.5 h before heating. During heating, with a rate of 10 - 20 °C/min., a pressure of 3 - 4 MPa was applied. The maximum pressure was applied at the top temperature.
62 Preparation, characterisation and properties of MgSiN2 ceramics
After cooling down, the resulting ceramic pellet (± ∅ 33 mm × 3 mm) was cleaned and ground using a 250 µm abrasive diamond wheel and when necessary cut and polished.
Table 3-1: Hot-pressing conditions used for processing the several MgSiN2 ceramic samples.
Code RB02 RB07 RB09 RB10
Starting MgSiN2 MgSiN2 MgSiN2 MgSiN2 Material (Merck & HCST) (Merck & HCST) (Alfa & Tosoh) (Cerac & Cerac) Reaction 1550 °C 1550 °C 1550 °C 1550 °C Conditions 2 h, 75 MPa 2 h, 75 MPa 2 h, 75 MPa 2 h, 75 MPa Code RB11 RB12 RB13 RB14
Starting MgSiN2 Mg3N2 (Alfa) Mg3N2 (Alfa) Mg3N2 (Alfa) Material (Alfa & Ube) Si3N4 (Cerac) Si3N4 (Cerac) Si3N4 (Cerac) Reaction 1550 °C, 1550 °C, 1600 °C, 1550 °C, Conditions 2 h, 75 MPa 2 h, 75 MPa 2 h, 75 MPa 5 h, 75 MPa Code RB25 RB30 RB31 RB32
Starting MgSiN2 Mg3N2 (Alfa) Mg3N2 (Alfa) Mg3N2 (Alfa) Material (Mg & HCST) Si3N4 (Cerac) Si3N4 (Cerac) Si3N4 (Cerac) Additive --- Si3N4 2.9 wt.% Mg3N2 2.1 wt.% Mg3N2 4.2 wt.% Reaction 1550 °C 1600 °C, 1600 °C, 1600 °C, Conditions 2 h, 75 MPa 2 h, 75 MPa 2 h, 75 MPa 2 h, 75 MPa Code RB33 RB34 RB35 RB36
Starting Mg3N2 (Alfa) Mg3N2 (Alfa) Mg3N2 (Alfa) Mg3N2 (Alfa) Material Si3N4 (Cerac) Si3N4 (Cerac) Si3N4 (SKW) Si3N4 (SKW) Reaction 1650 °C, 1700 °C 1600 °C 1750 °C Conditions 2 h, 75 MPa 2 h, 75 MPa 2 h, 75 MPa 2 h, 75 MPa Code RB37 RB38 RB39 RB40
Starting Mg3N2 (Alfa) Mg3N2 (Alfa) Mg3N2 (Alfa) MgSiN2 Material Si3N4 (SKW) Si3N4 (SKW) Si3N4 (SKW) (Alfa & SKW) Additive Y2O3 6 wt.% 2Al2O3.Si3N4 5.8 wt.% CaO 2.3 wt.% Mg3N2 5.0 wt.% Reaction 1600 °C 1600 °C 1600 °C 1600 °C Conditions 2 h, 75 MPa 1 h 4 MPa 1 h 4 MPa 1 h 4 MPa 2 h 20 MPa 2 h 20 MPa 2 h 20 MPa ½ h 25 MPa ½ h 25 MPa ½ h 25 MPa Code RB41 RB42 RB43 RB45
Starting Mg3N2 (Alfa) Mg3N2 (Alfa) Mg3N2 (Alfa) Mg3N2 (Alfa) Material Si3N4 (SKW) Si3N4 (SKW) Si3N4 (SKW) Si3N4 (SKW) Additive Mg3N2 5.0 wt.% Al2O3 4.3 wt.% ------Reaction 1600 °C 1600 °C 1500 °C 1450 °C Conditions 1 h 4 MPa 1 h 4 MPa 2 h, 75 MPa 2 h, 75 MPa 2 h 20 MPa 2 h 20 MPa ½ h 40 MPa ½ h 25 MPa
63 Chapter 3.
2.2. Characterisation
The ceramic samples were investigated with X-ray diffraction (XRD, Philips PW 1050/25) using Cu-Kα radiation. Standard continuous scans (1 °/min.) as well as step scans (0.1 °/min.) were recorded in the range of 10 - 140 ° 2θ for all samples.
The lattice parameters and unit cell volume of MgSiN2 were calculated with the computer program Refcel [26] using at least ten reflections of the orthorhombic cell (space group Pna21 [27]) including a zero point correction. The experimental accuracy of the lattice parameters and unit cell volume was estimated to equal 3 times the standard deviation of the calculated lattice parameters and unit cell volume. The density of the samples was determined by the Archimedes method in water and by using the lattice parameters as determined by XRD. The first method -3 results in the overall density (ρexp [kg m ]) whereas the second procedure gives the -3 crystallographic density (ρcryst [kg m ]) which does not take into account the porosity and secondary phases present in the sample. The accuracy of ρexp was estimated to equal the average standard deviation of 5 measurements and the accuracy of ρcryst was obtained from the accuracy of the unit cell volume obtained from the Refcel calculation. The oxygen, nitrogen and carbon content were measured using a O/N gas analyser (Leco TC 436). A small ceramic sample was powdered and mixed with carbon, after which the oxygen present in the sample is carbothermally converted at high temperatures in an inert atmosphere into CO, which after further oxidation to CO2 is measured with IR-absorption spectroscopy. By further decomposition of the sample at higher temperature the released N2 was measured with catharometry. The carbon content of sample RB35 (see Table 3-1) was determined by heating the sample at 1500 °C in pure O2. The carbon present in the sample is converted into
CO2, which is measured by IR spectroscopy. For this sample also the magnesium, silicon and boron content were determined with Inductive Coupled Plasma Optical
64 Preparation, characterisation and properties of MgSiN2 ceramics
Emission Spectroscopy (ICP/OES) by decomposing the sample in a Na2CO3 melt after which the sample was dissolved in water. Scanning electron microscopy (SEM, JEOL 840A) was used for microstructural analysis of the ceramic samples. The SEM samples were prepared by grinding a sample with abrasive diamond wheels (200, 63, 30 and 10 µm) and subsequently polishing it on nylon cloth with diamond paste (1 and 0.25 µm). The polished samples were thermally etched at 1300 °C for 18 minutes in vacuum using a heating rate of 300 °C h-1 and a cooling rate of 600 °C h-1. Subsequently, the samples were attached to a sample holder using conductive paste and finally sputtered with gold to obtain an electrically conducting surface layer. Transmission Electron Microscopy (TEM, JEOL 2000 FX) equipped with Noran Energy Dispersive Spectroscopy (EDS) suitable for light element analyses down to boron was used to study the microstructure in more detail, and moreover to determine qualitatively the chemical composition. The TEM samples were prepared by grinding a ceramic sample (8 × 6 × 2 mm) to a maximum thickness of 1 mm. Subsequently, a disk ∅ 3 mm was cut from the sample with an ultrasonic disk cutter. Then, both sides of the disk were ground and subsequently polished on a nylon cloth with diamond paste until ~ 100 µm thickness. With a dimple grinder the thickness of the sample was further reduced to 5 - 10 µm. Finally, the thickness of the sample was reduced using ion milling until a small hole appeared in the centre of the sample. Before investigating the sample, a thin carbon layer was sputtered on the samples to assure sufficient electrical conductivity. Atomic Force Microscopy (AFM, Topometrix, Santa Clara, CA, USA) was used to study simultaneously the topography and qualitatively the thermal conductivity by thermal contrast imaging. In general this method provides information about where the thermal barriers in a material are located (grains itself or grain boundary). The sample RB31 used for measuring the thermal diffusivity was investigated.
65 Chapter 3.
2.3. Properties
To check the oxidation behaviour of the densified MgSiN2 ceramics, DTA/TGA (differential thermal analysis/ thermo gravimetric analysis) measurements were performed (Netzsch STA 409). Samples with a high and low oxygen concentration were investigated.
The Knoop and Vickers hardness (HK [GPa] and HV [GPa]) were measured on several polished, fully dense ceramic samples. For each sample 5 - 10 measurements were performed. Small loads (< 500 g) did not result in a clear indent whereas larger loads (≥ 1000 g) caused cracking and chipping of material. The used load of 500 g is a compromise: the indentation is small but well shaped. The average standard deviation was about 2 GPa. The elastic constants (Young's modulus E [GPa] and Poisson's ratio ν [-]) at 293 K for sample RB43 were measured on a small ceramic disk (∅ 15.85 mm × 1.00 mm) using the impulse excitation method [28] (GrindoSonic, Lemmens Elektronica BV, Belgium). The fundamental natural flexural and torsional frequency of the sample was measured. From this, the sample dimensions and mass, the Young's modulus and Poisson's ratio were evaluated using the computer program E-mod (Lemmens Elektronica BV, Belgium) based on the work of Glandus [29]. The experimental accuracy for the Young's modulus and Poisson's ratio was estimated to equal 5 GPa and 0.01, respectively. The linear thermal expansion coefficient α [K-1] was measured with a dual rod dilatometer (Linseis L 75) in nitrogen from 300 to 1573 K, and in air from
300 to 1173 K on a MgSiN2 ceramic bar (2 mm × 2 mm × 10.00 mm). Al2O3 (sapphire) was used as a reference material. Three heating and cooling cycles were performed for each sample, one for setting of and two for measuring of the sample. The used heating rate for setting was 10 ºC/min. and 2 ºC/min. for the actual measuring of the sample. The experimental accuracy was estimated to be ± 0.4 10-6 K-1. The thermal diffusivity a [m2 s-1] was measured on small carbon coated ceramic samples (∅ 11 mm × 1 mm) with a uniform thickness and a low roughness
66 Preparation, characterisation and properties of MgSiN2 ceramics
using photo and/or laser flash equipment (Compotherm Messtechnik GmbH). The carbon coating was used to increase the absorptivity of the front surface, and the emissivity of the back surface. Some samples were coated with a thin layer of gold before the sample was coated with carbon. The thin gold layer prevents direct transmission of the laser beam and aids the energy transfer to the sample. The convective heat losses were minimised by measuring the samples in vacuum. The experimental accuracy of the measurement was estimated to be within 5%. The thermal conductivity κ [W m-1 K-1] of the samples was calculated from
-3 -1 -1 the density ρexp [kg m ], specific heat cV [J kg K ] and thermal diffusivity (a [m2 s-1]) using:
κ = ρ cV a (1) For each sample the density and thermal diffusivity data were taken from this work and the specific heat of 767.38 J kg-1 K-1 at 300 K as given in a previous study concerning the thermodynamic properties of MgSiN2 [30].
3. Results and discussion
3.1. Characterisation
3.1.1. Phase formation and lattice parameters of MgSiN2
The XRD results indicate that nearly single phase MgSiN2 ceramics were obtained (vide infra Table 3-3). No preferential orientation could be detected with cylindrical camera measurements indicating that isotropic materials were obtained.
The XRD data were compared with data previously obtained for MgSiN2 powders [17] and literature data [27, 31, 32]. In addition to the earlier presented results [17] the reflections between 100 and 140 ° 2θ (d-value range 1.0063 - 0.8204 Å) were established (Table 3-2). The tabulated d-values and intensities are evaluated from the average observed lattice parameters (viz. a = 5.2697 ± 0.0014 Å, b = 6.4734 ± 0.0011 Å, c = 4.9843 ± 0.0010 Å) and intensities of samples RB02, RB07, RB09-RB14, RB25, RB30-RB36, RB40, RB41, RB43 and RB45,
67 Chapter 3.
respectively. If a reflection cannot be ascribed to a single set of hkl values both possible sets are mentioned. An underlined set of hkl values indicates that this hkl set is thought to be the correctly indexed one based on experimentally observed d-value and/or theoretical intensity calculations.
Table 3-2: The hkl reflections and corresponding observed d-values and relative intensities (I/I0) for
MgSiN2 ceramics.
hkl d-value I/I0 hkl d-value I/I0 hkl d-value I/I0 [Å] [%] [Å] [%] [Å] [%] 110 4.09 9 241 1.3291 11 205 0.9323 2 011 3.949 10 400/033 1.3174/1.3164 1 531/522 0.9304/0.9298 2 111 3.160 1 322 1.3124 22 244 0.9245 1 120 2.758 88 401 1.2737 5 334/360 0.9194/0.9193 3 200 2.6348 45 150 1.2573 1 414/171 0.8965/0.8960 <1 002 2.4921 80 051 1.2531 1 171/225 0.8960/0.8959 <1 210 2.4404 3 004 1.2461 2 135/163 0.8919/0.8918 4 121 2.4132 100 420 1.2202 <1 532/154 0.8852/0.8850 <1 201 2.3294 23 151 1.2191 <1 353 0.8829 <1 211 2.1918 1 242 1.2066 5 600 0.8783 1 112 2.1277 1 332 1.1953 <1 610/443 0.8703/0.8703 4 220 2.0434 <1 114 1.1919 1 362 0.8625 4 130 1.9969 <1 313/421 1.1866/1.1852 <1 523 0.8581 1 031 1.9802 3 233 1.1776 1 235/263 0.8559/0.8558 <1 221 1.8907 <1 402 1.1647 1 172 0.8555 <1 122 1.8491 28 124 1.1355 2 254 0.8498 <1 202 1.8105 12 251 1.1316 1 434/460 0.8348/0.8347 <1 212 1.7436 2 204 1.1264 1 006 0.8307 1 310 1.6952 2 161 1.0340 6 602 0.8283 1 040 1.6183 25 403 1.0323 4 461/533 0.8232/0.8227 <1 013 1.6092 1 440/053 1.0217/1.0212 1 612 0.8216 <1 231 1.5830 1 153/520 1.0026/1.0021 2 132 1.5583 1 441 1.0009 5 320 1.5439 36 234/260 0.9986/0.9984 <1 321 1.4747 1 044 0.9873 2 123 1.4231 34 521 0.9825 2 203 1.4053 11 162 0.9731 1 240 1.3790 7 324 0.9696 3 213 1.3734 1 442 0.9453 1 042 1.3573 15 125 0.9375 5
68 Preparation, characterisation and properties of MgSiN2 ceramics
The lattice parameters and unit cell volume are the same within the experimental accuracy for all samples, except for RB39 that was sintered with CaO as an additive (see Table 3-3). For this sample somewhat smaller lattice parameters were observed. This might be caused by the presence of dissolved Ca and O in the
MgSiN2 lattice. Since Ca has a significantly larger ionic radius than Mg (~ 1.0 versus 0.57 Å [33], respectively) this observation cannot be explained by only 2+ 2+ replacing Mg by Ca in the MgSiN2 lattice. However, if Ca and O both dissolve in the MgSiN2 lattice, the increase of the lattice parameters, due to the incorporation of Ca on a Mg site, might be overcompensated by the substitution of N3- by the smaller O2- ion (1.46 versus 1.38 Å [33], respectively) in combination with the formation of cation vacancies, resulting as an overall effect in smaller lattice parameters for MgSiN2. In the CaO doped sample no Ca containing secondary phase could be detected supporting the assumption that CaO has dissolved into the MgSiN2 lattice as indicated by the lattice parameter results. In Table 3-3 also the observed secondary phases as detected with XRD are presented. Between brackets the relative intensity is presented of the strongest reflection of the detected secondary phase. In some of the samples without additives MgO (Periclase JCPDS 4-829), Mg2SiO4 (Forsterite JCPDS 34-189),
α-Si3N4 (JCPDS 41-360) and β-Si3N4 (JCPDS 33-1160) could be detected as a secondary phase. Only in RB43 a non-identified secondary phase was observed indicated with a ‘Y’. Mg2SiO4 was only detected in the ceramic samples with a high oxygen content (> 1.5 wt. % O) whereas MgO could be detected in oxygen poor samples. The presence of Si3N4 can be explained by the evaporation of magnesium [14, 16, 25] during hot-pressing, unreacted starting materials or the use of a non-stoichiometric Mg3N2 deficient starting mixture. It is noted that both
α-Si3N4 and β-Si3N4 were found as a secondary phase in samples hot-pressed at 1550 °C whereas in samples hot-pressed at higher temperatures only the thermodynamically more stable β-Si3N4 [34, 35] was found. In addition to the earlier mentioned secondary phases the Al2O3 doped samples (RB38 and RB42) contained AlN (JCPDS 25-1133) and a non-identified secondary phase indicated with an 'X' in Table 3-3. The presence of AlN indicates that reaction between
69 Chapter 3.
Table 3-3: The lattice parameters, unit cell volume and secondary phases observed for the several
MgSiN2 ceramic samples (the experimental accuracy of the lattice parameters is indicated between brackets and was estimated to equal 3 times the standard deviation of the calculated lattice parameters and unit cell volume; * : 5.2672(18) equals 5.2672 ± 0.0018; ** : a question mark indicates that the presence of that phase is questionable).
Code a b c V detected secondary phase(s) [Å] [Å] [Å] [Å3]
* RB02 5.2672(18) 6.4726(24) 4.9829(18) 169.88(12) Mg2SiO4 (2), MgO (1) ** RB07 5.2693(27) 6.4732(33) 4.9836(33) 169.98(24) Mg2SiO4 (1 ? ), MgO (2), α-Si3N4 (1)
RB09 5.2692(15) 6.4727(21) 4.9829(15) 169.95(12) Mg2SiO4 (2), α-Si3N4(2)
RB10 5.2695(21) 6.4721(27) 4.9842(24) 169.98(18) α-Si3N4(2), β -Si3N4 (1) RB11 5.2677(12) 6.4745(18) 4.9841(15) 169.99(9) MgO (2)
RB12 5.2699(15) 6.4711(21) 4.9833(15) 169.94(12) α-Si3N4 (2), β -Si3N4 (2) RB13 5.2691(15) 6.4737(21) 4.9848(15) 170.03(12) —
RB14 5.2701(12) 6.4743(15) 4.9859(12) 170.12(9) α-Si3N4 (1), β -Si3N4 (2) RB25 5.2676(21) 6.4732(27) 4.9826(18) 169.90(18) —
RB30 5.2689(9) 6.4738(9) 4.9838(9) 169.99(6) β-Si3N4 (3) RB31 5.2701(12) 6.4733(15) 4.9844(12) 170.04(9) MgO (1) RB32 5.2725(9) 6.4740(12) 4.9858(9) 170.19(6) MgO (3)
RB33 5.2684(12) 6.4734(15) 4.9833(12) 169.95(6) β-Si3N4 (1 ?) RB34 5.2699(9) 6.4745(9) 4.9840(9) 170.05(6) — RB35 5.2713(12) 6.4746(21) 4.9855(9) 170.15(12) MgO (< 1 ?) RB36 5.2692(15) 6.4750(27) 4.9836(15) 170.03(15) —
RB37 5.2705(27) 6.4771(48) 4.9852(27) 170.18(24) Y8Si4N4O14 (11)
RB38 5.2725(21) 6.4715(21) 4.9851(12) 170.09(15) β-Si3N4 (8), AlN (4), X(5)
RB39 5.2684(21) 6.4677(39) 4.9803(18) 169.70(21) β-Si3N4 (3)
RB40 5.2704(15) 6.4744(27) 4.9845(15) 170.08(15) β-Si3N4 (1), MgO (6) RB41 5.2701(9) 6.4737(18) 4.9860(9) 170.11(9) MgO (0 ?)
RB42 5.2745(30) 6.4702(48) 4.9849(30) 170.12(27) MgO (2), β -Si3N4 (1), AlN (4), X(6) RB43 5.2710(9) 6.4724(12) 4.9845(9) 170.05(6) Y (5)
RB45 5.2717(21) 6.4711(24) 4.9843(18) 170.03(15) α-Si3N4 (1)
Al2O3 and N2 atmosphere and/or Mg3N2 has occurred. The presence of AlN is in accordance with the fact that in RB38 also a substantial amount of (not reacted)
β-Si3N4 was detected and not as more likely expected a β-sialon. The Y2O3 doped
sample contained Y4Si2N2O7 (Y8Si4N4O14, JCPDS 32-1451) and not Y2Si3O3N4
(JCPDS 45-249) as previously reported [25] for pressureless sintered MgSiN2 with
70 Preparation, characterisation and properties of MgSiN2 ceramics
Y2O3 addition. The amount of secondary phase present in the samples sintered with an excess of Mg3N2 was not higher than for the undoped samples. This indicates that during hot-pressing the excess Mg3N2 and/or the formed MgO evaporates, as expected from a previous study [17] and the fact that the lattice parameters are independent of the weighed-in Mg/Si ratio.
3.1.2. Density The samples hot-pressed at low pressures (< 75 MPa) are not dense (relative density ρexp/ρcryst < 99.5 % (see Table 3-4)), except RB39 which was sintered using CaO as a sintering aid (Table 3-1). The high relative density of RB39 can be
Table 3-4: The overall density ρexp, crystallographic density ρcryst and relative density ρexp/ρcryst of the
MgSiN2 ceramic samples (between brackets the experimental accuracy is indicated).
Code ρexp ρcryst ρexp/ρcryst Code ρexp ρcryst ρexp/ρcryst [g cm-3] [g cm-3] [%] [g cm-3] [g cm-3] [%] RB02 3.154(3) 3.144(2) 100.3(2) RB34 3.144(1) 3.141(1) 100.1(1) RB07 3.141(2) 3.143(4) 99.9(2) RB35 3.144(1) 3.139(2) 100.2(1) RB09 3.143(3) 3.143(2) 100.0(2) RB36 3.142(1) 3.142(3) 100.0(2) RB10 3.148(3) 3.143(3) 100.2(2) RB37 3.168(1) 3.139(4) 100.9(2) RB11 3.147(1) 3.143(2) 100.1(1) RB38 3.060(3) 3.141(3) 97.4(2) RB12 3.145(2) 3.143(2) 100.1(1) RB39 3.131(2) 3.148(4) 99.5(2) RB13 3.145(1) 3.142(2) 100.3(1) RB40 3.074(3) 3.141(3) 97.9(2) RB14 3.149(1) 3.140(2) 100.3(1) RB41 3.074(1) 3.140(2) 97.9(1) RB25 — 3.142(3) — RB42 3.022(5) 3.140(5) 96.2(3) RB30 3.144(1) 3.143(1) 100.0(1) RB43 3.139(2) 3.141(1) 99.9(1) RB31 3.146(1) 3.141(2) 100.2(1) RB45 3.127(2) 3.142(3) 99.5(2) RB32 3.145(2) 3.139(1) 100.2(1) RB33 3.143(1) 3.143(2) 100.0(1)
explained by the fact that CaO most probably reacts with MgSiN2, as suggested by the XRD results, which enhances the sintering process at lower hot-pressing pressures. All samples hot-pressed at 75 MPa have a relative density
71 Chapter 3.
ρexp/ρcryst ≥ 99.5 % (see Table 3-4). As expected from the presence of a Y2O3 containing secondary phase (Y8Si4N4O14), with a higher density than MgSiN2, RB37 has an overall density higher than the crystallographic density. Dense samples (≥ 99.5 %) can be obtained at temperatures substantially below 1543 °C at which liquid phase formation in the MgO-SiO2 system is expected to occur [36] viz. RB43 (1500 °C) and RB45 (1450 °C). No systematic dependence of the sintering behaviour on the oxygen content was observed. So, it can be concluded that the applied pressure is more important than the temperature for obtaining fully dense samples.
3.1.3. Chemical composition For the ceramic samples it was quite difficult to obtain reliable oxygen and nitrogen content data. Investigation showed that the measured oxygen and especially nitrogen content varied with the method used to powder the ceramic sample indicating the necessity of very careful sample preparation.
Table 3-5: Measured oxygen and nitrogen content for several MgSiN2 powders and ceramic samples (* Oxygen and nitrogen content obtained after careful sample preparation (samples powdered without introduction of oxygen). Note that no reliable nitrogen content data were obtained; ** Nitrogen content for fully decomposed samples.). Code wt. % oxygen wt. % oxygen Code wt. % oxygen wt. % nitrogen powder ceramic ceramic ceramic RB02 1.6 3.8 RB12 1.3 — RB07 2.6 1.6 RB13 1.0 — RB09 3.4 3.1 RB14 1.7 (0.8)* — (25 - 31)* RB10 2.7 2.0 RB30 1.2 — RB11 2.7 1.8 RB31 1.1 — RB25 1.4 — RB32 1.0 — 34.1** RB33 1.1 — 35.2** RB34 1.0 — RB35 2.5 (0.0 - 0.3)* 31 (26 - 28)*
72 Preparation, characterisation and properties of MgSiN2 ceramics
The oxygen content generally decreases (except for RB02) for MgSiN2 ceramics as compared to the corresponding starting powder (Table 3-5). This decrease is probably due the graphite environment in which the samples are sintered resulting in a carbothermal nitridation reaction of the oxygen containing compounds present in the MgSiN2 starting powder. In the resulting ceramics still some oxygen is present due to the fact that the densification process, which reduces the contact surface of the MgSiN2 compact with the gas phase, is too fast for the carbothermal nitridation reaction to complete. After densification of the sample, the possibility of oxygen removal is hampered by solid state diffusion processes. The oxygen content of the reaction hot-pressed ceramics is lower than for the hot- pressed ceramics indicating that reaction hot-pressing results in purer samples. It is noted that the measured oxygen content for the ceramics might be too high because the method used to powder the ceramic sample can introduce oxygen impurities into the sample. The measured nitrogen content was in general much lower (25 - 31 wt. %) than expected for MgSiN2 (theoretical value 34.8 wt. %) due to incomplete decomposition of the sample. When careful sample preparation resulted in a full decomposition of the sample (RB32 and RB33), reliable nitrogen content data were obtained having values close to the theoretical value (see Table 3-5). A complete chemical analysis of sample RB35 was made. The measured magnesium and silicon content for this sample of 30.2 ± 0.9 wt. % Mg and 34.7 ± 1.0 wt. % Si matches very well with the theoretical amounts expected for
MgSiN2 (viz. 30.23 wt. % and 34.93 wt. % respectively). About 0.5 wt. % C was detected which originates from SiC impurity present in the used Si3N4 starting material and possibly from the graphite interior of the hot-press. Moreover, small traces (< 0.01 wt. %) of boron were detected that originates from the boron nitride coated graphite foils and dies used for hot-pressing of the sample. X-Ray Fluorescence (XRF) revealed the presence of small traces of Fe and W. Other impurities could not be detected.
73 Chapter 3.
These above results indicate that by suitable processing very pure MgSiN2 ceramics can be obtained with a purity comparable to that as previously reported for MgSiN2 powder viz. < 0.1 wt. % O and 34.2 wt. % N [17].
3.1.4. Microstructure The polished SEM samples showed grain boundaries only after thermal etching. No residual porosity could be observed in the investigated samples in agreement with the density measurements. The microstructural investigation with the SEM and TEM showed that as expected the grain size increased with the hot-pressing temperature from about 0.25 µm (Fig. 3-1, RB12, 1550 °C) to about 1.5 µm (Figs. 3-2 and , RB34, 1700 °C). The grains of most SEM samples appeared as if they were build up out of smaller grains but the TEM analyses showed that this observation is a result of the used thermal etching procedure. The grain size also increased with longer hot-pressing time (RB12: 1550 °C, 2 h about 0.25 µm and RB14: 1550 °C, 5 h about 1.8 µm).
Fig. 3-1: SEM photograph of thermally etched (1300 °C, 18 min.) surface of sample RB12 hot-pressed for 2 h at 75 MPa and 1550 °C, showing an average grain size of about 0.25 µm.
74 Preparation, characterisation and properties of MgSiN2 ceramics
Fig. 3-2: TEM photograph of sample RB34 showing a grain size of about 1.5 µm.
The microstructure was not influenced by the Mg/Si ratio in the starting mixture (RB30 and RB32) and the oxygen content of the samples. Clean grain boundaries were observed with TEM irrespective of the weighed-in Mg/Si ratio and the oxygen content of the samples. In Fig. 3-4 a typical picture of a grain boundary and triple point is shown. Only occasionally secondary phases could be detected at a triple point. Although the TEM study did not reveal any grain boundary phases, the AFM thermal contrast image study of RB31 revealed the
75 Chapter 3.
presence of a thermal resistance at the grain boundaries (see Fig. 3-5) resulting in clear image of the microstructure of the sample whereas the conventional topography image of the sample provided no information at all.
Fig. 3-3: SEM photograph of thermally etched surface of sample RB34 hot-pressed for 2 h at 75 MPa and 1700 °C, showing an average grain size of about 1.5 µm.
Fig. 3-4: TEM photograph of typical observed grain boundaries triple
points for MgSiN2 ceramics.
76 Preparation, characterisation and properties of MgSiN2 ceramics
Fig. 3-5: AFM thermal contrast image for sample RB31 showing a thermal barrier at the MgSiN2 grain boundaries.
3.1.5. TEM/EDS A qualitative EDS analysis of the phase at the triple points as observed with TEM indicated only sometimes the presence of Mg, Si and O (suggesting MgSiO3 or
Mg2SiO4) or Mg and O with some trace Si (suggesting MgO). Although EDS analysis with TEM is a qualitative method we concluded from the relative intensities of the Mg and Si signal of a MgSiN2 grain (Fig. 3-6) and several Mg-Si-
O containing grains, that MgSiO3 grains (Fig 3-7) and Mg2SiO4 grains (Fig. 3-8) are present as a secondary phase in the oxygen rich samples and Mg2SiO4 grains and MgO grains are present as a secondary phase in the oxygen poor samples. The
77 Chapter 3.
presence of MgO and Mg2SiO4 as a secondary phase was confirmed by the XRD
measurements. This suggests that MgSiO3 is present as a glassy secondary phase although crystalline MgSiO3 (proenstatite) is reported [16] as an oxidation product of MgSiN2. Although Si3N4 was observed with XRD the presence of this secondary phase could not be confirmed with EDS.
1600 C Si
1200 Mg
800 Counts [-]
400 N O
0 0.0 0.5 1.0 1.5 2.0 energy [keV]
Fig. 3-6: TEM/EDS analyses of a MgSiN2 grain (The "C" signal is caused by the carbon coating to enhance the electrical conductivity of the sample).
1600 Si C
1200 Mg
800
Counts [-] O 400
0 0.0 0.5 1.0 1.5 2.0 energy [keV]
Fig 3-7: TEM/EDS analyses of a Mg-Si-O grain probably MgSiO3 as suggested by the relative intensity of the Mg and Si peak as compared to Fig. 3-6.
78 Preparation, characterisation and properties of MgSiN2 ceramics
1600
C 1200 Mg Si 800 O Counts [-]
400
N 0 0.0 0.5 1.0 1.5 2.0 energy [keV]
Fig. 3-8: TEM/EDS analyses of a Mg-Si-O grain probably Mg2SiO4 as suggested by the relative intensity of the Mg and Si peak as compared to Fig. 3-6 (The "C" signal is caused by the carbon coating to enhance the electrical conductivity of the sample).
Mg Si
N O
Fig. 3-9: TEM-EDS mapping of an MgSiN2 samples (area about 1.8 × 1.8 µm) for Mg, Si, N and O showing the location of magnesium (Mg), silicon (Si), nitrogen (N) and oxygen (O) in the sample (in the upper right corner the video image of the sample is given).
In order to study the secondary phases in more detail a MgSiN2 sample from a previous study [14] with a high oxygen content (about 5 wt. % O) was
79 Chapter 3.
investigated. Also in this case neither grain boundary phases were observed nor did most triple points contain any secondary phase. Several mappings of a representative part of the sample (about 30 grains) for Mg, Si, N and O showed that most grains consisted of Mg, Si and N (MgSiN2), whereas a few grains consisted of
Mg and O (MgO) mostly in combination with Si (MgSiO3 or Mg2SiO4). In Fig. 3-9 a typical result is presented. So the secondary phases like MgSiO3 or Mg2SiO4, and
MgO are present as separate grains in the MgSiN2 matrix. This indicates that the wetting behavior of the MgSiN2 grains by secondary phases in the Mg-Si-N-O system is very poor.
3.2. Properties
3.2.1. Oxidation resistance
Irrespective of the oxygen content, the MgSiN2 ceramics prepared by hot uni-axial pressing are oxidation resistant in air up to about 1200 ºC (see Fig. 3-10,
Table 3-6), which is about 250 ºC higher than the value observed for MgSiN2
20 2.0
DTA 15 1.5 TGA
10 1.0 V] µ 5 0.5 [ × 100[%] × 0 T ∆ m /
0 0.0 m ∆
-5 -0.5
-10 -1.0 400 600 800 1000 1200 1400 1600 T [ºC]
Fig. 3-10: TGA/DTA plot of the oxidation behaviour of MgSiN2 ceramics in air showing the temperature difference (∆T ) and relative mass
difference (∆m/m0) as function of the temperature (T ).
80 Preparation, characterisation and properties of MgSiN2 ceramics
powders [17]. The difference between the oxidation resistance of MgSiN2 powder and ceramic can be ascribed to the much lower contact surface area of the ceramics and the formation of protective layer during oxidation. For comparison the oxidation resistance of AlN, γ -aluminium oxynitride (Alon) and Si3N4 ceramics are given in Table 3-6.
Table 3-6: Some properties of MgSiN2 ceramics as compared to Al2O3, Alon, AlN and β -Si3N4 ceramics.
MgSiN2 MgSiN2 [2] Al2O3 Alon AlN β-Si3N4
HV [GPa] 14 - 20 14 - 16 19.5 [37] 20 [38] 12 [39] 16 - 22 [40]
HK [GPa] 14 - 19 — 15.8 [41] — 12 [42] 10.9 [41] E [GPa] 284 235 393 [41] 322 [38] 315 [43] 304 [44] ν [-] 0.250 0.232 0.240 [41] 0.253 [38] 0.245 [43] 0.267 [44]
Toxidation [°C] 1200 > 920 — 1200 [45] 900 [42, 46] 1400 [47] α(293 K) [K-1] 3.8 10-6 — 5.4 10-6 [48] 5.8 10-6 [49] 2.7 10-6 [50] 1.4 10-6 [51] α(873 K) [K-1] 6.8 10-6 — 8.7 10-6 [48] 7.8 10-6 [49] 5.9 10-6 [50] 3.6 10-6 [51] α [K-1] 5.8 10-6 5.8 10-6 7.8 10-6 [48] 7.4 10-6 [49] 4.8 10-6 [50] 2.5 10-6 [51]
3.2.2. Hardness For the dense samples the measured Knoop and Vickers hardness varied from 13.9 to 19.9 GPa, in agreement with an earlier reported value for the Vickers hardness of about 15 GPa [2] (see Table 3-6). In general, the reaction hot-pressed samples starting from Mg3N2/Si3N4 mixtures have a lower hardness (∼ 15 GPa) than those prepared starting from MgSiN2 powder (∼ 19 GPa). This difference in hardness between the several MgSiN2 samples cannot be explained. The obtained values are fairly high and the best values are comparable with the hardness obtained for Al2O3, Alon and β-Si3N4 (see Table 3-6). The results indicate that
MgSiN2 ceramics with a high hardness can be quite easily obtained.
81 Chapter 3.
3.2.3. Young's modulus The flexural and torsional frequency of RB43 equalled 62.4 kHz and 39.0 kHz, respectively resulting in E = 284 GPa and ν = 0.250. The observed Young's modulus is somewhat lower as compared to Alon, AlN and β-Si3N4 but considerably higher than a previously observed value of 235 GPa [2] (see Table 3-6). This relative large difference may be partially ascribed to the low density (ρexp/ρcrys = 98.9 %) and purity (3.7 wt. % oxygen) of the sample described in [2].
3.2.4. Thermal expansion Within the experimental accuracy, the linear thermal expansion coefficient for the sample measured in air and nitrogen are the same. For heating and cooling of the sample no hysteresis was observed. The linear thermal expansion coefficient increases with temperature (see Fig. 3-11) and becomes almost constant at about 1000 K. Subsequently the thermal expansion starts to increase again with
9.5E-06
8.0E-06 ] -1 6.5E-06 [K α
5.0E-06
3.5E-06 300 500 700 900 1100 1300 1500 1700 T [K]
Fig. 3-11: The isotropic linear thermal expansion coefficient (α) of
MgSiN2 ceramics as a function of the absolute temperature (T ) (300 - 1573 K) as determined with dilatometry in a nitrogen atmosphere.
82 Preparation, characterisation and properties of MgSiN2 ceramics
increasing temperature. This increase at high temperatures is ascribed to the thermal generation of defects in the MgSiN2 crystal structure causing macroscopic length changes, which are measured with dilatometry. The value of 3.8 10-6 K-1 at 293 K, 6.7 10-6 K-1 at 827 K and the average value of 5.8 10-6 K-1 between 293 and 873 K (see Table 3-6) agrees reasonably well with the previously reported values of respectively 4.4 10-6 K-1 [52], 6.5 10-6 K-1 [8] and 5.8 10-6 K-1 [2]. For comparison the thermal expansion coefficients of Al2O3, AlN, Alon and Si3N4 are presented (Table 3-6) indicating that Al2O3 and Alon have a higher, and AlN and
Si3N4 a lower thermal expansion coefficient than MgSiN2.
3.2.5. Thermal diffusivity/conductivity In general the observed thermal conductivity of the fully dense samples -1 -1 (ρexp/ρcryst ≥ 99.5 %) equals 17 - 21 W m K whereas the thermal conductivity of the other samples is substantially smaller and varies between 12 - 16 W m-1 K-1 (Table 3-7).
Table 3-7: Measured thermal diffusivity (a) and resulting thermal conductivity (κ ) for several
MgSiN2 ceramic samples.
Code a κ Code a κ Code a κ 2 -1 -1 -1 2 -1 -1 -1 2 -1 -1 -1 [cm s ][W m K ][cms ][W m K ][cm s ][W m K ] RB02 0.076 18.4 RB30 0.076 18.3 RB38 0.058 13.6 RB07 0.071 17.1 RB31 0.073 17.6 RB39 0.057 13.7 RB09 0.082 19.8 RB32 0.076 17.8 RB40 0.069 16.3 RB10 0.067 16.2 RB33 0.074 17.8 RB41 0.071 16.7 RB11 0.061 14.7 RB34 0.082 19.2 RB42 0.052 12.1 RB12 0.066 15.9 RB35 0.079 19.1 RB43 0.062 14.9 RB13 0.080 19.3 RB36 0.086 20.7 RB45 0.054 13.0 RB14 0.077 18.6 RB37 0.086 20.9
Despite the high relative density, also the CaO doped sample has a low thermal conductivity, which is consistent with the assumption of incorporation of CaO into
83 Chapter 3.
the MgSiN2 lattice resulting in defects. The Y2O3 doped sample RB37 has a thermal conductivity (20 W m-1 K-1) comparable with that of the best samples indicating that the Y2O3 addition does not hamper the thermal conductivity.
The less dense Mg3N2 doped samples hot-pressed at a lower pressure (RB40 and RB41) have a low thermal conductivity (~ 16.5 W m-1 K-1) due to the lower relative density. The 2Al2O3.Si3N4 and Al2O3 doped samples RB38 and RB42 have the lowest thermal conductivity (12 - 14 W m-1 K-1), which is caused by the low relative density of the samples and the presence of secondary phases.
0.10
0.08 ]
-1 0.06 s 2
[cm 0.04 a
0.02
0.00 01234 overall oxygen content [wt. %]
Fig. 3-12: The thermal diffusivity (a) versus the overall oxygen
content for fully dense MgSiN2 ceramics (line is drawn as a guide to the eye).
For the fully dense samples, the influence of the overall oxygen content (RB02, RB07, RB09-RB14 and RB30-RB36) is very limited (Fig. 3-12) indicating that secondary phases are not hampering the heat transport between the MgSiN2 grains in agreement with the TEM investigation showing clean grain boundaries. Also the influence of the processing temperature (RB12, RB13, RB33 and RB34, and RB45, RB43, RB35 and RB36) on the thermal conductivity is small for
T ≥ 1600 ºC (Fig. 3-13). This indicates that the MgSiN2 grain size for the samples processed at T ≥ 1600 ºC does not limit the heat transport. These results suggest that the thermal conductivity of MgSiN2 is determined by phonon scattering
84 Preparation, characterisation and properties of MgSiN2 ceramics
0.10 Mg3N2 (alfa) & Si3N4 (cerac)
Mg3N2 (alfa) & Si3N4 (SKW) 0.08 ]
-1 0.06 s 2
[cm 0.04 a
0.02
0.00 1400 1500 1600 1700 1800 T [°C]
Fig. 3-13: The thermal diffusivity (a) versus the applied hot-pressing
temperature (T ) for fully dense MgSiN2 ceramics (line is drawn as a guide to the eye).
0.10
0.08 ]
-1 0.06 s 2
[cm 0.04 a
0.02
0.00 0.90 0.95 1.00 1.05 1.10 1.15 weighed-in Mg/Si ratio [-]
Fig. 3-14: The thermal diffusivity (a) versus the weighted-in Mg/Si
ratio for fully dense MgSiN2 ceramics processed at 1600 °C (line is drawn as a guide to the eye).
processes within the MgSiN2 grains itself. From the fact that manipulation of the defect chemistry (by changing the O concentration (Fig. 3-12) and the 'weighed-in' Mg/Si ratio (RB30, RB13, RB31 and RB32) (Fig. 3-14)) only has a limited influence it can be deduced that phonon-defect scattering is also not limiting the thermal conductivity. Therefore the thermal conductivity is probably determined by
85 Chapter 3.
intrinsic phonon-phonon scattering indicating that the thermal conductivity of
MgSiN2 cannot be increased substantially.
4. Theoretical considerations
The experimental results indicate that intrinsic scattering is determining the thermal -1 -1 conductivity of MgSiN2 limiting its value to about 25 W m K at 300 K. This conclusion can be supported by theoretical calculations considering the influence of secondary phases, grain size, and defects in the grains on the thermal conductivity.
4.1. Secondary phases
For simplicity the effect of a secondary phase at the grain boundary on the experimental thermal conductivity κexp is approximated by a two-phase serial system, which results in the most detrimental effect on the thermal conductivity, using [53]: 1 = V 1 + V 2 (2) κ exp κ 1 κ 2
-1 -1 where V1 (= 1 - V2) and V2 [-] are the volume fraction and κ1 and κ2 [W m K ] the intrinsic thermal conductivity of the MgSiN2 phase and the grain boundary phase, respectively. We can rewrite the above formula as:
κ -V κ κ = 2 2 2 κ (3) 1 κ κ exp 2 -V2 exp
-1 -1 V2 and κ2 was estimated to equal about 2 vol. % (≡ 1 wt. % O) and 2 W m K , -1 -1 -1 -1 respectively. For κexp = 25 W m K [10] we obtain κ1 = 33 W m K . From this it can be concluded that, even when secondary phases are present at the grain boundaries, the influence on the thermal conductivity is limited due to the small amount of secondary phase present.
86 Preparation, characterisation and properties of MgSiN2 ceramics
4.2. Grain size
For grain sizes in the range of the phonon mean free path, phonon-grain boundary scattering hampers effectively the heat transport. For the phonon mean free path, l [m], applies [12, 54]: 3κ l = exp (4) ρ vcV
-1 -1 -1 where κexp [W m K ] is the experimental thermal conductivity, v [m s ] the -1 -1 phonon group velocity, cV [J kg K ] the specific heat at constant volume and ρ [kg m-3] the density. The specific heat at constant volume can be approximated by the specific heat at constant pressure cp [55] and the group velocity was estimated to equal the sound velocity vs. Taking the sound velocity and the specific heat from 3 -1 -1 -1 our previous work (vs = 6.65 10 m s [56] and cp = 767 J kg K [30]), the resulting phonon mean free path at 300 K equalled about 4 - 5 nm. This is substantially below the observed grain size of the MgSiN2 ceramic samples suggesting that phonon-grain boundary scattering does not hamper the heat transport.
4.3. Defects
Assuming that the defect chemistry of MgSiN2 analogous to that of AlN [19, 57] incorporation of oxygen into the MgSiN2 lattice results in the formation of vacancies on the metal sites according to:
• '' '''' 2 MgSiO3 º 2 MgMg + 2 SiSi + 6 ON + VMg + VSi These vacancies are very effective in scattering the phonons and for the thermal conductivity then applies [19, 58, 59]:
2 2 2/3 δ m -m 1 1 (6π ) h 1 i i = + * ∑c (5) κ κ k 2θ 12 i m exp pp i i where κexp is the measured thermal conductivity, κpp the intrinsic thermal conductivity due to phonon-phonon scattering, h Planck’s constant (6.626 10-34
87 Chapter 3.
J s), δ 3 [m3] the average volume occupied by one atom, k the Boltzmann’s constant
-23 -1 (1.381 10 J K ), θ [K] the Debye temperature, ci [-] the site fraction of the isotope or foreign atom with mass mi at the ith lattice site, mi [g] the mass of impurity on the ith lattice site and mi [g] the average mass of the atoms at the ith lattice site. The average mass at the nitrogen ( M N ), magnesium ( M Mg ) and silicon
( M Si ) sites in the MgSiN2 lattice are taken constant and equal the atomic masses so, the scattering term, ∆WI, for MgSiN2 is given by:
π2 2/3 δ ∆ (6 ) h 1 ON W I = 2 * * k θ 12 ON + NN
2 2 2 (6) 1 1 M - M M - M M V Mg - M Mg O N + 6 V Si Si + 6 1 1 M 2 M 2 M N Si Mg
Introducing the input parameters for ∆WI and the Debye temperature of MgSiN2
(θ ≈ 830 K [8, 30]) we can write for the thermal conductivity of MgSiN2 with oxygen dissolved into the lattice: 1 1 = + 0.0202 [wt. % O] (7) κ exp κ pp
-1 -1 Taking for κexp 25 W m K and for [wt. % O] the maximum solubility of oxygen in the MgSiN2 lattice of 0.6 wt. % [17], an intrinsic thermal conductivity of 36 W m-1 K-1 results which indicates that a substantial improvement of the thermal conductivity by minimisation of phonon-defect scattering can be excluded.
4.4. Maximum influence of secondary phases, grain size and defects
From the above theoretical considerations it can be concluded that the maximum -1 -1 achievable thermal conductivity of MgSiN2 at 300 K will not exceed 35 W m K . These considerations confirm the experimental conclusion that the thermal conductivity can not be significantly improved. However, it should be noted that the two-phase serial and defect scattering formula are sensitive for the values used for V2 and [wt. % O], respectively.
88 Preparation, characterisation and properties of MgSiN2 ceramics
5. Conclusions
Dense MgSiN2 ceramic samples were prepared using the hot uni-axial pressing technique. The influence of additives on the sintering behaviour was very limited. The applied hot-pressing pressure appeared to be more important than the used sintering temperature indicating that liquid phase sintering was not the most important densification process. Oxygen poor samples (< 1.0 wt.%) could be prepared by using pure starting materials and/or using Mg3N2 as an additive. The grain size of the MgSiN2 grains increased with increasing hot-pressing temperature and time. The ceramic samples have a good oxidation resistance, a fairly high hardness and a low thermal expansion coefficient as compared to other ceramics. The thermal conductivity of the samples is not determined by the grain size or the presence of (inter-granular) secondary phases. From this it was concluded that phonon scattering processes within the MgSiN2 grains determine the thermal conductivity. It was concluded that the limiting factor is intrinsic phonon-phonon scattering resulting in a maximum achievable thermal conductivity of MgSiN2 ceramics at 300 K of 20 - 25 W m-1 K-1. From some simple theoretical considerations it was shown that the maximum achievable value does not exceed about 35 W m-1 K-1. This value is lower than the first reported theoretical estimate of 75 W m-1 K-1 with an approximate accuracy of about 30 %.
References
1. A. Roosen, Modern Substrate Concepts for the Microelectronic Industry, Electroceramics IV 2, Aachen (Germany), September 1994, edited by R. Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann (Augustinus Buchhandlung, 1994) 1089. 2. W.A. Groen, M.J. Kraan, and G. de With, Preparation, Microstructure and
Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413.
89 Chapter 3.
3. W.A. Groen, M.J. Kraan, and G. de With, New Ternary Nitride Ceramics:
CaSiN2, J. Mater. Sci. 29 (1994) 3166. 4. W.A. Groen, P.F. van Hal, M.J. Kraan, N. Sweegers and G. de With,
Preparation and Properties of ALCON (Al28C6O21N6) ceramics, J. Mater. Sci. 30 (1995) 4775. 5. G.A. Slack, Nonmetallic Crystals with High Thermal Conductivity, J. Phys. Chem. Solids 34 (1973) 321. 6. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent
Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non- Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh, 1994) 239. 7. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1979) 1. 8. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)-Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of Properties and Performances by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405. 9. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modelling of the Thermal th Diffusivity / Conductivity of MgSiN2 Ceramics, ITCC 24 and ITES 12, 24 International Thermal Conductivity Conference and 12th International Thermal Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 1997, edited by P.S. Gaal and D.E. Apostolescu (Technomics Publishing Co., Inc., Lancaster, Pennsylvania, 1999) 3.
10. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics 2, Basic Science - Developments in Processing of Advanced Ceramics - Part II, Faenza, Italy, October 1995,
90 Preparation, characterisation and properties of MgSiN2 ceramics
edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 289. 11. P.G. Klemens, The Thermal Conductivity of Dielectric Solids at Low Temperatures, Proc. Roy. Soc. London A208 (1951) 108. 12. F.R. Charvat and W.D. Kingery, Thermal Conductivity: XIII, Effect of Microstructure on Conductivity of Single-Phase Ceramics, J. Am. Ceram. Soc. 40 (1957) 306. 13. K. Watari, K. Ishizaki and F. Tsuchiya, Phonon Scattering and Thermal Conduction Mechanisms of Sintered Aluminum Nitride Ceramics, J. Mater. Sci. 28 (1993) 3709. 14. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan, Hot-
pressing of MgSiN2 ceramics, J. Mater. Sci. Lett. 13 (1994) 1314. 15. H.T. Hintzen, R.J. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Desification of MgSiN2, Ceram. Trans. 51, Int. Conf. Cer. Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995) 585. 16. H. Uchida, K. Itatani, M. Aizawa, F.S. Howell and A. Kishioka, Synthesis of Magnesium Silicon Nitride by the Nitridation of Powders in the Magnesium- Silicon System, J. Ceram. Soc. Japan 105 (1997) 934. 17. Chapter 2; R.J. Bruls, H.T. Hintzen and R. Metselaar, Preparation and
Characterisation of MgSiN2 Powders, J. Mater. Sci. 34 (1999) 4519. 18. M.P. Boron, G.A. Slack, and J.W. Szymazek, Thermal Conductivity of Commercial Aluminum Nitride, Bull. Am. Ceram. Soc. 51 (1972) 852. 19. G.A. Slack, R.A. Tanzilli, R.O. Pohl and J.W. Vandersande, The Intrinsic Thermal Conductivity of AlN, J. Phys. Chem. Solids 48 (1987) 641. 20. Y. Suwa, S. Naka and T. Noda, Preparation and Properties of Yttrium Magnesium Silicate with Apatite Structure, Mat. Res. Bull. 3 (1968) 139. 21. P.C. Martinengo, A. Giachello, P. Popper, A. Buri, F. Branda, Devitrification Phenomena of a Pressureless Sintered Silicon Nitride, Proc. of International
91 Chapter 3.
Symposium on Factors in Densification and Sintering of Oxide and Non-oxide Ceramics, Hakone, Japan, October 3 - 5, 1978, edited by S. Somiya and S. Saito (Gakujutsu Bunken Fukyu-Kai, Tokyo, 1979) 516. 22. S. Kuang, M.J. Hoffmann, H.L. Lukas and G. Petzow, Experimental Study
and Thermodynamic Calculations of the MgO-Y2O3-SiO2 System, Key Engineering Materials 89 - 91, Proceedings of the International Conference on Silicon Nitride-Based Ceramics, Silicon Nitride 93, Stuttgart, Germany, October 1993, edited by M.J. Hoffmann, P.F. Becker and G. Petzow (Trans Tech Publications, Switzerland, 1994) 399. 23. E.M. Levin, C.R. Robbins and F.H. McMurdie, Phase Diagrams for Ceramists, (comp. at the National Bureau of Standards and Technology -7th. comp.- Columbus, American Ceramic Society, 1964) p. 210. 24. Z.K. Huang, S.D. Nunn, I. Peterson and T.Y. Tien, Formation of N-phase and
Phase Relationships in MgO-Si2N2O-Al2O3 System, J. Am. Ceram. Soc. 77 (1994) 3251. 25. I.J. Davies, H. Uchida, M. Aizawa and K. Itatani, Physical and Mechanical Properties of Sintered Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 40. 26. Refcel, Calculation of cell constants and calculation of all possible lines in a powder diagram by H.M. Rietveld, October 1972.
27. J. David, Y. Laurent and J. Lang, Structure de MgSiN2 et MgGeN2, Bull. Soc. fr. Minéral. Cristallogr. 93 (1970) 153. 28. K. Heritage, C. Frisby and A. Wolfenden, Impulse excitation technique for dynamic flexural measurements at moderate temperatures, Rev. Sci. Instr. 59 (1988) 973. 29. J.C. Glandus, Rupture fragile et résistance aux chocs thermiques de céramiques a usage mécaniques, Ph.D. Thesis, University of Limoges, France, 1981.
92 Preparation, characterisation and properties of MgSiN2 ceramics
30. Chapter 5; R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,
Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998) 7871.
31. J. David, Étude sur Mg3N2 et quelques-unes de ses combinaisons, Rev. Chim. Miner. 9 (1972) 717. 32. M. Wintenberger, F. Tcheou, J. David und J. Lang, Verfeinerung der Struktur
des Nitrids MgSiN2 - eine Neutronenbeugungsuntersuchung, Z. Naturforsch. 35b (1980) 604. 33. R.D. Shannon, Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides, Acta. Cryst. A32 (1976) 751. 34. D.R. Messier, F.L. Riley and R.J. Brook, The α/β Silicon Nitride Phase Transformation, J. Mater. Sci. 13 (1978) 1199. 35. O.N. Carlson, The N-Si (Nitrogen-Silicon) System, Bull. Alloy Phase Diagrams 11 (1990) 569. 36. S.F. Kuang, Z.-K. Huang, W.-Y.Sun, T.-S. Yen, Phase Relationships in the
System MgO-Si3N4-AlN, J. Mater. Sci. Lett. 9 (1990) 69. 37. R. Morrell, Handbook of Properties of Technical & Engineering Ceramics (Part 2, Data Reviews) Her Majesty's Stationary Office, London, UK, 1987, pp. 37 - 57. 38. H.X. Willems, P.F. van Hal, G. de With and R. Metselaar, Mechanical properties of γ -aluminium oxynitride, J. Mater. Sci. 28 (1993) 6185. 39. Y. Kurokawa, K. Utsumi, H. Takamizawa, T. Kamata and S. Noguchi, AlN Substrates with High Thermal Conductivity, IEEE Trans. Compon., Hybrids, Manuf. Technol., CHMT-8 (1985) 247 - 252. 40. G. Ziegler, J. Heinrich and G. Wötting, Review Relationships between Processing, Microstructure and Properties of Dense and Reaction-Bonded Silicon Nitride, J. Mater. Sci. 22 (1987) 3041. 41. J. Piekarczyk, J. Lis and J. Bialoskórski, Elastic Properties, Hardness and Indentation Fracture Toughness of β-Sialons, Key Engineering Materials
93 Chapter 3.
89 - 91, Proceedings of the International Conference on Silicon Nitride-Based Ceramics, Silicon Nitride 93, Stuttgart, Germany, October 1993, edited by M.J. Hoffmann, P.F. Becher and G. Petzow, (Trans Tech Publications, Switserland, 1994) 541. 42. W. Werdecker and F. Aldinger, Aluminum Nitride - An Alternative Ceramic Substrate for High Power Applications in Microcircuits, 34th Electronic Components Conference, May 1984, New Orleans, Louisiana, USA (Institute of Electrical and Electronics Engineers, Inc., 1984) 402. 43. P. Boch, J.C. Glandus, J. Jarrige, J.P. Lecompte and J. Mexmain, Sintering, Oxidation and Mechanical Properties of Hot Pressed Aluminium Nitride, Ceram. Int. 8 (1982) 34.
44. I. Tomeno, High Temperature Elastic Moduli of Si3N4 Ceramics, Jpn. J. Appl. Phys. 20 (1981) 1751. 45. P. Lefort, G. Ado and M. Billy, Comportement a l'oxydation de l'oxynitrure d'aluminium transparant, J. Physique 47 (1986) C1, 521. 46. E.W. Osborne and M.G. Norton, Oxidation of aluminium nitride, J. Mater. Sci. 33 (1998) 3859. 47. J. Persson, P.-O. Käll and M. Nygren, Parabolic - Non-Parabolic Oxidation
Kinetics of Si3N4, J. Eur. Ceram. Soc. 12 (1993) 177. 48. Y.S. Touloukian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, Thermophysical Properties of Matter 13, Thermal Expansion of Nonmetallic Solids, (IFI/Plenum, New York, USA, 1977). 49. H.X. Willems, Preparation and Properties of Translucent γ-Aluminium Oxynitride, Ph. D. Thesis, Eindhoven University of Technology, The Netherlands (1992). 50. K. Wang and R.R. Reeber, Thermal Expansion of GaN and AlN, Mat. Res. Soc. Symp. Proc. 482, Nitride Semiconductors, Boston, Massachusetts, USA, December 1 - 5 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S. Nakamura and S. Strite, (Materials Reseasrch Society, Warrendale, Pennsylvania, 1998) 863.
94 Preparation, characterisation and properties of MgSiN2 ceramics
51. C.M.B. Henderson and D. Taylor, Thermal Expansion of the Nitrides and Oxynitride of Silicon in Relation to their Structures, Trans. J. Br. Ceram. Soc. 74 (1975) 49. 52. Chapter 4; R.J. Bruls, H.T. Hintzen, R. Metselaar and C.-K. Loong,
Anisotropic thermal expansion of MgSiN2 from 10 to 300 K as measured by neutron diffraction, J. Phys. Chem. Solids 61 (2000) 1285. 53. D.W. Richerson, Modern Ceramic Engineering; Properties, Processing, and Use in Design, second edition (Marcel Dekker, Inc., 1992) pp. 135 - 146. 54. P. Debije, in: Vorträge über die kinetische Theorie der Materie und Elektrizität, edited by B.G. Teubner, Leipzig and Berlin (1914) p. 16. 55. See for example: R.A. Swalin, Thermodynamics of Solids, second edition, John Wiley & Sons, New York, (1972) pp. 31 - 32 and 81 - 84. 56. Chapter 6; R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The
Temperature Dependence of the Young's Modulus of MgSiN2, AlN and Si3N4, accepted for publication in J. Eur. Ceram. Soc. 57. A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and Kinetic Effects of Oxygen Removal on the Thermal Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 72 (1989) 2031. 58. V. Ambegoakar, Thermal Resistance due to Isotopes at High Temperature, Phys. Rev. 114 (1959) 488.
59. G.A. Slack, Thermal Conductivity of MgO, Al2O3, MgAl2O3, and Fe3O4 Crystals from 3 ° to 300 °K, Phys. Rev. 126 (1962) 427.
95 Chapter 3.
96 Chapter 4.
Anisotropic thermal expansion of MgSiN2
Abstract
The lattice parameters of orthorhombic MgSiN2 as a function of the temperature have been determined from time-of-flight neutron powder diffraction. The results indicate that MgSiN2, just like several other adamantine-type crystals, exhibits a relatively small thermal expansion coefficient at low temperatures. This is ascribed to a strongly bonded three-dimensional, relatively open, crystal structure which is characteristic for highly covalent bonded materials. The anisotropic linear thermal expansion behaviour could be qualitatively related to the characteristics of the crystal structure. The least dense packed crystallographic direction showed the smallest anisotropic linear expansion coefficient.
1. Introduction
MgSiN2 is a relatively new ceramic nitride material that belongs to a class of compounds with potentially interesting thermal, mechanical and luminescence properties [1 - 6]. Therefore the crystal structure, the physical and the chemical properties of MgSiN2 are of technological and scientific importance. The thermal expansion is one of these properties, both for technological reasons (thermal expansion mismatch between substrate and coating) and scientific reasons (evaluation of the Grüneisen parameter, which is an important ingredient for modelling the thermal conductivity [7]). This makes it necessary to study the thermal expansion of MgSiN2 over a wide range of temperatures. The thermal expansion of a material can be determined with dilatometry, or diffraction measurements using X-rays or neutrons as a function of temperature.
97 Chapter 4.
Dilatometry provides only information about the change in the dimensions or volume of the specimen with respect to the temperature, whereas X-ray and neutron diffraction provides direct information about the lattice parameters and atomic positions in the crystalline unit cell. When single crystals are not available, as in the case for MgSiN2, the anisotropic behaviour cannot be studied with dilatometry. However, the Rietveld refinement of powder diffraction data [8] is a powerful and efficient method for crystal structure determination. An important advantage of neutrons over X-rays forms the comparable yet different neutron coherent scattering amplitudes among the elements in the periodic table (e.g., 0.5375, 0.4149 and 0.936⋅10-12 cm for Mg, Si and N, respectively) thereby providing the contrast and sensitivity needed for resolving the positions of different atoms. Furthermore, neutron time-of-flight (TOF) technique permits the measurement of the entire powder pattern at a fixed detector angle with a constant ∆d/d resolution (where d is atomic d-spacing) and minimal systematic errors. For these reasons the method of temperature dependent neutron TOF powder diffraction was employed for the measurement of the thermal expansion coefficients of MgSiN2. Structural refinement based on previous X-ray diffraction measurements showed a wurtzite-like structure of MgSiN2 (space group Pna21, no. 33) [9, 10] with Z = 4. Within this orthorhombic structure the Mg and Si metal atoms and the two crystallographically different N atoms occupy the general position 4a (x, y, z; x , y , z+½; x+½, y +½, z; x +½, y+½, z+½). Both metal atoms (Mg and Si) are tetrahedrally co-ordinated by N (2× N(1) and 2× N(2)) and vice versa both N atoms are tetrahedrally co-ordinated by the metal atoms (2× Mg and 2× Si). A neutron diffraction study of MgSiN2 using fixed incident neutron wavelength was reported in the literature [11]. However, in that study only a room-temperature measurement was performed and given the limited number of reflections only the lattice parameters and atomic positions were refined. The atomic positions for MgSiN2 from these previous measurements [9, 10, 11] are listed in Table 4-1.
Recently, the preparation technique of MgSiN2 was improved significantly resulting in MgSiN2 powder [12, 13] and ceramic [14, 15] with a very low impurity
98 Anisotropic thermal expansion of MgSiN2
content, which makes the refinement of the neutron diffraction data easier because no secondary phases have to be taken into account. Furthermore, the neutron diffraction and Rietveld refinement techniques have been improved substantially since the last neutron diffraction study of MgSiN2 [11].
Table 4-1: Atomic positions x, y, z for MgSiN2 as given in the literature [9 - 11] (*: positions obtained after translation (0, ¼, ½) and symmetry operation x +½, y+½, z+½).
XRD [9] XRD [10]* xyz xyz Mg 0.083 0.600 0.000 0.09 0.63 0.00 Si 0.070 0.130 0.000 0.07 0.12 0.00 N(1) 0.065 0.125 0.385 0.06 0.14 0.36 N(2) 0.083 0.650 0.400 0.11 0.56 0.45 Constant wavelength neutron diffraction [11] x y z Mg 0.076(2) 0.625(5) -0.005 Si 0.072(2) 0.131(5) 0.0 N(1) 0.0490(15) 0.095(2) 0.356(3) N(2) 0.110(1) 0.652(4) 0.414(2)
In this chapter the Rietveld refinements of temperature dependent TOF neutron diffraction data of a nearly single phase MgSiN2 powder is reported. The calculated lattice parameters as a function of the temperature were used to calculate the thermal expansion coefficients along the three crystallographic axes.
2. Experimental procedure
For the neutron diffraction measurements a nearly single phase MgSiN2 powder (0.1 wt.% O and 34.2 wt.% N (theoretical value 34.8 wt.%)) was used. The
MgSiN2 powder was prepared starting from a Si3N4 (SKW Trostberg)/Mg3N2 (Alfa) powder mixture, which was fired at a maximum temperature of 1500 °C in an
99 Chapter 4.
N2/H2 atmosphere. A more detailed description of the powder preparation method is given elsewhere [13]. The powder was first characterised with X-ray diffraction (Philips PW 1050/25, Cu-Kα, 20 - 50 ° 2θ-scan with step size 0.01 ° and scan rate 0.01 °/ 18 s) which identified the presence of some very small traces of MgO
(Periclase, JCPDS card 4-829, I/I0 < 0.5 %) as a secondary phase. The neutron diffraction experiments were performed on the Special Environment Powder Diffractometer (SEPD) of the Intense Pulsed Neutron Source
(IPNS) at the Argonne National Laboratories (ANL, U.S.A.). The MgSiN2 powder sample (3.87 g) was put into a cylindrical vanadium sample holder under a helium atmosphere that was used as a heat-exchange gas for thermal conduction. The sample was cooled by a closed-cycle helium refrigerator and maintained at a selected temperature within ∼ 0.5 K. Neutron TOF data were collected at 10, 20, 30, 40, 50, 75, 100, 150, 200, 250 and 300 K for about 1 h at each temperature in the d-value range of 0.268 - 4.02 Å (2 - 30 msec). Only the data measured in back- scattering configuration (of a mean detector angle 148 °) were refined because this detector bank provides the best spatial resolution, ∆d/d ≅ 0.34%. The neutron diffraction data were analysed in the d-spacing range of 0.5 - 3.88 Å using the General Structure Analysis System (GSAS) [16] computer code which is based on the Rietveld method. The experimental data below 0.5 Å were not included because of the high background and the data above 3.88 Å contained one reflection at about 3.96 Å which was not taken into account because of the lack of data points at high d-spacing for modelling the background. The initial input parameters for the refinement were taken from the literature [11]. The data were refined for 28 variables which include an overall scale factor, a background function and a neutron pulse-shape profile (the convolution of two “back-to-back” exponentials with a Gaussian), the lattice parameters, the atomic positions, an isotropic temperature factor, and a sample absorption coefficient for MgSiN2. Furthermore, the quality of the refinement was checked for disorder on the Mg and Si sites, strain broadening and preferential orientation. The anisotropic linear thermal expansion coefficients along the three crystallographic axes, αa, αb and αc, from 10 to 300 K were evaluated by
100 Anisotropic thermal expansion of MgSiN2
determination of the lattice parameters as a function of the absolute temperature using that αa = (da(T)/dT )/a(T ) [17], etc. The isotropic linear thermal expansion coefficient, α, was evaluated using α = ⅓ (αa + αb + αc) = ⅓ (dV(T )/dT )/V(T ). The lattice parameters and unit cell volume as a function of the absolute temperature were evaluated by a polynomial fit of the obtained data. The lattice parameters a(T), b(T ) and c(T ), and unit cell volume, V(T ) were described with a third degree polynomial: X(T ) = A + CT 2 + DT 3, where X(T ) denotes either a(T ), b(T ), c(T ) or V(T ), using the data between 10 and 300 K. A linear term (BT ) was not included for describing the lattice parameters and cell volume as a function of the absolute temperature since the thermal expansion coefficient at 0 K equals 0 K-1 [17, 18]. Statistical F-testing [19] with a 95 % confidence interval of the variance ratios showed that introduction of a linear or higher order terms to the polynomial fit did not improve the fit of the data points. The thermal expansion was evaluated by differentiation with respect to the temperature.
3. Results and discussion
3.1. Neutron diffraction data refinement
Refinement of the neutron data confirmed the space group Pna21 for MgSiN2 and the atomic positions of Mg and Si in the MgSiN2 lattice. Moreover, the results showed that introduction of (anisotropic) strain broadening, preferential orientation, or introduction of anisotropic temperature factors did not significantly improve the refinement statistics.
Fig. 4-1 shows a typical observed and calculated powder pattern of MgSiN2, and in Table 4-2 the lattice parameters and final refinement statistics at 10, 50, 250 and 300 K are presented. The observed statistics for all measurements are good. The ‘goodness of fit’ in terms of χ 2, was about 1.6 - 1.7 for all temperatures except for the measurement at 300 K for which the value was 1.98. The residual weighted
R-factor, wRp, equalled about 0.075 for all measurements except for the measurement at 300 K for which the value was 0.062. Although the statistics of the
101 Chapter 4.
300 K measurement deviate somewhat from the other measurements, it can be concluded that the observed statistics are, as expected, about the same for all measurements.
2 Table 4-2: Lattice parameters a, b, c and V and refinement statistics Rp, wRp and χ for MgSiN2 at 10, 50, 250 and 300 K.
Temperature 10 K 50 K 250 K 300 K Lattice parameters a [Å] 5.27078(5) 5.27068(5) 5.27171(5) 5.27249(4) b [Å] 6.46916(7) 6.46932(7) 6.47175(7) 6.47334(6) c [Å] 4.98401(5) 4.98419(5) 4.98542(5) 4.98622(4) V [Å3] 169.9425(28) 169.9495(28) 170.0883(28) 170.1827(24) Data points 5000 5000 5000 5000 Reflections 833 833 833 834
Rp 0.0462 0.0463 0.0456 0.0312 wRp 0.0752 0.0763 0.0745 0.0621 χ 2 1.618 1.667 1.639 1.977
MgSiN2 on SEPD at 300 K Hist 1 Bank 1, 2-Theta 144.8, L-S cycle 37 Obsd. and Diff. Profiles 4.0 2.0 .0 Norm. count/musec. Norm. count/musec. .5 1.0 1.5 2.0 2.5 3.0 D-spacing, A
Fig. 4-1: A typical TOF neutron diffraction profile for MgSiN2 at 300 K (+), fit (solid line) with tick marks indicated at the Bragg positions and the difference plot (at the bottom on the same scale).
102 Anisotropic thermal expansion of MgSiN2
The lattice parameters and cell volume as a function of the absolute temperature are presented in Fig. 4-2 and Fig. 4-3, respectively. The indicated error bars equal 3 times the standard deviation obtained from the structural refinement.
0.0070 (a - 5.2700) Å 0.0060 (b - 6.4670) Å 0.0050 (c - 4.9825) Å
[Å] 0.0040 c , b
, 0.0030 a
0.0020
0.0010
0.0000 0 50 100 150 200 250 300 350 T [K]
Fig. 4-2: The lattice parameters (a, b and c) of MgSiN2 as a function of the absolute temperature (T ) as determined by the Rietveld structure refinement using neutron diffraction data.
170.20
170.15
170.10
[ų] 170.05 V
170.00
169.95
169.90 0 50 100 150 200 250 300 350 T [K]
Fig. 4-3: The unit cell volume (V ) of MgSiN2 as a function of absolute temperature (T ) as determined by the Rietveld structure refinement using neutron diffraction data.
103 Chapter 4.
Table 4-3: Atomic positions x, y, z and isotropic temperature factor Uiso for MgSiN2 at 10 and 300 K.
MgSiN2 10 K
x y z 100 Uiso Mg 0.08448(34) 0.62255(30) -0.0134(5) 0.298(33) Si 0.0693(5) 0.1249(4) 0 0.14(4) N(1) 0.04855(17) 0.09562(15) 0.3472(4) 0.196(20) N(2) 0.10859(18) 0.65527(14) 0.4102(4) 0.234(20)
MgSiN2 300 K
x y z 100 Uiso Mg 0.08475(31) 0.62263(27) -0.0135(4) 0.628(32) Si 0.0687(4) 0.12535(34) 0 0.302(35) N(1) 0.04863(15) 0.09557(13) 0.34822(34) 0.358(19) N(2) 0.10873(16) 0.65519(13) 0.41130(37) 0.384(19)
In Table 4-3 the atomic positions and isotropic temperature factors are presented for Mg, Si, N(1) and N(2) at 10 and 300 K. To the extent of the accuracy of the refinement, the atomic positions of Mg, Si, N(1) and N(2) do not vary with temperature. The atomic positions at 300 K are in reasonable agreement with the values reported previously in the literature (Table 4-1).
In the MgSiN2 lattice the tetrahedral co-ordination of Mg, Si, N(1) and N(2) is distorted. Furthermore each cation (anion) is surrounded by 12 cations (anions) among which 4 occupy the same crystallographic site and 8 do not. For example, the nearest cation neighbours for Mg are 4 Mg and 8 Si atoms, for N(1) the nearest anion neighbours are 4 N(1) and 8 N(2), etc. The important bond lengths and bond angles were calculated at all temperatures and the results for 10 and 300 K are presented in Table 4-4 and Table 4-5, respectively. As expected, the average Mg–N and Si–N bond length (N = N(1), N(2)) increases with increasing temperature. The relative change for the average Si–N bond length is smaller than that for the Mg–N bond length. The relative variation between the 4 Si–N and 4 Mg–N bond lengths is smaller than that between the 4 N(1)–Me (Me = Mg, Si) and 4 N(2)–Me bonds. The tetrahedral co-ordination of the N atoms is heavily distorted due to the presence of two types of N–Me bonds.
104 Anisotropic thermal expansion of MgSiN2
Table 4-4: Si–N and Mg–N bond lengths, N–Si–N, N–Mg–N, Me–N(1)–Me and Me–N(2)–Me
(Me = Mg, Si) angle, and N–N and Me–Me distance for MgSiN2 at 10 K.
bond length [Å] bond length [Å] Mg–N(1)iv 2.0626 (1×) Si–N(1)v 1.7323 (1×) Mg–N(1)ii 2.0733 (1×) Si–N(1)ii 1.7442 (1×) Mg–N(2)I 2.1002 (1×) Si–N(2)vi 1.7612 (1×) Mg–N(2)iv 2.1256 (1×) Si–N(2)vii 1.7667 (1×) N–Me–N bond angle N–N dis. Me–N–Me bond angle Me–Me dis. [°] [Å] [°] [Å] N(1)ii–Mg–N(1)iv 106.173 3.3068 (2×) Mgx–N(1)–Siviii 104.935 3.0161 (1×) N(1)ii–Mg–N(2)i 105.396 3.3402 (1×) Mgix–N(1)–Siv 104.962 3.0345 (1×) N(1)iv–Mg–N(2)iv 108.726 3.3598 (1×) Mgx–N(1)–Siv 105.439 3.0351 (1×) N(1)ii–Mg–N(2)iv 121.865 3.3832 (1×) Siv–N(1)–Siviii 123.234 3.0586 (2×) N(2)i–Mg–N(2)iv 107.114 3.3995 (2×) Mgix–N(1)–Mgx 97.469 3.1088 (2×) N(1)iv–Mg–N(2)i 106.673 3.6478 (1×) Mgix–N(1)–Siviii 117.103 3.2515 (1×) N(1)v–Si–N(1)ii 108.919 2.8289 (2×) Mgix–N(2)–Mgv 93.778 3.0850 (2×) N(1)v–Si–N(2)vi 108.885 2.8332 (1×) Siix–N(2)–Sixi 122.480 3.0927 (2×) N(1)ii–Si–N(2)vi 108.953 2.8479 (1×) Mgv–N(2)–Siix 107.517 3.1422 (1×) N(1)v–Si–N(2)vii 107.844 2.8563 (1×) Mgv–N(2)–Sixi 107.337 3.1429 (1×) N(1)ii–Si–N(2)vii 111.244 2.8834 (1×) Mgix–N(2)–Sixi 108.938 3.1528 (1×) N(2)vi–Si–N(2)vii 110.942 2.9065 (2×) Mgix–N(2)–Siix 112.766 3.2211 (1×) av. bond length [Å] average angle [°] av. distance [Å] Mg–N 2.090 N–Mg–N 109.33 N(1)–N 3.129 Si–N 1.751 N–Si–N 109.46 N(2)–N 3.147 N(1)–Me 1.903 Me–N(1)–Me 108.86 Mg–Me 3.115 N(2)–Me 1.938 Me–N(2)–Me 108.80 Si–Me 3.108 Equivalent positions: (i) x, y, z-1 (v) x+½, y+½, z (ix) -x, -y+1, z+½ (ii) -x, -y, z+½-1 (vi) -x+½, y+½-1, z+½-1 (x) -x+½, y+½-1, z+½ (iii) x+½, -y+½, z-1 (vii) -x, -y+1, z+½-1 (xi) -x+½, y+½, z+½ (iv) -x+½, y+½, z+½-1 (viii) -x, -y, z+½
The average Mg-N bond length at 300 K (2.0916 Å) is somewhat larger than the previously reported value of 2.086 Å [11] but still substantially shorter than the
Mg–N distance (2.142 Å) in Mg3N2 [20]. Likewise, the average Si–N bond length at 300 K (1.7520 Å) is somewhat smaller than the previously reported value (1.760 Å) [11] but still substantially larger than the Si–N bond length observed in
β-Si3N4 (1.732 Å [21]). The average N–Mg–N angle (109.32 °) is slightly smaller
105 Chapter 4.
Table 4-5: Si–N and Mg–N bond lengths, N–Si–N, N–Mg–N, Me–N(1)–Me and Me–N(2)–Me
(Me = Mg, Si) angle, and N–N and Me–Me distance for MgSiN2 at 300 K.
bond length [Å] bond length [Å] Mg–N(1)iv 2.0598 (1×) Si–N(1)v 1.7322 (1×) Mg–N(1)ii 2.0731 (1×) Si–N(1)ii 1.7502 (1×) Mg–N(2)I 2.1013 (1×) Si–N(2)vi 1.7576 (1×) Mg–N(2)iv 2.1324 (1×) Si–N(2)vii 1.7679 (1×) N–Me–N bond angle N–N dis. Me–N–Me bond Angle Me–Me dis. [°] [Å] [°] [Å] N(1)ii–Mg–N(1)iv 106.371 3.3087 (2×) Mgx–N(1)–Siviii 105.017 3.0153 (1×) N(1)ii–Mg–N(2)i 105.279 3.3428 (1×) Mgix–N(1)–Siv 104.688 3.0332 (1×) N(1)iv–Mg–N(2)iv 108.849 3.3609 (1×) Mgx–N(1)–Siv 105.425 3.0370 (1×) N(1)ii–Mg–N(2)iv 121.943 3.3845 (1×) Siv–N(1)–Siviii 123.091 3.0617 (2×) N(2)i–Mg–N(2)iv 106.896 3.4016 (2×) Mgix–N(1)–Mgx 97.596 3.1095 (2×) N(1)iv–Mg–N(2)i 106.575 3.6500 (1×) Mgix–N(1)–Siviii 117.388 3.2561 (1×) N(1)v–Si–N(1)ii 108.718 2.8301 (2×) Mgix–N(2)–Mgv 93.662 3.0879 (2×) N(1)v–Si–N(2)vi 108.566 2.8351 (1×) Siix–N(2)–Sixi 122.508 3.0910 (2×) N(1)ii–Si–N(2)vi 108.933 2.8484 (1×) Mgv–N(2)–Siix 107.356 3.1420 (1×) N(1)v–Si–N(2)vii 107.848 2.8564 (1×) Mgv–N(2)–Sixi 107.142 3.1456 (1×) N(1)ii–Si–N(2)vii 111.549 2.8855 (1×) Mgix–N(2)–Sixi 109.200 3.1598 (1×) N(2)vi–Si–N(2)vii 111.148 2.9080 (2×) Mgix–N(2)–Siix 112.860 3.2209 (1×) av. Bond length [Å] Average angle [°] Av. [Å] Distance Mg–N 2.092 N–Mg–N 109.32 N(1)–N 3.120 Si–N 1.752 N–Si–N 109.46 N(2)–N 3.149 N(1)–Me 1.904 Me–N(1)–Me 108.87 Mg–Me 3.117 N(2)–Me 1.940 Me–N(2)–Me 108.79 Si–Me 3.110 Equivalent positions:
(i) x, y, z-1 (v) x+½, y+½, z (ix) -x, -y+1, z+½ (ii) -x, -y, z+½-1 (vi) -x+½, y+½-1, z+½-1 (x) -x+½, y+½-1, z+½ (iii) x+½, -y+½, z-1 (vii) -x, -y+1, z+½-1 (xi) -x+½, y+½, z+½ (iv) -x+½, y+½, z+½-1 (viii) -x, -y, z+½ than the average observed N–Si–N angle (109.46 °). Although these average bond angles are close to that for an ideal tetrahedron (109.47 °), the individual bond angles, especially for the N–Mg–N angles deviate considerably from the ideal value. From these results it can be concluded that the Mg–N4 tetrahedra are more distorted than the Si–N4 tetrahedra. Similarly, individual Me–N–Me angles deviate substantially from the mean angles (108.86 ° and 108.80 °) as well as from the
106 Anisotropic thermal expansion of MgSiN2
ideal value of 109.47 °. The N–N distances vary from 2.83 to 3.65 Å whereas the Me–Me (Me = Mg, Si) distances vary only from 3.02 to 3.26 Å. This also indicates that the anion sublattice is distorted whereas the cation sublattice is still quite regular. The N–(Si–)N and N–(Mg–)N distances in MgSiN2 (2.83 - 2.91 Å and 3.31 - 3.65 Å, respectively) are comparable with the N–(Me–)N distances observed in Si3N4 (2.77 - 2.90 Å [21]) and in Mg3N2 (3.31 - 3.75 Å [20]). The same applies for the Me-Me distances (3.06 - 3.09 Å for Si – Si, 3.09 - 3.11 Å for Mg – Mg and
3.02 - 3.26 Å for Si – Mg in MgSiN2) that vary between 3.00 and 3.05 Å [21] in
Si3N4 and 2.72 and 3.29 Å [20] in Mg3N2.
In a recent study of the crystal structures of Mg3N2 and Zn3N2 [20] the Brese-O’Keeffe nitride bond valence parameters [22] for Mg–N were discussed. In this study it is noted that it would be desirable to have a database of well-refined nitride structures so that bond valence parameters could be directly determined. For
MgSiN2 the bond valence parameters for Mg–N and Si–N were calculated by using the obtained bond lengths at 300 K. By varying the bond valence parameters for Mg–N and Si–N the difference between the calculated and expected bond valence sums of Mg, Si, N(1) and N(2) was minimised using a least-squares method. For the Mg–N and Si–N bonds a bond valence parameter of 1.833 and 1.752 respectively was obtained. Both values are in reasonable agreement with the bond valence parameters proposed by Brese and O’Keeffe for Mg–N (1.85) and Si–N (1.77) [22].
3.2. Thermal expansion
Since the experimental diffraction data were collected under the identical experimental configuration (other than the temperature change) without interruption, and the refinements were performed in a consistent manner, it is expected that the systematic errors, experimental or computational, to have very small, if any, effects on the calculation of the thermal expansion coefficients. This implies that the error in the observed dependence of the lattice parameters and cell
107 Chapter 4.
volume as a function of the absolute temperature (see Table 4-6) is very small and equals about the accuracy of the structure refinement.
Table 4-6: The lattice parameters a, b and c and unit cell volume V of
MgSiN2 as a function of the absolute temperature T between 0 and 300 K.
T a-axis b-axis c-axis V [K] [Å] [Å] [Å] [Å3] 10 5.27078(5) 6.46916(7) 4.98401(5) 169.9425(28) 15 5.27080(5) 6.46917(7) 4.98406(5) 169.9448(28) 20 5.27064(5) 6.46914(7) 4.98415(5) 169.9422(28) 25 5.27075(5) 6.46919(7) 4.98405(5) 169.9435(29) 30 5.27072(5) 6.46924(7) 4.98408(5) 169.9449(29) 35 5.27071(5) 6.46924(7) 4.98409(5) 169.9448(28) 40 5.27079(5) 6.46917(7) 4.98404(5) 169.9439(28) 45 5.27073(5) 6.46926(7) 4.98404(5) 169.9443(28) 50 5.27068(5) 6.46932(7) 4.98419(5) 169.9495(28) 75 5.27082(5) 6.46927(7) 4.98410(5) 169.9498(28) 100 5.27083(5) 6.46953(7) 4.98424(5) 169.9617(28) 150 5.27083(5) 6.46992(7) 4.98441(5) 169.9776(28) 200 5.27118(5) 6.47065(7) 4.98473(5) 170.0189(28) 250 5.27171(5) 6.47175(7) 4.98542(5) 170.0883(28) 300 5.27249(4) 6.47334(6) 4.98622(4) 170.1827(24)
The coefficients of the third degree polynomial to describe the lattice parameters and unit cell volume for all data points are presented in Table 4-7. These coefficients were used to calculate the thermal expansion coefficients from
10 to 300 K for MgSiN2. The lattice parameters and cell volume at 293 K evaluated from the polynomial fit are 5.27237 Å, 6.47308 Å, 4.98609 Å and 170.167 Å3 for a(293 K), b(293 K), c(293 K) and V(293 K), respectively. These are in good agreement with those obtained previously by XRD measurements at 293 K a(293 K) = 5.2698 Å, b(293 K) = 6.4734 Å, c(293 K) = 4.9843 Å and V(293 K) = 170.04 Å3 [13]. The isotropic and anisotropic linear thermal expansion
108 Anisotropic thermal expansion of MgSiN2
coefficients are presented in Table 4-8. The accuracy of the presented thermal expansion data was estimated to be about 0.3 10-6 K-1.
Table 4-7: The coefficients used in the polynomial A + CT 2 + DT 3 for describing the lattice
parameters a, b and c and unit cell volume V of MgSiN2 as a function of the absolute temperature T between 10 and 300 K.
coefficient a-axis b-axis c-axis V A 5.27075 6.46919 4.98407 169.9440 C -6.16924 10-09 1.87691 10-08 5.83402 10-09 4.91604 10-07 D 8.53434 10-11 9.07302 10-11 6.04839 10-11 7.20613 10-09
From Fig. 4-3 and Table 4-8 it can be concluded that the unit cell volume increases monotonically as a function of the absolute temperature. The data points at 50 and 100 K seem to be somewhat scattered but are well within the experimental uncertainty. Similar to the case of Si3N4 [23] the observed thermal expansion for MgSiN2 is quite small. The small thermal expansion coefficient in these materials was attributed to an almost symmetric potential well for atomic bonding [17]. It can be seen from Fig. 4-2 that the b and c-axis monotonically increase with increasing temperature, whereas the a-axis in the temperature range of 10 to 150 K remains about constant. A low or negative thermal expansion coefficient is often found for tetrahedrally bonded solids (adamantine type materials) below a reduced temperature T/θ = 0.2 [24, 25] (where θ is the Debye temperature). E.g., Si, Ge, GaAs, GaSb, InAs, InSb and AlN exhibit a negative thermal expansion coefficient at low temperatures [24, 26, 27]. The negative or small thermal expansion coefficient in these tetrahedrally bonded solids may be related to structural features such as [26, 28, 29]: i) a strongly covalently bonded three- dimensional polyhedron network which hinders changes in bond length; and ii) a relatively open structure which can absorb thermal energies via transverse modes with atomic displacements perpendicular to the bond directions. In general, the observed negative thermal expansion coefficient of these materials usually does not
109 Chapter 4.
exceed the value of 0.3 - 0.4 10-6 K-1 which is comparable to the accuracy of the presented thermal expansion data for MgSiN2. Thus MgSiN2 exhibits a very low, or possibly, negative thermal expansion coefficient for the a-axis in the temperature range of 10 to 150 K.
Table 4-8: The anisotropic thermal expansion coefficients αa, αb and αc, and
linear isotropic thermal expansion coefficient α (= ⅓(αa + αb +
αc)) of MgSiN2 as a function of the absolute temperature T between 0 and 300 K (*: The fit for α = ⅓(dV/dT )/V yields identical results.).
T αa αb αc ⅓(αa + αb + αc)* [K] [K-1] [K-1] [K-1] [K-1] 0 0 0 0 0 10 - 0.02 10-6 0.06 10-6 0.03 10-6 0.02 10-6 50 0.00 10-6 0.40 10-6 0.20 10-6 0.20 10-6 100 0.25 10-6 1.0 10-6 0.60 10-6 0.62 10-6 150 0.74 10-6 1.8 10-6 1.2 10-6 1.2 10-6 200 1.5 10-6 2.8 10-6 1.9 10-6 2.1 10-6 250 2.5 10-6 4.1 10-6 2.9 10-6 3.1 10-6 300 3.7 10-6 5.5 10-6 4.0 10-6 4.4 10-6
The difference in the thermal expansion behaviour along the a and b-axis can be understood in terms of the crystal structure of MgSiN2. Fig. 4-4 shows the crystal structure projected on the a-b plane. It can be seen that the MgSiN2 structure can be deduced from a hexagonal structure (with lattice parameters a' and c') in which a ≈ √3 a', b ≈ 2 a'and c = c' ≈ √8/3a' [10, 30]. Along the b-axis zigzag Mg–N–Si–N chains with bond angles of ∼ 120 ° are observed whereas along the a-axis Mg–N····Mg–N·· and Si–N····Si–N·· chains are observed. This gives rise to a low packing density along the a-axis. Since stretching force constants are in general higher than bending forces [29, 31], elongation along the b-axis will take place under a thermal load, resulting in some bending of the bonds along the b-axis which counteracts the thermal expansion along the a-axis.
110 Anisotropic thermal expansion of MgSiN2
Mg
Si
N(1)
N(2)
b
c a
Fig. 4-4: The crystal structure of MgSiN2 showing the a-b plane in which along the a-axis Mg–N····Mg–N·· and Si–N····Si–N·· chains are observed and along the b-axis Mg–N–Si–N– chains.
In the literature only the average thermal expansion coefficient of MgSiN2 between 293 and 873 K (5.8 10-6 K-1 [2]) and the thermal expansion coefficient at T = θ = 827 K (6.5 10-6 K-1 [32]) were reported. A check for self-consistency of the linear thermal expansion coefficient of MgSiN2 at 300 K can be made using the -1 Grüneisen relation γ = 3α Vm/βT CV. Here γ [-] is the Grüneisen parameter, α [K ] 3 -1 -1 the linear thermal expansion coefficient, Vm [m mol ] the molar volume, βT [Pa ] -1 -1 the isothermal compressibility and CV [J mol K ] the heat capacity at constant volume. If the temperature is sufficiently high, the Grüneisen parameter is nearly constant [25] and the thermal expansion can be evaluated from Vm, βT and CV. -5 3 -1 Using the (almost) temperature independent values for Vm of 2.57 10 m mol , βT -12 -1 -1 -1 of 6.84 10 Pa and the average CV of 83.61 J mol K in the temperature range of 293 and 873 K [33], the average Grüneisen parameter between 293 and 873 K
111 Chapter 4.
was calculated to be 0.78, in complete agreement with the γ of 0.78 at 827 K [32].
Therefore, using a constant Grüneisen parameter of 0.78 and CV at 300 K of 61.713 J mol-1 K-1 [33], the linear thermal expansion coefficient at 300 K was estimated as 4.3 10-6 K-1. This calculated value agrees very well with the observed value of 4.4 10-6 K-1.
4. Conclusions
Neutron TOF powder-diffraction data obtained between 10 and 300 K were used to determine the volumetric thermal expansion coefficient and the linear thermal expansion coefficient along the three different crystallographic directions of orthorhombic MgSiN2. Rietveld analyses confirmed the crystal structure of
MgSiN2, space group Pna21. The results indicate that in-situ neutron diffraction is very effective for accurate characterisation of the thermal expansion behaviour. A relatively small thermal expansion coefficient is observed for MgSiN2 due to the strongly bonded three-dimensional tetrahedra network and the open crystal structure as a consequence of the high degree of covalency. The variation of the Si–N bond lengths with temperature is smaller than that for the Mg–N bond lengths. The anisotropic thermal expansion behaviour can be explained qualitatively in terms of the characteristics of the crystal structure of MgSiN2. The neutron diffraction data reveal a very low, or possibly, negative linear thermal expansion coefficient along the a-axis at low temperatures which is the least dense packed crystallographic direction.
References
1. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent
Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non-Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh, 1994) 239.
112 Anisotropic thermal expansion of MgSiN2
2. W.A. Groen, M.J. Kraan, and G. de With, Preparation, Microstructure and
Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413. 3. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan, Hot-
pressing of MgSiN2 ceramics, J. Mater. Sci. Lett. 13 (1994) 1314. 4. G.K. Gaido, G.P. Dubrovskii and A.M. Zykov, Photoluminescence of
Europium-Activated MgSiN2, Inorganic Materials 10 (1974) 485 (translated from Izv. Akad. Nauk. SSSR, Neorg. Mater. 10 (1974) 564). 5. G.P. Dubrovskii, A.M. Zykov and B.V. Chernovets, Luminescence of Rare
Earth Activated MgSiN2, Inorganic Materials 17 (1981) 1059 (translated from Izv. Akad. Nauk. SSSR, Neorg. Mater. 17 (1981) 1421). 6. S. Lim, S.S. Lee and G.K. Chang, Photoluminescence Characterization of
MgxZn1-xSiN2:Tb for Thin Film Electroluminescent Devices Application, Wissenschaft und Technik, Inorganic and organic electroluminescence, Verlag, Berlin (1996), p. 363. 7. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1979) 1. 8. R.A. Young (ed), The Rietveld Method, Oxford University Press, (1996).
9. J. David, Y. Laurent and J. Lang, Structure de MgSiN2 et MgGeN2, Bull. Soc. Fr. Minéral. Cristallogr. 93 (1970) 153. 10. S. Wild, P. Grieveson and K.H. Jack, The Crystal Chemistry of New Metal- Silicon-Nitrogen Ceramic Phases, Spec. Ceram. 5 (1972) 289. 11. M. Wintenberger, F. Tcheou, J. David und J. Lang, Verfeinerung der Struktur
des Nitrids MgSiN2 - eine Neutronenbeugungsuntersuchung, Z. Naturforsch. 35b (1980) 604. 12. H. Uchida, K. Itatani, M. Aizawa, F.S. Howell and A. Kishioka, Synthesis of Magnesium Silicon Nitride by the Nitridation of Powders in the Magnesium- Silicon System, J. Ceram. Soc. Japan 105 (1997) 934. 13. Chapter 2; R.J. Bruls, H.T. Hintzen and R. Metselaar, Preparation and
Characterisation of MgSiN2 Powders, J. Mater. Sci. 34 (1999) 4519.
113 Chapter 4.
14. H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Densification of MgSiN2, Ceram. Trans. 51, Int. Conf. Cer. Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995) 585.
15. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics 2, Basic Science – Developments in Processing of Advanced Ceramics – Part II, Faenza, Italy, October 1995, edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 289. 16. GSAS, General Structure Analysis System written by A.C. Larson and R.B. von Dreele, LANSCE, MS-H805, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (1989). 17. Y.S. Touloukian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, Thermophysical Properties of Matter 13, Thermal Expansion of Nonmetallic Solids: Theory of Thermal Expansion of Solids (IFI/Plenum, New York, USA, 1977), pp. 3a - 16a. 18. R.A. Swalin, Thermodynamics of Solids, second edition (John Wiley & Sons, Inc., New York, USA, 1972). 19. W.C. Hamilton, Statistics in Physical Science: Estimation, Hypothesis Testing and Least Squares (The Ronald Press Company, New York, USA, 1964), pp. 168 - 173.
20. D.E. Partin, D.J. Williams and M. O’Keeffe, The Crystal Structures of Mg3N2
and Zn3N2, J. Solid State Chem. 132 (1997) 56.
21. R. Grün, The Crystal Structure of β-Si3N4; Structural and Stability
Considerations Between α- and β-Si3N4, Acta Cryst. B35 (1979) 800. 22. N.E. Brese and M. O’Keeffe, Bond-Valence Parameters for Solids, Acta Cryst. B47 (1991) 192. 23. Y.S. Touloukian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, Thermophysical Properties of Matter 13, Thermal Expansion of Nonmetallic Solids, (IFI/Plenum, New York, USA, 1977), pp. 1140 - 1141.
114 Anisotropic thermal expansion of MgSiN2
24. P.W. Sparks and C.A. Swenson, Thermal Expansion from 2 to 40 ºK of Ge, Si, and Four III-V Compounds, Phys. Rev. 163 (1967) 779. 25. W.B. Daniels, The amomalous thermal expansion of germanium, silicon and compounds crystallizing in the zinc blende structure, Int. Conf. on the Physics of Semiconductors, Exeter, UK, July 1962, edited by A.C. Stickland (Institute of Physics, London, 1962) 482. 26. G.A. Slack and S.F. Bartram, Thermal expansion of some diamondlike crystals, J. Appl. Phys. 46 (1975) 89. 27. K. Wang and R.R. Reeber, Thermal Expansion of GaN and AlN, Mat. Res. Soc. Symp. Proc. 482, Nitride Semiconductors, Boston, Massachusetts, USA, December 1 - 5, 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S. Nakamura and S. Strite (Materials Research Society, Warrendale, Pennsylvania, 1998) 863. 28. R. Roy, D.K. Agrawal and H.A. McKinstry, Very Low Thermal Expansion Coefficient Materials, Annu. Rev. Mater. Sci. 19 (1989) 59. 29. C.M.B. Henderson and D. Taylor, Thermal Expansion of the Nitrides and Oxynitride of Silicon in Relation to their Structure, Trans. J. Br. Ceram. Soc. 74 (1975) 49. 30. F.F. Grekov, G.P. Dubrovskii and A.M. Zykov, Structure and Chemical Bonding in Ternary Nitrides, Inorganic Materials 15 (1979) 1546 (translated from Izv. Akad. Nauk. SSSR, Neorg. Mater. 15 (1979) 1965). 31. H.D. Megaw, Crystal Structures and Thermal Expansion, Mat. Res. Bull. 6 (1971) 1007. 32. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)-Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science – Optimisation of Properties and Performance by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) p. 405. 33. Chapter 5, R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,
Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998) 7871.
115 Chapter 4.
116 Chapter 5.
The heat capacity of MgSiN2
Abstract o The heat capacity at standard pressure (Cp ) of MgSiN2 was determined by adiabatic calorimetry in the range of 8 - 400 K and differential scanning o calorimetry in the range of 300 - 800 K. The measured Cp data for T < 24 K
3 o 3 can be described using the Debye T approximation: Cp = AT with
-5 -1 -4 o A = 1.3632 10 J mol K . For temperatures between 350 and 650 K the Cp can be described with the Debye equation using a constant Debye temperature of 996 K. For temperatures between 24 and 350 K the Debye temperature is a o function of temperature and has a minimum value of 740 K at about 55 K. The Cp o data for T ≥ 300 K were compared with those of AlN. As expected, the Cp data of o MgSiN2 were about a factor 2 larger than those of AlN. The entropy ST , enthalpy o o o o (HT - H0 ) and the energy function (GT - H0 ) in the range of 0 - 800 K were o calculated using standard thermodynamic formulas. By extrapolating the Cp data o o to high temperatures at which GT is known, H0 was estimated to equal - 534 kJ mol-1.
1. Introduction
MgSiN2 is a ternary adamantine type compound with tetrahedral coordination of Mg and Si. It can be deduced from the well-known AlN by systematically replacing two Al ions with one Mg and one Si ion. The properties of MgSiN2 ceramics have recently been reported [1]. Because the thermal and mechanical
117 Chapter 5.
properties of MgSiN2 ceramics look promising, an investigation of the preparation, characterisation and properties of MgSiN2 was started. This chapter focuses in more detail on one thermal property, viz. the heat capacity. For understanding the thermal properties of a material, it is necessary to have accurate and reliable specific heat data as a function of the sample temperature. E.g., the heat capacity is needed for calculating the thermal conductivity from thermal diffusivity data [2]. The heat capacity can also be used to estimate the Debye temperature, which is an important parameter for theoretical modelling of the thermal conductivity [3]. Furthermore, heat capacity data can be used to o o o o o evaluate the thermodynamic functions ST , (HT - H0 ) and (GT - H0 ). o o o o The (GT - H0 ) function can be used to evaluate the Gibbs energy, GT , if H0 o o o o is known, or vice versa, if (GT - H0 ) and GT are known H0 can be evaluated. The o GT is necessary for predicting the thermodynamic stability of a compound in a system. Therefore, it is a very strong tool when optimising the synthesis or processing of a material. As far as the (present) author knows, specific heat data and thermodynamic functions for MgSiN2 have never been extensively reported. Only the room temperature value of the specific heat has been published (738 J kg-1 K-1 [4]). o Concerning the thermodynamic functions of MgSiN2, only estimates of GT at high temperatures (T > 1600 K) have been reported [5, 6]. Specific heat data of MgSiN2 below 300 K have never been reported, and high temperature data (T ≥ 300 K) are not available in the open literature [7]. o In this chapter standard specific heat data at constant pressure (Cp ) of
MgSiN2 in the temperature range of 0 - 800 K are reported. These data were used o to calculate the Debye temperature and the thermodynamic functions ST , o o o o (HT - H0 ) and (GT - H0 ) of MgSiN2 in the same temperature range. Also the o o o value of H0 was estimated by extrapolation of the (GT - H0 ) function into the o o temperature range where GT data are known. In order to do this the Cp curve was extrapolated up to 2000 K using standard thermodynamic functions.
118 The heat capacity of MgSiN2
2. Experimental
2.1. Adiabatic calorimeter measurements
The specific heat at constant pressure (Cp) for MgSiN2 in the range of 8 - 400 K was measured with an adiabatic calorimeter. The experimental set-up of the adiabatic calorimeter (CAL V) is described elsewhere [8]. For the measurement, isostatically pressed MgSiN2 powder pellets were used. The synthesis of the
MgSiN2 powders is described elsewhere [9]. Isostatically pressed powder pellets (diameter 6.5 mm, thickness 1 - 2 mm) with a total mass of about 12.12 g were put into a sample holder (copper vessel) with a mass of about 20 g and an internal volume of about 11 cm3. Before sealing the copper vessel, it was evacuated and filled with 1000 Pa of He gas as a heat exchanger. The sample plus sample holder was heated to the highest measuring temperature, before starting the Cp measurement, to enhance possible energy relaxation. Subsequently, the sample plus sample holder was cooled to the lowest measuring temperature. Then, stepwise a known quantity of energy was added to the sample plus sample holder and the temperature increase was measured. The temperature was measured with a Pt resistance thermometer (100 Ω at 298.15 K, Oxford instruments) with an accuracy of ± 0.003 K between 5 - 30 K and ± 0.005 K above 30 K. A temperature increase of about 2 - 3 K was measured after the energy was added to the system. From the temperature increase and the amount of energy added the specific heat of the sample plus the sample holder was calculated. Another independent specific heat measurement of the empty sample holder was used to correct for the specific heat of it, and subsequently to determine the specific heat of the MgSiN2 sample. When the powder pellets were first heated to the highest measuring temperature energy relaxation was observed. This energy was introduced into the system during isostatic pressing of the powder into pellets. After this energy relaxation the measured specific heat of the sample was not influenced by the thermal history of the measurement.
119 Chapter 5.
The reliability of the measured adiabatic calorimeter data was estimated from the impurity content of the MgSiN2 pellets and the absolute accuracy of the calorimeter. The major phase impurity in the MgSiN2 sample was MgO as a secondary phase. The impurity content of the MgSiN2 sample was estimated to be less than 1.0 wt. % based on the measured oxygen content and the measured X-ray diffraction pattern of the sample. The internal precision of the adiabatic calorimeter was about 0.02 %, and the absolute accuracy was estimated to be 0.1 %. From this and the purity of the sample, the reliability of the measured adiabatic calorimeter data was estimated to deviate no more than 1 % from the true values of pure
MgSiN2.
2.2. Differential scanning calorimeter measurement
For the DSC measurement MgSiN2 ceramic disks (diameter 4.90 mm, thickness
1.6 - 1.8 mm) cut from a sintered MgSiN2 tablet were used. The tablet was prepared by hot uni-axial pressing. The hot-pressing method for obtaining the
MgSiN2 ceramics has been reported earlier [10, 11]. Nearly single phase, glassy phase free, fully dense MgSiN2 ceramics with an isotropic microstructure were obtained [11]. The Cp of the ceramic samples in the range of 300 - 800 K was measured with a differential scanning calorimeter (DSC) Setaram 111. The sample was first heated to the highest measuring temperature before starting the Cp measurement to check for possible energy relaxation. Subsequently, the sample was cooled to the lowest measuring temperature. Then, the measurements were performed under a nitrogen atmosphere using a heating rate of 10 K/min. During heating, the differences in energy input (q, Q, Q') to keep the temperature constant were measured between: the reference sample holder and the sample holder (q); the reference sample holder and the sample holder plus sample with mass m (Q); and the reference sample holder and the sample holder plus a reference material with mass m' (Q'). The differences between the differences in energy input (Q - q) and (Q' - q) are directly related to the specific heat capacity in J g-1 K-1 of the sample
(cp) and the reference material (cp') according to:
120 The heat capacity of MgSiN2
(Q - q) m' c = c ' (1) p p (Q' - q) m Sapphire rods (Calorimeter Conference Sample 720), supplied by the National Bureau of Standards (NIST), were used as a reference material. The mass of the MgSiN2 sample was 0.38659 g, and the mass of the sapphire reference material was 0.19497 g. At every degree the Cp was determined. No energy relaxation was observed in the ceramic sample for the DSC measurement when it was heated to the highest measuring temperature. The measured specific heat of the sample was not influenced by the thermal history of the measurement. The reliability of the DSC data was estimated from the impurity content of the ceramic sample and the absolute accuracy of the DSC. The major phase impurity in the MgSiN2 ceramic sample was MgO as a secondary phase. The impurity content of the MgSiN2 sample was estimated to be less than 2.0 wt. % based on the measured oxygen content and the measured X-ray diffraction pattern of the sample. The absolute accuracy of the DSC was estimated to be 2 %. From this and the purity of the ceramic sample, the reliability of the measured DSC data was estimated to deviate no more than 3 % from the true values of pure MgSiN2.
3. Results and discussion
o 3.1. Cp of MgSiN2 For the results of both, adiabatic calorimeter and DSC, measurements it was
2 2 assumed that [∂Cp /∂p]T = -T [∂ V/∂T ]p ≈ 0. This implies that the measured Cp is pressure independent. So we may assume that the measured Cp equals the heat o capacity at standard pressure Cp . o In Fig. 5-1 the measured Cp values for MgSiN2 in the range of 8 - 400 K and 300 - 800 K are presented for the adiabatic calorimeter and DSC measurement, respectively.
121 Chapter 5.
o The Cp data of the adiabatic calorimeter and the DSC in the overlapping temperature range of 300 - 400 K are in excellent agreement with each other considering the experimental accuracy of the equipment (largest deviation
o -1 -1 ≈ 0.5 %). The Cp value measured at 293 K of 60.5 ± 0.6 J mol K (= 752 ± 8 J kg-1 K-1) is comparable with the earlier published value of 738 J kg-1
-1 o K [4]. The S-shaped Cp curve in Fig. 5-1 shows, as expected, a gradually increase with the absolute temperature till 760 K. At temperatures above 760 K an o unexpected (small) decrease, as function of the temperature, is observed. The Cp for electronic insulators, like MgSiN2, is expected to increase only slowly at high temperatures. So, it can be concluded that the DSC data are less reliable at high o temperatures. The systematic measurement of too low Cp values can most probably be ascribed to the difference in mass, shape and thermal properties like thermal conductivity of the sapphire reference and the MgSiN2 sample.
100
Adiabatic 80 DSC ] -1 60 K -1
40 [J mol p C 20
0 0 100 200 300 400 500 600 700 800 T [K]
Fig. 5-1: The heat capacity at constant pressure (Cp) of MgSiN2 as a function of the absolute temperature (T ) in the range of 0 - 800 K.
o The Cp values between 0 and 8 K were obtained by extrapolation using the o Cp data between 16 and 24 K and the Debye theory of the specific heat. This 3 theory states that if the temperature is sufficiently low the CV = AT [12]. At low
122 The heat capacity of MgSiN2
o 3 o temperatures Cp ≈ CV. So, if the Cp exhibits this T behavior then a Cp /T versus
2 o T plot results in a straight line through the origin with slope A. In Fig. 5-2 a Cp /T versus T 2 plot in the range of 0 - 50 K is shown. From the figure it can be
o 2 2 2 o concluded that Cp /T is proportional to T if T ≤ 600 K (T < 24 K). The Cp values measured below 16 K (T 2 < 250 K2) are less accurate due to the small contribution (< 1 %) of the MgSiN2 sample to the measured specific heat of the o sample plus sample holder. Therefore the Cp /T values between 16 and 24 K were used to evaluate the heat capacity data for MgSiN2 between 0 and 20 K, using that
o 2 -5 -1 -4 Cp /T = AT with A = 1.3632 10 J mol K for temperatures between 16 and
o -5 3 24 K. This resulted in Cp (T ) = 1.3632 10 T for T ≤ 20 K. For T > 20 K the o o Cp (T ) function at every degree was constructed by polynomial fitting the Cp data over several small temperature ranges.
0.05
0.04 ] 2 - K
1 0.03 -
-5 2 0.02 Cp /T = 1.36315 10 T [J mol [J T / p C 0.01
0 0 500 1000 1500 2000 2500 T 2 [K2]
2 Fig. 5-2: Cp /T versus T for MgSiN2 at T ≤ 50 K
o o The Cp data of MgSiN2 at T ≥ 300 K were compared with tabulated Cp values of AlN [13]. Because AlN and MgSiN2 have both a wurtzite-like crystal structure, the same average atomic mass, about the same average volume per atom, and about the same sound velocity [14], the Debye temperatures are about the same and so are the specific heats per mole atoms. So it is expected that
123 Chapter 5.
o o o Cp (MgSiN2) ≈ 2 Cp (AlN). In Table 5-1 the Cp values of MgSiN2 and AlN are presented from 300 to 800 K for every 100 K. It can be seen that, as expected, o o Cp (MgSiN2)/Cp (AlN) ≈ 2.0. For T ≥ 600 K a systematic decrease of the o o Cp (MgSiN2)/Cp (AlN) ratio is observed. This is probably caused by systematically o measuring too low Cp values at T ≥ 600 K.
o Table 5-1: Cp values of MgSiN2 and AlN from 300 to 800 K for every 100 K.
o o o o TCp (AlN) Cp (MgSiN2) Cp (MgSiN2)/Cp (AlN) [K] [J mol-1 K-1][J mol-1 K-1][-] 300 30.291 61.71 2.04 400 36.402 74.78 2.05 500 40.448 82.43 2.04 600 43.683 87.09 1.99 700 45.719 90.17 1.96 800 47.083 90.14 1.91
3.2. Debye temperature of MgSiN2
For calculating the Debye temperature we assumed that Cp = CV over the whole temperature range of 0 to 800 K. In order to check this assumption we estimated the maximum difference between the Cp and the CV for T ≤ 800 K using [15]: 9α 2V T C - C = m (2) p V β T
-1 3 -1 in which α [K ] is the linear thermal expansion coefficient, Vm [m mol ] the -1 molar volume, T [K] the absolute temperature and βT [Pa ] the isothermal compressibility. Vm and βT are almost constant as function of temperature, and from the Grüneisen relation it is known that α is about proportional with CV. So the (relative) difference between Cp and CV increases with temperature 2 (Cp - CV ~ CV T). The maximum difference between Cp and CV occurs at the
124 The heat capacity of MgSiN2
highest measuring temperature, viz. 800 K. For MgSiN2 this results in, taking the -6 -1 values of [14] α(T = 800 K) ≈ α(T = 827 K) = 6.5 10 K , Vm(T = 800 K) ≈ -5 3 -1 Vm(T = 293 K) = 2.57 10 m mol (= M/ρ of Ref. 14) and βT (T = 800 K) -12 -1 ≈ βT (T = 293 K) ≈ βS(T = 293 K) = 6.84 10 Pa (calculated with equation (12) of -1 -1 Ref. 14), a maximum difference between Cp - CV of about 1.2 J mol K . So the relative difference between Cp and CV at 800 K equals about 1.3 %. This is well within the experimental accuracy of the measured DSC data, so it may be assumed that Cp = CV.
1200
Adiabatic 1100 DSC
1000 [K] θ 900
800
700 0 100 200 300 400 500 600 700 800 T [K]
Fig. 5-3: The Debye temperature (θ ) as a function of the absolute
temperature (T ) of MgSiN2 in the range of 0 - 800 K.
In Fig. 5-3 the Debye temperature is presented as a function of the absolute temperature. The Debye temperature was determined using tabulated values of the specific heat per atom as function of T/θ [16]. The shape of the curve is similar to that determined for other adamantine type compounds [17]: a decrease of the Debye temperature with increasing temperature to a minimum value and then an increase with temperature to a constant value. At T > 650 K an unexpected steep increase of the Debye temperature is observed. This increase can be totally o ascribed to the less reliable measurement of the Cp data at high temperatures as will be discussed later in this section.
125 Chapter 5.
The Debye temperature at 0 K (θ 0) in Fig. 5-3 was evaluated from the Debye T 3 expression for the specific heat per mol atoms at low temperatures [12]:
4 12π T 3 C = R ( ) = AT 3 (3) V 5 θ in which R is the gas constant 8.314 J mol-1 K-1. Using the value for A of -6 -1 -4 -5 3.408 10 J mol K (= 1.3632 10 /4; there are 4 mol atoms per mol MgSiN2) the value for θ 0 was calculated and equals 829 K. As expected, this value is very close to the Debye temperature obtained from elastic data, θ E, of 827 K [14]. At about 55 K the Debye temperature has a minimum value of 740 K. If we express these temperatures in terms of reduced values T /θ 0 and θ /θ 0 we obtain for the location of the minimum T /θ 0 = 0.07 with θ /θ 0 = 0.9. The location of the minimum is comparable with that of other adamantine like compounds [17, 18]
(T/θ 0 ~ 0.05 - 0.07) whereas the minimum value is larger than for most other adamantine type compounds like Si and Ge [17, 18] (θ /θ 0 ~ 0.7). This difference is most probably caused by the relative low mean mass of MgSiN2 [18]. As Fig. 5-3 shows, the Debye temperature at 350 K ≤ T ≤ 650 K has a constant value of about 996 ± 4 K. At T > 650 K an increase of the Debye temperature is observed. This increase cannot be explained by the assumption made that Cp = CV because it results in a decrease of the Debye temperature as a function of temperature. So, this increase of the Debye temperature at T > 650 K is totally caused by systematically o measuring of too low Cp values indicating that the DSC measurement is less reliable at higher temperatures. o Using the constant Debye temperature of 996 K the Cp of MgSiN2 at T ≥ 350 K was calculated and compared with the experimentally measured values (Table 5-2). From Table 5-2 it can be seen that between 350 - 700 K the measured and calculated values are in good agreement with each other, and between o 700 - 800 K the measured Cp of MgSiN2 becomes less reliable. The maximum o deviation of the measured from the expected Cp value is about 2.5 %. If the assumption that Cp = CV is also considered, a maximum deviation of about 4 % is o o expected between the true Cp and the measured Cp .
126 The heat capacity of MgSiN2
o o Table 5-2: Comparison between the measured (Cp m) and calculated (Cp c) heat capacity using the Debye equation with θ = 996 K. o o o o o o o T Cp m Cp c Cp m - Cp c (Cp m - Cp c)/Cp m * 100% [K] [J mol-1 K-1] [J mol-1 K-1] [J mol-1 K-1] [%]
350 69.06 68.69 0.37 0.5 400 74.78 74.58 0.20 0.3 450 79.19 79.04 0.15 0.2 500 82.44 82.47 - 0.03 - 0.0 550 85.01 85.14 - 0.13 - 0.2 600 87.09 87.24 - 0.15 - 0.2 650 88.72 88.96 - 0.24 - 0.3 700 89.76 90.36 - 0.40 - 0.4 750 90.17 91.52 - 1.35 - 1.5 800 90.14 92.47 - 2.33 - 2.6
o o o o o 3.3. Thermodynamic functions ST , (HT - H0 ) and (GT - H0 ) of MgSiN2
o o o o For T ≤ 20 K, the thermodynamic functions (ST - S0 ) and (HT - H0 ) as a function of the absolute temperature, follow from the heat capacity function as o o o o o 1 3 1 4 (ST - S0 ) = 3 AT and (HT - H0 ) = 4 AT . A non-zero S0 due to the (partially) random occupation of the cation sites in MgSiN2 by Mg and Si is not expected because Mg and Si are complete ordered in the MgSiN2 lattice [19]. So, the
o o -1 -1 o absolute entropy, ST , can be calculated by taking S0 = 0 J mol K . The ST and o o (HT - H0 ) function at T > 20 K were calculated by numerical integration of the o o o Cp (T )/T and Cp (T ) function. The Cp (T )/T function was constructed using the o polynomial fit of the Cp (T) curve in the corresponding temperature range. The o o o o o o o Gibbs energy, (GT - H0 ), was calculated using (GT - H0 ) = (HT - H0 ) - T ST . o o o o o o The Cp , ST , (HT - H0 ) and (GT - H0 ) of MgSiN2 are presented in Table 5-3 for every 10 K in the range of 0 - 800 K. It is noted that for T ≥ 700 K
127 Chapter 5.
o o o o the Cp and the thereof calculated thermodynamic functions ST , (HT - H0 ) and o o (GT - H0 ) are less reliable.
o o o o o o Table 5-3: The Cp , ST , (HT - H0 ) and (GT - H0 ) of MgSiN2 for every 10 K between 0 and 800 K.
o o o o o o T Cp ST (HT - H0 ) (GT - H0 ) [K] [J mol-1 K-1] [J mol-1 K-1] [J mol-1] [J mol-1] 0 0 0 0 0 10 0.014 0.0045 0.034 -0.011 20 0.109 0.0364 0.545 -0.183 30 0.403 0.120 2.716 -0.890 40 1.133 0.325 9.997 -2.990 50 2.367 0.701 27.069 -7.957 60 4.088 1.275 58.87 -17.66 70 6.206 2.062 110.20 -34.18 80 8.593 3.046 184.13 -59.56 90 11.154 4.206 282.81 -95.68 100 13.912 5.521 408.91 -144.20 110 16.650 6.975 560.66 -206.62 120 19.484 8.545 741.30 -284.13 130 22.337 10.217 950.41 -377.86 140 25.183 11.977 1188.03 -488.77 150 28.001 13.811 1453.98 -617.65 160 30.774 15.707 1747.9 -765.2 170 33.488 17.654 2069.3 -932.0 180 36.135 19.644 2417.4 -1118.4 190 38.708 21.667 2791.7 -1324.9 200 41.201 23.716 3191.3 -1551.8 210 43.612 25.785 3615.5 -1799.3 220 45.938 27.868 4063.3 -2067.6 230 48.181 29.959 4533.9 -2356.7 240 50.340 32.056 5026.6 -2666.8 250 52.416 34.153 5540.5 -2997.8 260 54.410 36.248 6074.7 -3349.8 270 56.325 38.338 6628.4 -3722.8 280 58.161 40.420 7200.9 -4116.6 290 59.921 42.491 7791.4 -4531.1 300 61.713 44.551 8399.1 -4966.4 310 63.32 46.60 9024 -5422 320 64.86 48.63 9665 -5898 330 66.32 50.65 10321 -6395 340 67.73 52.65 10993 -6911 350 69.06 54.63 11676 -7448 360 70.33 56.60 12373 -8004 370 71.53 58.55 13082 -8580 380 72.67 60.47 13803 -9174 390 73.75 62.37 14535 -9789 400 74.78 64.25 15278 -10422
128 The heat capacity of MgSiN2
o o o o o o Table 5-3: (Continued) The Cp , ST , (HT - H0 ) and (GT - H0 ) of
MgSiN2 for every 10 K between 0 and 800 K.
o o o o o o T Cp ST (HT - H0 ) (GT - H0 ) [K] [J mol-1 K-1] [J mol-1 K-1] [J mol-1] [J mol-1] 410 75.86 66.15 16044 -11076 420 76.77 67.98 16807 -11747 430 77.62 69.80 17579 -12436 440 78.42 71.59 18359 -13143 450 79.19 73.37 19147 -13868 460 79.91 75.11 19942 -14610 470 80.59 76.84 20745 -15370 480 81.24 78.54 21554 -16147 490 81.85 80.23 22370 -16941 500 82.43 81.88 23191 -17751 510 83.00 83.52 24019 -18578 520 83.53 85.14 24851 -19421 530 84.04 86.73 25689 -20281 540 84.54 88.31 26532 -21156 550 85.01 89.87 27380 -22047 560 85.46 91.40 28232 -22953 570 85.89 92.92 29089 -23875 580 86.31 94.42 29950 -24812 590 86.71 95.90 30815 -25763 600 87.09 97.36 31684 -26730 610 87.46 98.80 32557 -27710 620 87.80 100.22 33433 -28706 630 88.13 101.63 34313 -29715 640 88.43 103.02 35196 -30738 650 88.72 104.39 36082 -31775 660 88.98 105.75 36970 -32826 670 89.21 107.09 37861 -33890 680 89.42 108.41 38754 -34968 690 89.61 109.72 39649 -36058 700 89.76 111.01 40546 -37162 710 89.89 112.29 41445 -38279 720 90.00 113.54 42344 -39408 730 90.08 114.79 43244 -40549 740 90.13 116.01 44145 -41703 750 90.17 117.22 45047 -42870 760 90.18 118.42 45949 -44048 770 90.18 119.60 46851 -45238 780 90.17 120.76 47752 -46440 790 90.15 121.91 48654 -47653 800 90.14 123.04 49555 -48778
129 Chapter 5.
o 3.4. H0 of MgSiN2
o In order to make fully use of the thermodynamic data, H0 should be known. o Because estimates of GT values at high temperatures (T > 1600 K) are known o o o [5, 6] H0 , can be evaluated from (GT - H0 ) data in the same temperature range by o o o matching the (GT - H0 ) function with the known estimates of the GT function. o o o In order to obtain (GT - H0 ) data at high temperatures, the Cp curve at high temperatures was calculated (up to 2000 K) using the Debye equation for the CV with θ = 996 K and the expression for the difference between Cp - CV, assuming o that α and Vm/βT are constant for T ≥ 800 K. In order to obtain a smooth Cp curve between 500 and 800 K, and to minimize the error in the calculated thermodynamic o functions introduced by measuring to low Cp values at T > 700 K, the experimental data between 650 and 800 K were not used in the polynomial fit to o o describe the Cp data below 800 K. From the Cp curve the thermodynamic o o o functions ST and (HT - H0 ) for temperatures between 500 and 2000 K were o o recalculated by numerical integration of the Cp (T )/T and Cp (T ) function using the o o polynomial fit of the Cp (T) curve for construction of the Cp (T )/T function. The o o (GT - H0 ) function was calculated in the same way as described before. o In the literature two estimates for the GT of MgSiN2 at high temperatures are reported [5, 6]. The first estimate results in a minimum and maximum value for o GT , and is based on the following two reactions [5]:
4Si2N2O + 2 MgSiN2 → 3Si3N4 + Mg2SiO4
Si3N4 + 4 MgO → Mg2SiO4 + 2MgSiN2 According to Müller [5] both reactions proceed to the right for temperatures o between 1673 and 2073 K. This results in the following conditions for the GT of
MgSiN2 in the corresponding temperature range:
o MgSiN o Si N o Mg SiO o Si N O GT 2 > 1½ GT 3 4 + ½ GT 2 4 - 2 GT 2 2
o MgSiN o Si N o MgO o Mg SiO GT 2 < ½ GT 3 4 + 2 GT - ½ GT 2 4
130 The heat capacity of MgSiN2
o o o Table 5-4: (GT - H0 ) and the estimates of H0 as function of the temperature.
o o o o o o o o o * o o T (GT - H0 ) GT max - (GT - H0 ) GT min - (GT - H0 ) GT - (GT - H0 ) [K] [kJ mol-1] [kJ mol-1] [kJ mol-1] [kJ mol-1] 800 -48.840 -523.000 -538.228 -539.791 900 -61.730 -521.280 -537.162 -539.168 1000 -75.680 -519.397 -536.061 -538.509 1100 -90.595 -517.407 -534.938 -537.829 1200 -106.397 -515.346 -533.800 -537.134 1300 -123.017 -513.246 -532.655 -536.431 1400 -140.398 -511.127 -531.508 -535.726 1500 -158.488 -509.008 -530.361 -535.022 1600 -177.243 -506.900 -529.222 -534.326 1700 -196.625 -504.816 -528.089 -533.635 1800 -216.601 -502.760 -526.964 -532.953 1900 -237.136 -500.744 -525.854 -532.285 2000 -258.202 -498.773 -524.762 -531.636 o average H0 — -511 ± 8 -532 ± 4 -536 ± 3
o o o From this a minimum and maximum value for the GT function, GT min and GT max, o of MgSiN2 were obtained (Table 5-4). The second estimate is based on the ∆GT R of the following reaction [6]:
Si3N4 + 4 MgO → Mg2SiO4 + 2MgSiN2 with:
o -1 o MgSiN o Mg SiO o MgO o Si N ∆GT R = 3953 - 8.35 T [J mol ] = GT 2 + ½GT 2 4 - 2GT - ½GT 3 4
Kaufman et al. [6] used this expression, and the Gibbs energy of Mg2SiO4, MgO and Si3N4 for thermodynamic calculations at temperatures above 1900 K resulting
o o * in the GT function of MgSiN2 based on Ref. 6 which will be referred to as GT . It was assumed that both estimates are valid for temperatures between 800 and
o o o * o 2000 K. For the calculation of GT max, GT min and GT , tabulated GT values of
o Si N o Si N O o MgO o Mg SiO Si3N4 (GT 3 4), Si2N2O (GT 2 2 ), MgO (GT ) and Mg2SiO4 (GT 2 4) were
131 Chapter 5.
o Si N o Si N O o MgO o Mg SiO used. GT 3 4 and GT 2 2 were taken from Ref. [20] and GT and GT 2 4 were taken from Ref. [21].
-5.0E+05 -5.0E+04
-5.5E+05 -1.0E+05 ] -1 × → ] -6.0E+05 -1.5E+05 -1 J mol [
-6.5E+05 -2.0E+05 ) o J mol 0 [
H o - T o T G -7.0E+05 ←{ I -2.5E+05 G
∆ (
-7.5E+05 -3.0E+05
-8.0E+05 -3.5E+05 800 1000 1200 1400 1600 1800 2000 T [K]
o o o * Fig. 5-4: The Gibbs energies GT max (□), GT min (■) and GT (∆), and the o o energy function (GT - H0 ) (×) as a function of the absolute
temperature T for MgSiN2 from 800 to 2000 K.
o o o o * o o The estimates for GT (GT max, GT min and GT ) and (GT - H0 ) of MgSiN2 between 800 and 2000 K are graphically presented in Fig. 5-4. Although it is clear
o * o from the figure that the GT function is not within the range of the GT max and o o GT min function, but just below the GT min function, both estimates are in favourable
o o o * agreement with each other. As expected the GT max, GT min and GT function have o o a similar shape as the (GT - H0 ) function, indicating that the extrapolation of the o o (GT - H0 ) function is reliable and has been done correctly. In Table 5-4 the
o o o o o o * (GT - H0 ) function and the estimates of H0 , based on the GT max, GT min and GT o function, are presented as a function of the absolute temperature. Indeed, the H0
o o * values are nearly constant, as expected. The H0 values obtained from the GT function are the most constant and have the smallest standard deviation, whereas o o the H0 values obtained from the GT max function fluctuate the most and have the
o o * largest standard deviation. If it is assumed that the GT min and GT function are the
132 The heat capacity of MgSiN2
o o -1 best estimates for the real GT function, then a value for H0 of -534 ± 3 kJ mol is obtained.
4. Conclusions
o The heat capacity Cp , the Debye temperature θ , and the thermodynamic functions o o o o o ST , (HT - H0 ) and (GT - H0 ) of MgSiN2 were determined for temperatures between 0 and 800 K. o The experimental Cp data for T < 24 K can be described by the Debye
3 o T approximation. The measured Cp data for T ≥ 300 K were compared with those o of AlN. As expected the Cp data of MgSiN2 were about a factor 2 larger than for AlN. The Debye temperature at 0 K equals 829 K and is comparable with the Debye temperature obtained from elastic constants (827 K). The Debye temperature below 350 K is a function of the absolute temperature and has a minimum value of 740 K at about 55 K. A constant Debye temperature of 996 K can be used to o describe the experimental Cp data for T ≥ 350 K using the Debye equation. o o By extrapolation of the Cp data to high temperatures (> 1600 K), H0 was estimated to equal -534 kJ mol-1.
References
1. W.A. Groen, M.J. Kraan, and G. de With, Preparation, Microstructure and
Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413. 2. P. Debye, Zustandsgleichung und Quantenhypothese mit einem Anhang über Wärmeleitung, in: Vorträge über die Kinetische Theorie der Materie und der Electrizität (Teubner, Berlin, 1914), pp. 19 - 64.
133 Chapter 5.
3. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1979) 1. 4. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent
Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non-Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh, 1994) 239. 5. R. Müller, Kostitutionsuntersuchungen und thermodynamischen Berechnungen im system Mg, Si/N, O, Ph. D. Dissertation University of Stuttgart, 1981; pp. 32 - 34. 6. L. Kaufman, F. Hayes, and D. Birnie, Calculation of quasibinary and quasiternary oxynitride systems, High Temp. High Pres. 14 (1982) 619. o 7. W.A. Groen, Personal communication (Cp values of MgSiN2 had been determined between 300 - 850 K at Philips Research Laboratories but these data have not been published). 8. J.C. van Miltenburg, G.J.K. van den Berg and M.J. van Bommel, Construction of an adiabatic calorimeter. Measurements on the molar heat capacity of synthetic sapphire and n-heptane. J. Chem. Thermodyn. 19 (1987) 1129. 9. Chapter 2; R.J. Bruls, H.T. Hintzen and R. Metselaar, Preparation and
characterisation of MgSiN2 powders, J. Mater. Sci. 34 (1999) 4519. 10. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan,
Hot-pressing of MgSiN2 ceramics, J. Mat. Sci. Lett. 13 (1994) 1314. 11. Chapter 3; R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar,
Preparation, Characterisation and Properties of MgSiN2 Ceramics, to be published. 12. See for example C. Kittel, Introduction to Solid State Physics, fifth edition (John Wiley & Sons, Inc., New York, 1976), pp. 136 - 140, or R.A. Swalin, Thermodynamics of Solids, second edition (John Wiley & Sons, Inc., New York, 1972), pp. 57 - 62.
134 The heat capacity of MgSiN2
13. I. Barin, Thermochemical Data for Pure Substances, Part I, second edition (VCH Verlagsgesellschaft mbH, Weinheim, FRG, 1993), p. 42. 14. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)-Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of Properties and Performance by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) p. 405. 15. See for example Swalin, R.A. Thermodynamics of Solids, second edition (John Wiley & Sons, Inc., New York, 1972). 16. W.M. Rogers and R.L. Powell, Tables of Transport Integrals, Natl. Bur. Stand. Circ. 595 (1958) 1. 17. J.C. Phillips, Vibration Spectra and Specific Heats of Diamond-Type Lattices, Phys. Rev. 113 (1959) 147. 18. T.C. Cetas, C.R. Tilford and C.A. Swenson, Specific Heats of Cu, GaAs, InAs, and InSb from 1 to 30 °K, Phys. Rev. 174 (1968) 835. 19. R.K. Harris, M.J. Leach and D.P. Thompson, Nitrogen-15 and Oxygen-17 NMR Spectroscopy of Silicates and Nitrogen Ceramics, Chem. Mater. 4 (1992) 260. 20. M. Hillert, S. Jonsson and B. Sundman, Thermodynamic Calculation of the Si-N-O System, Z. Metalkd. 83 (1992) 648. 21. I. Barin, Thermochemical Data for Pure Substances, second edition (VCH Verlagsgesellschaft mbH, Weinheim, FRG, 1993).
135 Chapter 5.
136 Chapter 6.
The Young's modulus of MgSiN2, AlN and Si3N4
Abstract
The temperature dependence of the Young's modulus of MgSiN2 and AlN was measured between 293 and 973 K using the impulse excitation method and compared with literature data reported for Si3N4. The data could be fitted with
E = E0 - B·T exp (-T0/T ). The values of the fitting parameters E0 and T0 are related to the Debye temperature and the parameter B to the harmonic character of the bond.
1. Introduction
The relatively new ternary adamantine type compound MgSiN2, which can be deduced from the well known AlN by replacing two Al3+ ions by one Mg2+ and one Si4+ ion, might be interesting for specific applications because of its favourable chemical, mechanical and thermal properties [1 - 8]. We pointed out [6] that in order to understand the (thermal) properties of
MgSiN2, AlN and other (new) potentially interesting materials more insight is needed in the parameters that determine the intrinsic thermal conductivity. Two important parameters that determine the intrinsic thermal conductivity are the Debye temperature θ [K] and Grüneisen parameter γ [-] [6, 9 - 12]. The Debye temperature can be evaluated from elastic constants E (Young's modulus) [Pa] and ν (Poisson's ratio) [-] [13]. For evaluation of the Grüneisen parameter elastic constants as a function of the temperature are needed. So far, only room temperature values for the elastic constants have been published (MgSiN2:
137 Chapter 6.
E = 235 GPa and ν = 0.232 [2], and AlN: E = 308 - 315 GPa and ν = 0.179 - 0.245 [14 - 16]).
In this chapter the Young's modulus of MgSiN2 and AlN as a function of the temperature between 293 - 973 K is reported. The temperature dependence of the
Young's modulus was described with the empirical expression E = E0 - B·T exp (-T0/T ) which was previously shown to be valid by Wachtman for several oxides [17]. Also temperature dependent Young's modulus data for the related nitride compound Si3N4 [18] were fitted using this expression. For MgSiN2, AlN and Si3N4 the fitting parameter E0 was used for calculating the Debye temperature
θ0. The values obtained for B and T0 from fitting of the experimental data are discussed in view of the analytical expressions of Anderson [19] for B and T0.
2. Experimental section
The preparation of the MgSiN2 ceramic disks (∅ 33 mm × 3 mm) with hot- pressing (1550 - 1650 °C, 75 MPa, N2 atmosphere, 2 h) is described elsewhere [5, 8, 20]. Three fully dense (ρ = 3.14 - 3.15 g cm-3) samples (RB10, RB31 and RB33) processed in somewhat different ways, were selected to measure the temperature dependence of the Young's modulus. X-ray diffraction (XRD) revealed that they contain some (< 2 wt. %) α- and β-Si3N4 (RB10), MgO (RB31), and β-Si3N4 (RB33) as a secondary phase [8]. Clean grain boundaries were observed between the MgSiN2 grains (~ 0.3 - 1.0 µm) with transmission electron microscopy (TEM). The AlN ceramics were obtained from Xycarb ceramics (Helmond, The Netherlands). The fully dense (ρ = 3.29 g cm-3) AlN ceramic disk (∅ 250 × 20 mm) was prepared by hot-pressing (1830 °C, 35 MPa, N2 45 min.) AlN powder
(ART, grade A100) containing about 4 wt. % Y2O3 as an additive. The resulting ceramics contain some YAP (YAlO3, JCPDS 33-41) and YAG (Y3Al5O12, JCPDS 33-40) as detected with XRD, which are commonly found secondary phases for
AlN sintered with Y2O3 addition. The grain size of the AlN ceramics was about 4
138 The Young's modulus of MgSiN2, AlN and Si3N4
µm as observed on a fractured surface with a field emission scanning electron microscopy (FESEM).
For a fully dense MgSiN2 disk (∅ 15 mm × 2.89 mm) the room temperature -1 -1 longitudinal vl [m s ] and transverse sound velocity vt [m s ] were measured at 10 and 20 MHz, respectively, using the pulse-echo method. From the sound velocities and the density ρ [kg m-3] the room temperature Young's modulus E [Pa] and Poisson's ratio ν [-] were calculated using the formulas for isotropic materials [21]:
2 2 ν = vl vl - 2 2 - 2 (1) vt vt
(1+ν )(1- 2ν ) E = 2v 2 ρ()1+ν = v 2 ρ (2) t l (1-ν ) The Young's modulus was measured from 293 to 973 K on three hot-pressed, fully dense, MgSiN2 ceramic materials processed under somewhat different conditions (for details see Ref. 8) and one hot-pressed, fully dense, AlN ceramic material using the impulse excitation method [22] (GrindoSonic, Lemmens Elektronica BV,
Belgium). For MgSiN2 and AlN two different sample sizes of the same material were measured (rectangular bars l × b × h ~ 18 mm × 8 mm × 2 mm and ~ 18 mm × 5 mm × 2 mm). For comparison also some room temperature measurements on larger AlN bars (rectangular bars ~ 50 mm × 8 mm × 3 mm and ~ 50 mm × 6 mm × 3 mm) were performed. Each measurement was performed twice in order to obtain an impression of the accuracy of the data points and of the resulting fitting parameters. The fundamental natural flexural frequency of the samples was measured every 5 K during heating and cooling. From this frequency, the sample dimensions and mass, the Young's modulus E [Pa] was evaluated using [23, 24]:
mf 2l 3 E = 0.9465 A (3) bh3 in which m [kg] is the sample mass, f [s-1] the flexural frequency, l [m] the sample length, b [m] the sample width, h [m] the sample height and A a dimensionless shape factor dependent on sample length, sample width and Poisson's ratio. As the
139 Chapter 6.
dependence of A on the Poisson's ratio is very limited, A can be approximated by: A = 1 + 6.585(h/l)2 [24]. The sample dimensions were corrected for thermal expansion in order to calculate the Young's modulus. The resolution of the flexural frequency measurement was 10 Hz. For a typical resonance frequency of about 30 kHz this results in an experimental error in the Young's modulus introduced by the frequency measurement of ∆E/E ≈ 2(∆f /f ) ≈ 0.07 %.
For comparison temperature dependent literature data for Si3N4 (β-modification according to the processing temperature of 1750 °C mentioned [18]) were taken for sample H-1 with 0.5 wt. % MgO addition. The Young's modulus data for this sample with the least amount of secondary phase were
-3 corrected for porosity (E = Emeas/(2ρ meas/ρ the - 1) [18]) with ρ meas (3.104 g cm ) and
Emeas (varying between 302 GPa and 291 GPa for temperatures between ~ 300 K and ~ 1200 K, respectively) the experimental density and Young's modulus, -3 respectively, and ρ the (3.19 g cm ) the theoretical density of β-Si3N4. The data obtained as a function of the absolute temperature were described using the empirical formula of Wachtman [17]: = ⋅ E E0 - B T exp (-T0 /T ) (4) -1 in which E0 [Pa] is the Young's modulus at 0 K, B [Pa K ] and T0 [K] are fitting parameters.
3. Results and discussion
3.1. Evaluation of the measurements
3 -1 For MgSiN2 a longitudinal sound velocity of 10.17 10 m s and transverse sound velocity of 5.90 103 m s-1 were measured. This resulted in a room temperature value for the Poisson's ratio ν of 0.246 and the Young's modulus E of 273 GPa. The Poisson's ratio is comparable with ν = 0.232 given in the literature measured with the same pulse-echo technique [2]. The value of the Young's modulus reported before is considerably lower E = 235 GPa [2], which may be (partially)
140 The Young's modulus of MgSiN2, AlN and Si3N4
ascribed to the lower density (98.9 %) and purity (3.7 wt. % oxygen) of the sample described in the literature [2].
325
Si3N4 320
315 AlN 310
305
300
295 [GPa] E 290
285 MgSiN2
280
275
270
265 0 200 400 600 800 1000 1200 T [K]
Fig. 6-1: A typical result obtained for the Young's modulus (E ) as a function of the absolute temperature (T ) between 293 and
973 K for a MgSiN2 and AlN ceramic sample with fit E = 284.8 - 0.0228⋅T exp (-424/T ) and E = 310.2 - 0.0247⋅T exp (-533/T ), respectively. For comparison literature data
for Si3N4 [18] between 300 and 1200 K with fit E = 320.4 - 0.0151⋅T exp (-445/T ) are included.
For the impulse excitation experiments no hysteresis in the resonance frequency was observed during heating and cooling. The reproducibility of the measurements using the same sample was excellent (± 0.3 GPa). The slight difference between the observed Young's moduli for the same material having different dimensions is caused by experimental errors in the sample dimension measurement, ∆l, ∆b and ∆h ≈ 0.02 mm leading to ∆E ≈ (3(∆l /l )2 + (∆b/b)2 + 3(∆h/h)2 + 2(∆f /f )2)1/2 E ≈ 0.016 E = 4.5 GPa). Considering the experimental accuracy the Young's modulus at 293 K was the same for the various samples and
141 Chapter 6.
equal about 279 ± 4 GPa and 312 ± 4 GPa for MgSiN2 and AlN, respectively. The room temperature value for MgSiN2 is in good agreement with the value measured using the pulse-echo method (273 GPa) and the values for AlN (312 GPa) and
Si3N4 (319 GPa) are in excellent agreement with previously reported values (AlN:
308 - 315 GPa [14 - 16] and Si3N4: 290 - 335 GPa [25, 26]).
With increasing temperature the Young's modulus of MgSiN2, AlN and
Si3N4 slightly decreases (Fig. 6-1). As compared with the room temperature value the Young's modulus at 973 K for the MgSiN2 samples has decreased with
12.6 ± 0.2 GPa (∆E/E293 = 0.045), for the AlN samples with 12.9 ± 0.2 GPa
(∆E/E293 = 0.041) and for the Si3N4 sample with 8.3 GPa (∆E/E293 = 0.026). So, the temperature dependences of MgSiN2 and AlN are similar whereas Si3N4 shows a smaller temperature dependence. As expected [17], the temperature dependence of the experimental data is very well described by E = E0 - B·T exp (-T0/T ) (Fig. 6-1). As the Young's modulus shows no anomalies, this indicates that for all three materials the influence of microstructure and secondary phases on the temperature dependence of the Young's modulus can be assumed to be negligible.
In Table 6-1 the values of the fitting parameters E0, B and T0 are presented.
The average E0 value for AlN of 314 GPa was calculated from the average observed E293 and the average values of B and T0. Within the experimental accuracy the values of E0, B and T0 of the AlN samples are the same as all samples originate from one large homogeneous ceramic bar. Also for the several MgSiN2 samples processed under somewhat different conditions (see Ref. 8) the values of
E0, B and T0 are within the experimental error the same (see Table 6-1).
A relatively large variation in T0 is observed for the measurements performed on the same sample having the same size. For T / T0 we can write for
E = E0 - B·T exp (-T0/T ) ≈ E0 - B·T (1-T0/T ) = (E0 + B·T0) - B·T resulting in a linear relation between E and T (as observed in Fig. 6-1) showing that the slope B can be easily evaluated whereas the constants E0 and B·T0 are correlated. As E0 >> B·T0 the fitting parameter T0 is relatively sensitive to small errors as compared to
E0 and B.
142 The Young's modulus of MgSiN2, AlN and Si3N4
Table 6-1: Fitting parameters E0, B and T0 for describing the Young's modulus as a function of the
absolute temperature, and room temperature value of the Young's modulus for MgSiN2,
AlN and Si3N4. Between brackets the 95 % confidence interval of the fitting parameters are presented. The experimental error was estimated to equal the standard deviation of the average values.
Material / Sample E0 B T0 E293 [GPa] [GPa K-1] [K] [GPa]
MgSiN2 RB10 (17.29 × 8.05 × 2.15 mm) 277.93 (7) 0.02237(18) 450(12) 276.5 278.24(16) 0.02190(38) 403(25) 276.6 (17.30 × 5.69 × 2.15 mm) 286.64(10) 0.02134(16) 347(14) 284.7 286.76 (9) 0.02136(15) 332(11) 284.7 RB31 (17.76 × 8.06 × 2.14 mm) 284.78 (6) 0.02281(14) 424 (9) 283.2 285.01(10) 0.02241(20) 397(15) 283.3 (17.76 × 5.85 × 2.15 mm) 276.45 (7) 0.02251(17) 422(11) 274.9 276.58 (8) 0.02203(19) 390(13) 274.9 RB33 (17.54 × 8.06 × 2.15 mm) 280.87(14) 0.02133(24) 349(19) 279.0 280.98(10) 0.02156(24) 366(16) 279.1 (17.55 × 5.88 × 2.16 mm) 277.87(12) 0.02161(25) 386(18) 276.2 277.94 (5) 0.02227(11) 403 (8) 276.3
Average value MgSiN2 281 ± 4 0.0220 ± 0.0005 389 ± 34 279 ± 4 AlN (17.82 × 8.12 × 2.12 mm) 310.11(12) 0.02419(35) 487(21) 308.8 310.14(14) 0.02451(39) 488(23) 308.8 (17.80 × 5.89 × 2.12 mm) 310.20 (6) 0.02468(24) 533(12) 309.0 310.64(12) 0.02404(37) 473(21) 309.2 (50.23 × 8.11 × 2.99 mm) — — — 318.8 — — — 318.8 (50.06 × 5.88 × 2.99 mm) — — — 312.9 — — — 312.9 Average value AlN 314 ± 4 0.0244 ± 0.0003 495 ± 26 312 ± 4
Si3N4 H-1 (∅ 30 mm × 12 mm) [18] 320.41(13) 0.01508(24) 445(28) 319.4
143 Chapter 6.
3.2. Interpretation of the fitting parameters
3.2.1. E0
The average E0 value (Young's modulus at 0 K) for MgSiN2, AlN and Si3N4 was used to evaluate the Debye temperature at 0 K, θ 0 [K]. The Debye temperature can -1 be calculated from the average sound velocity (vs [m s ]) obtained from the -1 -1 longitudinal vl [m s ] and transverse sound velocity vt [m s ] using the elastic constants E and ν, and the density ρ [kg m-3] [12, 21, 27]:
E (1-ν ) v = (5) l ρ (1+ν )(1- 2ν )
E 1 v = (6) t ρ 2(1+ν )
1 1 1 2 - = + 3 vs ( [ 3 3 ]) (7) 3 vl vt Subsequently the average sound velocity can be used to calculate the Debye temperature using [12, 13, 27 - 29]:
hv 8s N ρ 1 θ = s ( A ) 3 (8) 2k 4π M 3 in which h is Planck's constant (6.626 10-34 J s), k the Boltzmann's constant -23 -1 (1.381 10 J K ), s [-] the number of atoms per formula unit, NA Avogadro's number (6.023 1023 mol-1) and M [kg mol-1] the mole mass.
Using the Young's modulus and density at 0 K (E0 and ρ 0, respectively) and the room temperature value of the Poisson's ratio ν, the Debye temperature θ 0 was calculated (see Table 6-2). The resulting Debye temperatures of all three compounds are in the same range (900 - 950 K) with θ MgSiN2 = 900 K
. θ AlN = 940 K . θ Si3N4 = 955 K. The values agree reasonably well with previously reported values for MgSiN2, AlN and Si3N4 determined in different ways (vide infra Table 6-3).
144 The Young's modulus of MgSiN2, AlN and Si3N4
Table 6-2: The sound velocities and Debye temperatures at 0 K for MgSiN2, AlN and Si3N4 (*:
assuming that ρ 0 = ρ 293 lacking the availability of low temperature data).
Compound E0 ρ 0 ν vl vt vs s θ 0 [GPa] [kg m-3] [-][m s-1][m s-1][m s-1][-][K]
3 4 3 3 MgSiN2 281 3.142 10 [30] 0.246 1.033 10 5.99 10 6.65⋅10 4 900 AlN 314 3.258 103 [31] 0.245 [14] 1.071 104 6.22 103 6.90 103 2 940 3 * 4 3 3 Si3N4 320 3.202 10 [32] 0.267 [18] 1.115 10 6.28 10 7.00 10 7 955
3.2.2. B and T0 Anderson [19] quantified the suggestion of Wachtman that the fitting parameters B and T0 are related to the Grüneisen parameter and Debye temperature, respectively. Using the equation of Anderson [19] and assuming that the Poisson's ratio is temperature independent (dν/dT = 0) we can calculate B and T0 using: 3Rγ δ B = 3(1- 2ν ) (9) V0
The parameter T0 is very approximately given as: ≈ θ T0 0/2 (10) in which ν [-] is the Poisson's ratio, R (8.314 J mol-1 K-1) the gas constant, γ [-] the
Grüneisen constant [33], δ [-] the Anderson-Grüneisen constant [34] and V0 [m3 mol-1] the specific volume per atom at absolute zero. Using the expressions for γ and δ [19] the equation for B can be written as: s3R ∂E B = (11) ∂ C p T
-1 -1 in which s [-] is the number of atoms per formula unit and Cp [J mol K ] the heat capacity at constant pressure. It is directly clear that for calculating the value of the fitting parameter B to describe the temperature dependence of the Young's modulus these data themselves are needed. So, this makes an independent evaluation of the fitting parameter B from the present data impossible.
145 Chapter 6.
For comparison with other compounds only few experimental data are available. The experimentally observed value for the fitting parameter B of -1 -1 -1 0.0220 GPa K for MgSiN2, 0.0244 GPa K for AlN and 0.0151 GPa K for Si3N4 are somewhat lower than the values reported for the three oxides investigated by -1 -1 Wachtman (0.048 GPa K for Al2O3 [17], 0.027 GPa K for ThO2 [17] and 0.037 GPa K-1 for MgO [19]). The somewhat smaller value for B indicates that the nitrides show a more harmonic bond character as compared to the oxides, as expected from the more covalent nature of nitrides, and considering that the Young's modulus of a fully harmonic bond is temperature independent.
θ θ Table 6-3: The Debye temperature 0 and T0 of MgSiN2, AlN and Si3N4 obtained from
the fitting parameters E0 and T0, respectively as compared to previously reported values obtained from specific heat measurements (θ C), lattice dynamic calculations (θ LD) and elastic constants (θ E).
θ θ θ C θ E θ LD Compound 0 T0 [K] [K] [K] [K] [K]
MgSiN2 900 778 829 [35] 827 [12] AlN 940 990 950 [1], 1010 [36] 800 [37]
Si3N4 955 890 754 [38], 900 [39] 900 -1005 [40]
For MgSiN2, AlN and Si3N4 the experimentally obtained average T0 value (see Table 6-1) was used to estimate the Debye temperature, resulting in 778, 990 θ and 890 K for MgSiN2, AlN and Si3N4, respectively. These T0 values are in rough agreement with the θ 0 value (obtained from E0) and the other reported Debye temperatures for MgSiN2, AlN and Si3N4, respectively (see Table 6-3), indicating the approximate nature of the Anderson equation θ 0 ≈ 2T0 [19].
4. Conclusions
The temperature dependence of the Young's modulus of MgSiN2, AlN and Si3N4 can be described very well with E = E0 - B·T exp (-T0/T ). The Debye temperatures
146 The Young's modulus of MgSiN2, AlN and Si3N4
estimated from E0 and T0 are in rough agreement with each other, and previously reported values obtained in different ways. The values of the fitting parameter B determined for our nitrides are lower than those previously reported for oxides. This is ascribed to the more harmonic nature of bonds in nitrides as compared to oxides resulting in a relatively temperature independent Young's modulus.
References
1. G.A. Slack, Nonmetallic Crystals with High Thermal Conductivity, J. Phys. Chem. Solids 34 (1973) 321. 2. W.A. Groen, M.J. Kraan, and G. de With, Preparation, Microstructure and
Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413. 3. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent
Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non- Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh, 1994) 239. 4. H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Densification of MgSiN2, Ceram. Trans. 51, Int. Conf. Cer. Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995) 585 .
5. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics 2, Basic Science - Developments in Processing of Advanced Ceramics - Part II, Faenza, Italy, October 1995, edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 289. 6. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modelling of the Thermal
Diffusivity/Conductivity of MgSiN2 Ceramics, Thermal Conductivity 24 and Thermal Expansion 12, Pittsburgh, Pennsylvania, USA, October 1997, edited
147 Chapter 6.
by P.S. Gaal and D.E. Apostolescu (Technomics Publishing Co., Inc., Lancaster, Pennsylvania 1999) 3. 7. I.J. Davies, H. Uchida, M. Aizawa and K. Itatani, Physical and Mechanical Properties of Sintered Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 40. 8. Chapter 3; R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar,
Preparation, Characterisation and Properties of MgSiN2 Ceramics, to be published. 9. G. Leibfried and E. Schlömann, Wärmeleitung in elektrisch isolierenden Kristallen, Nach. Akad. Wiss. Göttingen, Math. Phys. Klasse 4 (1954) 71. 10. C.L. Julian, Theory of Heat Conduction in Rare-Gas Crystals, Phys. Rev. 137 (1965) A128. 11. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1979) 1. 12. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)- Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of Properties and Performances by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405. 13. P. Debye, Zur Theorie der spezifischen Wärmen, Ann. Physik 39 (1912) 789. 14. P. Boch, J.C. Glandus, J. Jarrige, J.P. Lecompte and J. Mexmain, Sintering, Oxidation and Mechanical Properties of Hot Pressed Aluminium Nitride, Ceram. Int. 8 (1982) 34. 15. D. Gerlich, S.L. Dole and G.A. Slack, Elastic Properties of Aluminum Nitride, J. Phys. Chem. Solids 47 (1986) 437. 16. A.A. Khan, J.C. Labbe, Aluminium Nitride - Molybdenum Ceramic Matrix Composites. Influence of Molybdenum Addition on Electrical, Mechanical and Thermal Properties, J. Eur. Ceram. Soc. 17 (1997) 1885.
148 The Young's modulus of MgSiN2, AlN and Si3N4
17. J.B. Wachtman, Jr., W.E. Tefft, D.G. Lam, Jr., and C.S. Apstein, Exponential Temperature Dependence of Young's Modulus for Several Oxides, Phys. Rev. 122 (1961) 1754.
18. I. Tomeno, High Temperature Elastic Moduli of Si3N4 Ceramics, Jpn. J. Appl. Phys. 20 (1981) 1751. 19. O.L. Anderson, Derivation of Wachtman's Equation for the Temperature Dependence of Elastic Moduli of Oxide Compounds, Phys. Rev. 144 (1966) 553. 20. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan,
Hot-pressing of MgSiN2 ceramics, J. Mat. Sci. Lett. 13 (1994) 1314. 21. see for example E. Schreiber, O.L. Anderson and N. Soga, Elastic Constants and Their Measurement (McGraw-Hill, Inc., 1973). 22. K. Heritage, C. Frisby and A. Wolfenden, Impulse excitation technique for dynamic flexural measurements at moderate temperatures, Rev. Sci. Instr. 59 (1988) 973. 23. S. Spinner, T.W. Reichard and W.E. Tefft, A Comparison of Experimental and Theoretical Relations Between Young's Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars, Journal of Research, National Bureau of Standards 64A (1960) 147. 24. S. Spinner and W.E. Tefft, A Method for Determining Mechanical Resonance Frequencies and for Calculating Elastic Moduli from these Frequencies, Proceedings ASTM 61 (1961) 1221. 25. G. Woetting, G. Leimer, H.A. Lindner and E. Gugel, Silicon Nitride Materials and Components for Industrial Application, Industrial Ceramics 15 (1995) 191. 26. D.-S. Park, H.-D. Kim, S.-Y. Lee and S. Kim, Gas Pressure Sintering of
Si3N4-Based Particulate Composites and their Wear Behavior, Key Eng. Mater. 89 - 91 (1994) 439. 27. R.H. Fowler and E.A. Guggenheim, Statistical Thermodynamics (Cambridge University Press, 1939), p. 126.
149 Chapter 6.
28. O.L. Anderson, A Simplified Method for Calculating the Debye Temperature from Elastic Constants, J. Phys. Chem. Solids 24 (1963) 909. 29. D. Singh and Y.P. Varshni, Debye temperatures for hexagonal crystals, Phys. Rev. B 24 (1981) 4340. 30. Chapter 4; R.J. Bruls, H.T. Hintzen, R. Metselaar and C.-K. Loong,
Anisotropic thermal expansion of MgSiN2 from 10 to 300 K as measured by neutron diffraction, J. Phys. Chem. Solids 61 (2000) 1285. 31. K. Wang and R.R. Reeber, Thermal Expansion of GaN and AlN, Mat. Res. Soc. Symp. Proc. 482, Nitride Semiconductors, Boston (Massachusetts, USA), December 1-5, 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S. Nakamura and S. Strite (Materials Research Society, Warrendale, Pennsylvania, 1998) 863. 32. C.M.B. Henderson and D. Taylor, Thermal Expansion of the Nitrides and Oxynitride of Silicon in Relation to their Structure, Trans. J. Br. Ceram. Soc. 74 (1975) 49. 33. E. Grüneisen, Zustand des festen Körpers, Handbuch der Physik 10, edited by H. Geiger and K. Scheel (Springer, Berlin, Germany, 1926) 1. 34. Y.A. Chang, On the Temperature Dependence of the Bulk Modulus and the Anderson-Grüneisen Parameter δ of Oxide Compounds, J. Phys. Chem. Solids 28 (1967) 697. 35. Chapter 5; R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,
Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998) 7871. 36. V.I. Koshchenko, Ya. Kh. Grinberg and A.F. Demidenko, Thermodynamic Properties of AlN (5 - 2700 ºK), GaP (5 - 1500 ºK), and BP (5 - 800 ºK), Inorg. Mater. 20 (1984) 1550 (Translated from Izv. Akad. Nauk SSSR, Neorg. Mater. 20 (1984) 1787). 37. J.C. Nipko and C.-K. Loong, Phonon Excitations and Related Thermal Properties of Aluminum Nitride, Phys. Rev. B 57 (1998) 10550. 38. I.Ya. Guzman, A.F. Demidenko, V.I. Koshchenko, M.S. Fraifel'd and Yu. V.
Egner, Specific Heats and Thermodynamic Functions of Si3N4 and Si2ON2,
150 The Young's modulus of MgSiN2, AlN and Si3N4
Inorg. Mater. 12 (1976) 1546 (Translated from Izv. Akad. Nauk SSSR, Neorg. Mater. 12 (1976) 1879). 39. Personal communication J.C. van Miltenburg (unpublished results). 40. K. Watari, Y. Seki and K. Ishizaki, Temperature Dependence of Thermal Coefficients for HIPped Sintered Silicon Nitride, J. Ceram. Soc. Jpn. Inter. Ed. 97 (1989) 170.
151 Chapter 6.
152 Chapter 7.
The Grüneisen parameters of MgSiN2, AlN and
β-Si3N4
Abstract
The temperature dependence of the Grüneisen parameter of MgSiN2 (80 - 1600 K),
AlN (90 - 1600 K), and β-Si3N4 (300 - 1300 K) was evaluated from thermal expansion, elastic constants and heat capacity data of these materials. For all compounds the Grüneisen parameter increases as a function of the reduced temperature approaching a constant value at high temperatures (T /θ ≥ 0.8). The high temperature limit of the Grüneisen parameter of the wurtzite type materials
MgSiN2 and AlN is about the same (0.98 and 0.95, respectively) whereas these are much higher than that of the phenakite β-Si3N4 (0.63). This behaviour can be understood quantitatively from the relation between the Grüneisen parameter and the bond parameter W as established by Slack.
1. Introduction
The ternary adamantine type compound MgSiN2, which can be deduced from the well known AlN structure by replacing two Al3+ ions systematically by one Mg2+ and one Si4+ ion, is considered a potentially interesting material because of its mechanical and thermal properties [1 - 9]. AlN is known for its high intrinsic thermal conductivity (320 W m-1 K-1 [1, 10] at 300 K) and therefore intensively studied. Recently a very high thermal conductivity was reported for β-Si3N4 ceramics (> 100 W m-1 K-1 at 300 K) [11 - 16]. In order to understand the thermal conductivity of these nitrides it is important to obtain more insight in the (thermal)
153 Chapter 7.
properties of these materials. Furthermore, a better understanding of the thermal properties might result in more reliable models to predict the intrinsic thermal conductivity of new potentially interesting materials. An important parameter for the theoretical prediction of the intrinsic thermal conductivity κ is the Grüneisen parameter at the Debye temperature γθ [-] [17 - 20] as: M δθ 3 κ ∝ (1) 2 2 3 γ θ n in which M [kg mol-1] is the mean atomic mass, δ 3 [m3] the average volume of one atom in the primitive unit cell, θ [K] the Debye temperature, γθ [-] the Grüneisen parameter at T = θ and n [-] the number of atoms per primitive unit cell. Using the procedure as described by Slack [17] the Grüneisen parameter can be evaluated from thermodynamic properties resulting in the so-called thermodynamic Grüneisen parameter [21]: 3α V 3α V γ = lat m = lat m (2) β β S C p T CV
-1 3 -1 in which α lat [K ] is the lattice linear thermal expansion coefficient, Vm [m mol ] -1 -1 -1 the molar volume, βS [Pa ] the adiabatic compressibility, Cp [J mol K ] the heat -1 capacity at constant pressure, βT [Pa ] the isothermal compressibility and CV [J mol-1 K-1] the heat capacity at constant volume. The Grüneisen parameter, which is related to the anharmonicity of the crystal structure, within the quasi-harmonic approximation [22] is temperature independent. Therefore, the Grüneisen parameter is often calculated at a temperature (in most cases room temperature) for which the thermal expansion coefficient, molar volume, compressibility and heat capacity are known. The resulting Grüneisen parameter is then assumed to equal the Grüneisen parameter at the Debye temperature γθ. However, the few experimental data that are available show that the Grüneisen parameter is a function of temperature [23 - 26] and only becomes constant at high temperatures [27]. Therefore it is essential to evaluate γ as a function of the temperature in order
154 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
to investigate the temperature range for which the Grüneisen parameter can be considered as a constant. As far as the present author knows only one value for the Grüneisen parameter of MgSiN2 was reported (viz. γθ = 0.78 with θ = 827 K [18]). For AlN three values for γ are reported of which one deviates substantially from the others viz. 0.71 (at T = θ = 941 K) [18], 0.77 (at T = θ = 950 K) [10] and 0.45 (room temperature) [28]. The Grüneisen parameter of β-Si3N4 as a function of the absolute temperature (300 - 1300 K) was already reported by Slack [26] and at T =
θ = 1140 K, γθ = 0.72. However, lacking the availability of reliable elastic constants (Young's modulus E [GPa] and Poisson's ratio ν [-]) as a function of the temperature, an estimate of the temperature dependence of the compressibility -1 -4 -1 (-∂lnβS /∂T = 1.0 10 K ) was used. For MgSiN2 all data required for evaluation of the Grüneisen parameter as a function of the temperature have been previously reported (thermal expansion [9, 29] providing also the values for the molar volume
Vm, elastic constants [30] resulting in values for the adiabatic compressibility βS, and heat capacity [31]). For AlN thermal expansion and heat capacity data are known from literature (αlat [32 - 35] and Cp [36 - 42]) and recently elastic constants as a function of the temperature were reported [30]. In the meantime, for β-Si3N4 the temperature dependence of the elastic constants has been determined [43] and modelled [30], and in the literature data for the thermal expansion coefficient
[44, 45] and specific heat are reported [40 - 48]. So, for MgSiN2, AlN and β-Si3N4 all required input parameters as a function of the temperature are available for evaluation of the temperature dependence of the Grüneisen parameter. In this chapter the experimental determination of the thermodynamic
Grüneisen parameter of MgSiN2, AlN and β-Si3N4 is reported. The choice of the used input parameters is briefly discussed and using my best judgement some irregularities were smoothed in order to obtain the most reliable data.
155 Chapter 7.
2. Evaluation of the input parameters
2.1. Lattice linear thermal expansion coefficient α lat
-1 For MgSiN2 the lattice linear thermal expansion coefficient α lat [K ] at T < 300 K was taken from a previous neutron diffraction study [29] concerning the temperature dependence of the lattice parameters. From 300 to 1573 K the linear thermal expansion coefficient α lin of fully dense MgSiN2 ceramics was measured with a dual rod dilatometer in nitrogen using Al2O3 (sapphire) as a reference material. More experimental details are given elsewhere [9]. The length of the ceramic rod as a function of the absolute temperature l(T ) [m] was fitted by a 2 3 4 5 polynomial l(T ) = A0 + A1·T + A2·T + A3·T + A4·T + A5·T (see Table 7-1). Statistical F-testing with a 95% confidence interval of the variance ratios showed that introduction of higher order terms to the polynomial did not improve the fit.
-1 The linear thermal expansion coefficient, α lin [K ], for T ≥ 300 K was calculated from:
α lin(T ) = (dl(T)/dT)/l(T )(3)
Table 7-1: The coefficients and statistics used for describing the dilatometer
MgSiN2 sample length l [mm] as a function of the absolute temperature T [K] between 300 and 1573 K. Between brackets the 95 % confidence intervals are given.
Coefficient Statistics
A0 9.997729(1) Data points 999 -5 2 A1 -1.3375(66) 10 R 0.99999938 -8 2 -14 A2 8.1942(16) 10 χ 7.15 10 -11 A3 3.5418(19) 10 Tolerance 0.0005 -15 A4 1.510(10) 10 Confidence 0.95 -18 A5 3.9463(22) 10
First the thermal expansion coefficient increases with temperature (Fig. 7-1) to become almost constant at about 1000 K, and subsequently it increases again with temperature. In order to obtain a smooth thermal expansion curve around
156 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
room temperature, the values between 250 K (measured by neutron diffraction) and 300 K (measured by dilatometry) were obtained by interpolation. The further increase of the thermal expansion coefficient above 1000 K was ascribed to the thermal generation of vacancies in the MgSiN2 crystal structure [9] causing macroscopic length changes for constant unit cell size, which are measured with dilatometry. Until now this point was only briefly discussed [9]. For evaluation of the Grüneisen parameter discrimination between the lattice unit cell and vacancy thermal expansion is required.
1.0E-05 MgSiN2 8.0E-06 AlN β-Si3N4 6.0E-06 ] -1 4.0E-06 [K α
2.0E-06
0.0E+00
-2.0E-06 0 500 1000 1500 2000 T [K]
Fig. 7-1: The linear thermal expansion coefficient (α ) of MgSiN2,
AlN and β-Si3N4 ceramics as a function of the absolute temperature (T ). The dots indicate the calculated values and the solid lines are drawn as a guide to the eye. The dashed
line represents the measured data for MgSiN2.
In order to do this the macroscopic linear thermal expansion coefficient α lin for T ≥ 750 K was separated in a lattice contribution α lat and a vacancy contribution α vac using the following expression: α =α +α lin lat vac (4) CQv - Qv with α = α ∞ F(θ /T ) and α = exp lat D vac 3kT 2 kT
157 Chapter 7.
-1 in which α ∞ [K ] is the high temperature limit of the lattice linear thermal expansion coefficient, F(θ D/T ) [-] the normalised Debye function (so, for T >> θ D applies F(θ D/T ) = 1), θ D [K] the Debye temperature, C [-] a constant, Qv [eV] the energy required for the formation of a vacancy and k is the Boltzmann's constant (8.62 10-5 eV K-1). An excellent fit of the data is obtained (Table 7-2) with -6 -1 reasonable values for α ∞ (7.502 10 K ), θ D (1248 K), C (19.2) and Qv (1.365 eV). 1 3 For comparison C and Qv have typical values of 10 - 10 (estimated from [49]) and 1 eV [50], respectively. This indicates that the increase of the macroscopic thermal expansion coefficient above 1000 K can indeed be ascribed to thermal generation of vacancies in the MgSiN2 lattice. It is noted that at 750 K the contribution of vacancies to the thermal expansion coefficient is negligibly small.
Table 7-2: The values for α ∞, θ D, C and Qv and statistics used for -1 describing the linear thermal expansion coefficient α lin [K ]
for MgSiN2 as a function of the absolute temperature T [K]. Between brackets the 95 % confidence intervals are given.
Parameter Statistics
-1 -6 α ∞ [K ] 7.502(3) 10 Data points 182 2 θ D [K] 1248(3) R 0.9999 C [-] 19.2(4) χ 2 2.8833 10-17
Qv [eV] 1.365(3) Tolerance 0.0005 Confidence 0.95
For AlN [32 - 35] and β-Si3N4 [44, 45] several thermal expansion data sets are reported. After evaluation, for AlN the thermal expansion data between 0 - 2000 K reported by Wang [35] based on lattice parameters were used. For
β-Si3N4 the thermal expansion coefficient between 293 - 1300 K was calculated from the dilatometer data of Huseby [45]. Also for AlN and β-Si3N4 the lattice thermal expansion coefficient increases as a function of the temperature approaching a constant value at high temperatures (Fig. 7-1).
158 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
2.2. Molar volume Vm
The molar volume Vm of MgSiN2, AlN and β-Si3N4 as a function of the temperature was evaluated from literature data of the molar volume at temperature
T (Vm T) and the lattice linear thermal expansion (α lat) using:
T* V * m T = α ln ∫3 latdT (5) V m T T
-5 3 -1 -5 3 -1 For MgSiN2 Vm0 = 2.559 10 m mol [29], for AlN Vm0 = 1.258 10 m mol [35] -5 3 -1 and for β-Si3N4 Vm293 = 4.381 10 m mol [44]. Since for β-Si3N4 thermal expansion data are reported for room temperature and above the molar volume for
β-Si3N4 could be calculated only in the same temperature region.
2.3. Adiabatic compressibility βS
For evaluation of the adiabatic compressibility βS elastic constants are needed. The temperature dependence of the Young's modulus of MgSiN2 and AlN between 293 - 973 K [30] has been measured with the impulse excitation method [51] and for β-Si3N4 literature data (293 - 1223 K) [43] were used. These Young's modulus data obtained for MgSiN2 and AlN, and the literature data for β-Si3N4, can be very well described [30] using the empirical formula [52]: = ⋅ E E0 - B T exp (-T0 /T ) (6)
-1 in which E0 [GPa] is the Young's modulus at 0 K, and B [GPa K ] and T0 [K] are fitting parameters. For all temperatures this expression was used to describe the -1 Young's modulus of MgSiN2 (E0 = 281 GPa, B = 0.0220 GPa K and T0 = 389 K), -1 AlN (E0 = 314 GPa, B = 0.0244 GPa K and T0 = 495 K) and β-Si3N4 -1 (E0 = 320 GPa, B = 0.0151 GPa K and T0 = 445 K). The Poisson's ratio for these materials was assumed to be constant (for MgSiN2: ν = 0.246 [30], AlN: ν = 0.245
[53] and β-Si3N4: ν = 0.267 [43] at room temperature).
159 Chapter 7.
For all temperatures (80 - 1600 K: MgSiN2, 90 - 1600 K: AlN and -1 300 - 1300 K β-Si3N4) the adiabatic compressibility βS [Pa ] was calculated from the Young's modulus and the Poisson's ratio using: 3(1 - 2ν) β = (7) S E
The resulting adiabatic compressibility βS of MgSiN2, AlN and β-Si3N4 shows a slight increase as a function of the absolute temperature (Fig. 7-2).
6.5E-12
MgSiN2 6.0E-12 AlN β-Si3N4
] 5.5E-12 -1 [Pa S
β 5.0E-12
4.5E-12
4.0E-12 0 500 1000 1500 2000 T [K]
Fig. 7-2: The adiabatic compressibility (βS) of MgSiN2, AlN and
β-Si3N4 as a function of the absolute temperature (T ). The dots indicate the calculated values and the lines are drawn as a guide to the eye.
2.4. Heat capacity at constant pressure Cp
Below 400 K the heat capacity at constant pressure Cp has been previously determined for MgSiN2 (8 - 400 K) [31], AlN and Si3N4 (20 - 400 K) [42]) with adiabatic calorimetry (CAL V [54]). The Cp of MgSiN2 was measured on pure (oxygen content < 0.1 wt. %), iso-statically pressed powder compacts (~ 35 % of theoretical density), that of AlN on crushed hot-pressed pellets (fully dense) prepared from Dow Chemical Aluminum Nitride Powder - XUS 35560, and that of
Si3N4 (Ube SN-E10) on crushed slip cast compacts sintered at high temperature in
160 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
N2 pressure furnace (~ 50 % of theoretical density). The sample was put into a sample holder (copper vessel) and sealed using He gas as a heat exchanger. After cooling of the sample and sample holder to the lowest measuring temperature, stepwise a known quantity of energy was added and the temperature increase was measured.
For low temperatures (T ≤ 350 K) these experimental Cp data for MgSiN2,
AlN and β-Si3N4 were used. For T ≥ 350 K the Cp for MgSiN2, AlN and β-Si3N4 was calculated with the Debye expression for the heat capacity at constant volume
CV [55, 56] and the expression for Cp - CV : 9V Tα 2 C - C = m lat (8) p V β T
The isothermal compressibility βT that was needed for obtaining Cp - CV was calculated from the adiabatic compressibility βS with: 9V Tα 2 β = β + m lat T S (9) C p
The Debye temperatures (θ∞) needed to describe the CV were evaluated using the same procedure as previously described for MgSiN2 [31] resulting in 996 K, 989 K and 1200 K for MgSiN2, AlN and β-Si3N4, respectively.
The Cp data calculated for MgSiN2 are in good agreement with experimentally observed data (300 - 800 K), measured with differential scanning calorimetry [31]. For AlN and Si3N4 the heat capacity is reported in a broad temperature range (AlN: 5 - 2700 K [36]; 291 - 577 K [37]; 300 - 773 K [38];
0 - 800 K [39]; and β-Si3N4: 55 - 310 K [46]; 100 - 1250 K [47]; 0 - 680 K [48]) and presented in standard thermodynamic handbooks (AlN: 298 - 2000 K [40];
0 - 3000 K [41]; and Si3N4: 298 - 2200 K [40]; 298 - 3000 K [41]). No large discrepancy was observed between the several heat capacity data of AlN and the here presented experimental and calculated data. The data of β-Si3N4 deviate substantially from the data presented in the standard thermodynamic handbooks [40, 41] but agree very well with the data presented by Guzman [46], Watari [47]
161 Chapter 7.
and Rocabois [48]. The resulting Cp curves of MgSiN2, AlN and β-Si3N4 show the expected S shaped increase with the absolute temperature (Fig. 7-3).
180
β-Si3N4
150 MgSiN2 AlN ]
-1 120 K -1 90 [J [J mol
p 60 C
30
0 0 500 1000 1500 2000 T [K]
Fig. 7-3: The heat capacity at constant pressure (Cp) of MgSiN2,
AlN and β-Si3N4 as a function of the absolute temperature (T ). The dots indicate the calculated values and the lines are drawn as a guide to the eye.
3. Evaluation of the Grüneisen parameter γ
The Grüneisen parameters of MgSiN2 between 80 and 1600 K, AlN between
90 and 1600 K, and β-Si3N4 between 300 and 1300 K were calculated from α lat,
Vm, βS and Cp. The Grüneisen parameter of all three materials roughly has the same temperature dependence, showing an increase of the Grüneisen parameter with increasing temperature approaching a constant value at high temperatures (Fig. 7-4). For all three materials the reliability of the absolute value of the Grüneisen parameter was estimated to be within 10 % for T ≥ 300 K and 15 - 20 % for T < 300 K considering the accuracy of the used input parameters. However, the internal consistency of the resulting Grüneisen parameter is considered to be within 5 % providing a good indication of the true temperature dependence.
For MgSiN2 the Grüneisen parameter shows a minimum at about 125 K, subsequently it significantly increases till 300 K and finally it becomes constant for
162 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
T ≥ 600 K having a value of 0.96 ± 0.02 (Table 7-3 and Fig. 7-4). The high temperature limit is significantly higher than the previously reported value of 0.78 [18]. This difference is mainly caused by the too high input value used for the isothermal compressibility that originates from a too low value measured for the Young's modulus (see also [30]).
Table 7-3: The lattice linear thermal expansion coefficient α lat, molar volume Vm,
the heat capacity at constant pressure Cp, adiabatic compressibility βS,
and the Grüneisen parameter γ for MgSiN2 at several temperatures.
T α lat Vm Cp βS γ [K] [K-1] [m3 mol-1] [J mol-1 K-1] [Pa-1] [-] 80 0.426 10-6 2.5590 10-5 8.59 5.421 10-12 0.701 100 0.617 10-6 2.5591 10-5 13.91 5.422 10-12 0.628 110 0.725 10-6 2.5592 10-5 16.65 5.422 10-12 0.617 120 0.842 10-6 2.5592 10-5 19.48 5.423 10-12 0.612 130 0.967 10-6 2.5593 10-5 22.34 5.424 10-12 0.613 140 1.10 10-6 2.5594 10-5 25.18 5.425 10-12 0.619 150 1.24 10-6 2.5595 10-5 28.00 5.426 10-12 0.628 200 2.08 10-6 2.5601 10-5 41.20 5.433 10-12 0.714 250 3.13 10-6 2.5611 10-5 52.42 5.444 10-12 0.843 300 3.82 10-6 2.5625 10-5 61.71 5.456 10-12 0.873 350 4.30 10-6 2.5641 10-5 69.06 5.470 10-12 0.876 400 4.73 10-6 2.5658 10-5 75.04 5.486 10-12 0.884 450 5.11 10-6 2.5677 10-5 79.57 5.503 10-12 0.899 500 5.44 10-6 2.5697 10-5 83.07 5.520 10-12 0.915 550 5.74 10-6 2.5719 10-5 85.85 5.539 10-12 0.931 600 5.99 10-6 2.5742 10-5 88.08 5.558 10-12 0.945 650 6.20 10-6 2.5765 10-5 89.92 5.577 10-12 0.956 700 6.38 10-6 2.5789 10-5 91.45 5.597 10-12 0.965 800 6.65 10-6 2.5840 10-5 93.85 5.638 10-12 0.975 900 6.83 10-6 2.5892 10-5 95.63 5.681 10-12 0.977 1000 6.95 10-6 2.5946 10-5 96.99 5.725 10-12 0.975 1200 7.11 10-6 2.6057 10-5 98.88 5.816 10-12 0.966 1400 7.22 10-6 2.6174 10-5 100.05 5.936 10-12 0.958 1600 7.27 10-6 2.6307 10-5 100.72 6.012 10-12 0.947
163 Chapter 7.
For AlN the Grüneisen parameter is negative at 100 K and shows a steep increase till 200 K, subsequently it slowly increases till 700 K and finally for higher temperatures T ≥ 700 K it is about constant equalling 0.95 ± 0.02 (Table 7-4 and Fig. 7-4). The Grüneisen parameters determined in this work for AlN (0.95 at T = 940 - 950 K and 0.70 at T = 300 K) deviates significantly from the value of 0.71 at 941 K previously reported by de With et al. [18], and the values reported by Slack et al. of 0.77 at 950 K [10] and 0.45 at room temperature [28]. The value of -6 -1 γ = 0.71 [18] was based on a mistake taking α lat = 4.8 10 K instead of -6 -1 α lat = 5.9 10 K at T = θ as reported in his reference for the thermal expansion coefficient [33].
1.2
MgSiN2 AlN 1.0 β-Si3N4
0.8
0.6
[-] 0.4 γ
0.2
0.0 300 600 900 1200 1500 1800
-0.2 T [K]
-0.4
Fig. 7-4: The Grüneisen parameter (γ ) of MgSiN2, AlN and β-Si3N4 as a function of the absolute temperature (T ). The dots indicate the calculated values and the lines are drawn as a guide to the eye.
Correction results in γ = 0.87 which is in much closer agreement with the value of 0.95 determined in this work. The low values of 0.77 [10] and 0.45 [28] can be ascribed to the too low Poisson's ratio (ν = 0.179 [28]) reported by Slack resulting
164 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
in a too high value for βS, and consequently, in a too small Grüneisen parameter. Taking a more realistic value for the Poisson's ratio (ν = 0.245 [53, 57]) results in γ = 0.99 at 950 K and γ = 0.58 at room temperature, which is in far better agreement with our value of 0.95 at 950 K and 0.70 at 300 K.
Table 7-4: The lattice linear thermal expansion coefficient α lat, molar volume Vm, the
heat capacity at constant pressure Cp, adiabatic compressibility βS, and the Grüneisen parameter γ for AlN at several temperatures.
T α lat Vm Cp βS γ [K] [K-1] [m3 mol-1] [J mol-1 K-1] [Pa-1] [-] 90 -0.140 10-6 1.2580 10-5 4.52 4.873 10-12 -0.241 100 -0.009 10-6 1.2580 10-5 5.77 4.873 10-12 -0.012 150 0.84 10-6 1.2580 10-5 12.57 4.875 10-12 0.520 200 1.62 10-6 1.2583 10-5 19.50 4.879 10-12 0.643 250 2.24 10-6 1.2586 10-5 25.58 4.886 10-12 0.677 300 2.77 10-6 1.2591 10-5 30.64 4.895 10-12 0.698 350 3.25 10-6 1.2597 10-5 34.66 4.905 10-12 0.723 400 3.68 10-6 1.2603 10-5 37.68 4.917 10-12 0.753 450 4.07 10-6 1.2611 10-5 39.83 4.930 10-12 0.785 500 4.42 10-6 1.2619 10-5 41.56 4.944 10-12 0.814 550 4.72 10-6 1.2627 10-5 42.92 4.959 10-12 0.840 600 4.98 10-6 1.2637 10-5 44.01 4.974 10-12 0.863 650 5.21 10-6 1.2646 10-5 44.92 4.990 10-12 0.882 700 5.41 10-6 1.2656 10-5 45.67 5.007 10-12 0.899 750 5.58 10-6 1.2667 10-5 46.30 5.024 10-12 0.912 800 5.73 10-6 1.2677 10-5 46.85 5.041 10-12 0.923 900 5.98 10-6 1.2700 10-5 47.72 5.077 10-12 0.940 1000 6.17 10-6 1.2723 10-5 48.40 5.115 10-12 0.951 1100 6.32 10-6 1.2747 10-5 48.93 5.153 10-12 0.958 1200 6.43 10-6 1.2772 10-5 49.34 5.193 10-12 0.962 1300 6.53 10-6 1.2796 10-5 49.67 5.234 10-12 0.964 1400 6.60 10-6 1.2821 10-5 49.93 5.276 10-12 0.964 1500 6.67 10-6 1.2846 10-5 50.12 5.318 10-12 0.964 1600 6.72 10-6 1.2872 10-5 50.26 5.362 10-12 0.963
165 Chapter 7.
Table 7-5: The lattice linear thermal expansion coefficient α lat, molar volume Vm,
the heat capacity at constant pressure Cp, adiabatic compressibility βS,
and the Grüneisen parameter γ for β-Si3N4 at several temperatures.
T α lat Vm Cp βS γ [K] [K-1] [m3 mol-1] [J mol-1 K-1] [Pa-1] [-] 300 1.19 10-6 4.3807 10-5 90.68 4.377 10-12 0.393 350 1.59 10-6 4.3816 10-5 104.93 4.383 10-12 0.454 400 1.94 10-6 4.3828 10-5 115.91 4.390 10-12 0.500 450 2.23 10-6 4.3842 10-5 125.77 4.398 10-12 0.529 500 2.47 10-6 4.3857 10-5 133.48 4.406 10-12 0.552 550 2.66 10-6 4.3874 10-5 139.59 4.414 10-12 0.569 600 2.83 10-6 4.3892 10-5 144.52 4.423 10-12 0.582 650 2.96 10-6 4.3911 10-5 148.54 4.432 10-12 0.592 700 3.07 10-6 4.3931 10-5 151.88 4.441 10-12 0.600 750 3.17 10-6 4.3952 10-5 154.67 4.450 10-12 0.607 800 3.25 10-6 4.3973 10-5 157.04 4.460 10-12 0.612 900 3.37 10-6 4.4016 10-5 160.82 4.479 10-12 0.618 1000 3.47 10-6 4.4061 10-5 163.67 4.499 10-12 0.622 1100 3.54 10-6 4.4108 10-5 165.88 4.519 10-12 0.624 1200 3.59 10-6 4.4155 10-5 167.62 4.540 10-12 0.625 1300 3.64 10-6 4.4203 10-5 169.01 4.561 10-12 0.626
For β-Si3N4 the Grüneisen parameter increases up to 700 K whereas above this temperature it is about constant and equals 0.63 ± 0.02 (Table 7-5 and Fig. 7-4). The observed temperature dependence of the Grüneisen parameter is in good agreement with the previously reported temperature dependence [26]. Also the observed Grüneisen parameter at T = θ∞ = 1200 K of 0.63 is in reasonable agreement with the previously reported value of 0.72 [26].
4. Discussion
4.1. The temperature dependence of the Grüneisen parameter
For comparison, the reduced Grüneisen parameter (γ /γθ) as a function of the reduced temperature (T/θ ) is presented in Fig. 7-5 for MgSiN2, AlN and β-Si3N4.
166 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
More or less a similar dependence of γ /γθ on the reduced temperature is found indicating its universal behaviour, as it has also been observed for Ar [58], ZnTe and RbI [24], Ge and Si [59], and several phenakites (Be2SiO4, Zn2SiO4 and
α-LiAlSiO4) [26].
1.2 MgSiN2 AlN
1.0 β-Si3N4
0.8
0.6 [-] θ
γ 0.4 / γ
0.2
0.0 0.5 1.0 1.5 2.0 -0.2 T/θ [-]
-0.4
Fig. 7-5: The reduced Grüneisen parameter (γ /γθ) of MgSiN2, AlN and β-
Si3N4 as a function of the reduced temperature (T /θ ), with γθ equal to 0.98, 0.95 and 0.63 and θ equal to 996, 989 and 1200 K for
MgSiN2, AlN and Si3N4, respectively. The dots indicate the calculated values and the lines are drawn as a guide to the eye.
The Grüneisen parameters of MgSiN2, AlN and β-Si3N4 are almost constant for T /θ ≥ 0.8, at lower reduced temperatures (0.3 ≤ T /θ ≤ 0.8) it slowly decreases followed by a faster decrease below about T /θ ≈ 0.3, and finally at still lower temperatures (T /θ ≤ 0.15) different temperature dependencies are observed
(Fig. 7-5). For MgSiN2 the Grüneisen parameter shows a (positive) minimum value of about 0.61 at T/θ ≈ 0.13. Also the Grüneisen parameter for AlN shows a (negative) minimum at lower temperatures (0.03 < T/θ < 0.08) as the thermal
167 Chapter 7.
expansion coefficient near 0 K has again a positive value. For several other adamantine type compounds (Si, Ge, GaAs, GaSb, InAs and InSb) this (negative) minimum in the Grüneisen parameter was also observed at about T /θ ≈ 0.04 - 0.07
[23]. It is important to note that the usual assumption that γθ = γ300 is not valid for
MgSiN2, AlN and β-Si3N4 (Fig. 7-5) as these materials have a high Debye temperature (~ 1000 - 1200 K [30]).
Table 7-6: The average atomic mass M , density ρ, average volume per atom δ 3, coordination number of the anions η, the average volume per anion bond W
and the Grüneisen parameter at T = θ γθ for several materials.
3 Material M ρ δ η W γθ [kg] [kg m3] [Å3] [-] [Å3] [-]
β-Si3N4 0.020041 3202 10.636 [44] 3 6.056 0.63 AlN 0.020495 3256 10.453 [35] 4 5.226 0.95
MgSiN2 0.020103 3138 10.391 [29] 4 5.313 0.98
α-Al2O3 0.02039 3986 [52] 8.494 4 3.539 1.34 [52] MgO 0.02016 3581 [52] 9.345 6 3.115 1.52 [52]
ThO2 0.08801 9991 [52] 14.626 4 5.485 1.78 [52]
4.2. The absolute value of the Grüneisen parameter at the Debye temperature
The Grüneisen parameters at the Debye temperature γθ for MgSiN2 (0.98) and AlN
(0.95) are about the same whereas for Si3N4 (0.63) it is considerably smaller
(Fig. 7-4). Furthermore, for MgSiN2 and AlN the absolute value of the Grüneisen parameter as a function of the reduced temperature (T /θ ) is more or less similar for T /θ ≥ 0.2 (Fig. 7-5). This is not surprising if we consider the similarity in crystal structure, bond character, mean atomic volume and average mole mass per atom.
168 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
The differences between γθ for MgSiN2, AlN and Si3N4 can be quantitatively explained in view of the empirical linear relation between the Grüneisen parameter 3 at T = θ (γθ) and the average volume per anion bond (W [Å ]) as found by Slack [26]:
γθ = Γ∞[1 - W/W0] (10) 3 with Γ∞ [-] and W0 [m ] constants and the parameter W defined as: 3 W = stδ /saη (11) 3 3 in which st [-] is the number of atoms per formula unit, δ [Å ] is the average volume per atom, sa [-] the number of anions per molecule and η [-] the number of bonds per anion. Using the data of Slack [26], some literature data for Al2O3, MgO and ThO2 [52], and our data for MgSiN2, AlN and Si3N4 an updated plot of γθ versus W was constructed (Table 7-6, Fig. 7-6). Again a linear relation is observed
2.0 Slack [26] ThO2 Wachtman [52] This work 1.5 MgO
α-Al2O3 BeO
[-] Be2SiO4
θ 1.0 MgSiN2
γ AlN ZnO
β-Si3N4 { β-SiAlON α-LiAlSiO 0.5 4 Zn2SiO4 CdAl2O4 Zn2GeO4
β-SiO α-SiO2 2 0.0 246810 W [Å3]
Fig. 7-6: The Grüneisen parameter at T = θ, γθ versus the volume per anion bond W for several materials.
between γθ and W (disregarding the point of ThO2) with Γ∞ = 2.11 and 3 -3 W0 = 9.45 Å (dγθ /dW = -0.22 Å ). The value of W0 equals the reported value by 3 Slack (W0 = 9.45 Å [26]). However, his reported values of Γ∞ = 2.91 and
169 Chapter 7.
-3 dγθ /dW = -0.31 Å appear to be erroneous as his figure of γθ versus W results in a
-3 value for Γ∞ = 2.41 ± 0.05 and dγθ /dW = -0.26 ± 0.01 Å which is close to the values here presented. So, the general validity of the relation observed by Slack is supported by the inclusion of the data for the oxides Al2O3 and MgO, and the nitride materials MgSiN2, AlN and Si3N4. The point for ThO2 does most probably not fit the relation due to the large ionic radius of Th resulting in an inverse structure where the cation is larger than the anion providing a too high value for δ 3.
5. Conclusions
The reduced Grüneisen parameters (γ /γθ) of MgSiN2, AlN and β-Si3N4 show a similar behaviour as a function of the reduced temperature (T /θ ). The Grüneisen parameter increases as a function of the temperature approaching a constant value at a reduced temperature of T /θ ≥ 0.8 indicating that the usual assumption that
γθ = γ300 is not valid for these materials as θ ≈ 1000 - 1200 K. The absolute value of the Grüneisen parameter of AlN at the Debye temperature (0.96) equals that of the structurally related MgSiN2 (0.98), whereas it is much higher than that of Si3N4 (0.63). This can be quantitatively understood from the relationship between the Grüneisen parameter and the average volume per anion bond, established for other (most oxide) compounds.
References
1. G.A. Slack, Nonmetallic Crystals with High Thermal Conductivity, J. Phys. Chem. Solids 34 (1973) 321. 2. W.A. Groen, M.J. Kraan, and G. de With, Preparation, Microstructure and
Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413. 3. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent
Ceramics MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non- Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S.
170 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh, 1994) 239. 4. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen and M.J. Kraan, Hot-
pressing of MgSiN2 ceramics, J. Mater. Sci. Lett. 13 (1994) 1314. 5. H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Densification of MgSiN2, Ceram. Trans. 51, Int. Conf. Cer. Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995) 585.
6. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics 2, Basic Science - Developments in Processing of Advanced Ceramics - Part II, Faenza, Italy, October 1995, edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 289. 7. I.J. Davies, H. Uchida, M. Aizawa and K. Itatani, Physical and Mechanical Properties of Sintered Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 40. 8. I.J. Davies, T. Shimazaki, M. Aizawa, H. Suemasu, A. Nozue and K. Itatani, Physical Properties of Hot-Pressed Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 276. 9. Chapter 3; R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar,
Preparation, Characterisation and Properties of MgSiN2 ceramics, to be published. 10. G.A. Slack, R.A. Tanzilli, R.O. Pohl and J.W. Vandersande, The Intrinsic Thermal Conductivity of AlN, J. Phys. Chem. Solids 48 (1987) 641. 11. K. Hirao, K. Watari, M.E. Brito, M. Toriyama and S. Kanzaki, High Thermal Conductivity of Silicon Nitride with Anisotropic Microstructure, J. Am. Ceram. Soc. 79 (1996) 2485. 12. N. Hirosaki, Y. Okamoto, M. Ando, F. Munakata and Y. Akimune, Thermal Conductivity of Gas-Pressure-Sintered Silicon Nitride, J. Am. Ceram. Soc. 79 (1996) 2878.
171 Chapter 7.
13. N. Hirosaki, M. Ando, Y. Okamoto, F. Munakata, Y. Akimunde, K. Hirao, K. Watari, M.E. Brito, M. Toriyama and S. Kanzaki, Effect of Alignment of Large Grains on the Thermal Conductivity of Self-Reinforced β-Silicon Nitride, J. Ceram. Soc. Jpn. Int. Ed. 104 (1996) 1187. 14. K. Watari, M.E. Brito, M. Toriyama, K. Ishizaki, S. Cao and K. Mori,
Thermal conductivity of Y2O3-doped Si3N4 ceramics at 4 to 1000 K, J. Mater. Sci. Lett. 18 (1999) 865. 15. K. Watari, K. Hirao, M.E. Brito, M. Toriyama and S. Kanzaki, Hot Isostatic
Pressing to Increase Thermal Conductivity of Si3N4 Ceramics, J. Mater. Res. 14 (1999) 1538. 16. B. Li, L. Pottier, J.P. Roger, D. Fournier, K. Watari and K. Hirao, Measuring the Anisotropic Thermal Diffusivity of Silicon Nitride Grains by Thermoreflectance Microscopy, J. Eur. Ceram. Soc. 19 (1999) 1631. 17. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1979) 1. 18. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)-Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of Properties and Performance by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405. 19. G. Leibfried and E. Schlömann, Wärmeleitung in elektrisch isolierenden Kristallen, Nachr. Akad. Wiss. Göttingen, Math. Physik. Kl. IIa (1954) 71. 20. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modelling of the Thermal th Diffusivity/Conductivity of MgSiN2 Ceramics, ITCC 24 and ITES 12, 24 International Thermal Conductivity Conference and 12th International Thermal Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 1997, edited by P.S. Gaal and D.E. Apostolescu (Technomics Publishing Co., Inc., Lancaster, Pennsylvania, 1999) 3. 21. E. Grüneisen, Zustand des festen Körpers, Handbuch der Physik 10, edited by H. Geiger and K. Scheel (Springer, Berlin, Germany, 1926), 1.
172 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
22. J.C. Slater, Introduction to Chemical Physics (McGraw-Hill, New York, 1939) Chap. XIII. 23. P.W. Sparks and C.A. Swenson, Thermal Expansions from 2 to 40 ºK of Ge, Si, and Four III-V Compounds, Phys. Rev. 163 (1967) 779. 24. J.F. Vetelino, K.V. Namjoshi and S.S. Mitra, Mode-Grüneisen Parameters and Thermal-Expansion Coefficient of NaCl, CsCl, and Zinc-Blende-Type Crystals, J. Appl. Phys. 41 (1970) 5141. 25. J.F. Vetelino, S.S. Mitra and K.V. Namjoshi, Lattice Dynamics of ZnTe: Phonon Dispersion, Multiphonon Infrared Spectrum, Mode Grüneisen Parameters, and Thermal Expansion, Phys. Rev. B 2 (1970) 967.
26. G.A. Slack and I.C. Huseby, Thermal Grüneisen parameters of CdAl2O4,
β -Si3N4, and other phenacite-type compounds, J. Appl. Phys. 53 (1982) 6817. 27. W.B. Daniels, The anomalous thermal expansion of germanium, silicon and compounds crystallizing in the zinc blende structure, International Conference on the Physics of Semiconductors, Exeter, UK, July 1962, edited by A.C. Stickland (Institute of Physics, London, 1962) 482. 28. D. Gerlich, S.L. Dole and G.A. Slack, Elastic Properties of Aluminum Nitride, J. Phys. Chem. Solids 47 (1986) 437. 29. Chapter 4; R.J. Bruls, H.T. Hintzen, R. Metselaar and C.-K. Loong,
Anisotropic thermal expansion of MgSiN2 from 10 to 300 K as measured by neutron diffraction, J. Phys. Chem. Solids. 61 (2000) 1285. 30. Chapter 6; R. J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The
temperature dependence of the Young's modulus of MgSiN2, AlN and Si3N4, accepted for publication in J. Eur Ceram. Soc. 31. Chapter 5; R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,
Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998) 7871. 32. G.A. Slack and S.F. Bartram, Thermal expansion of some diamondlike crystals, J. Appl. Phys. 46 (1975) 89.
173 Chapter 7.
33. Y.S. Touloukian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, Thermophysical Properties of Matter 13, Thermal Expansion of Nonmetallic Solids, IFI/Plenum, New York, USA, 1977. 34. W. Werdecker and F. Aldinger, Aluminum Nitride - An Alternative Ceramic Substrate for High Power Applications in Microcircuits, IEEE Trans. Components, Hybrids, Manuf. Technol. CHMT-7 (1984) 399. 35. K. Wang and R.R. Reeber, Thermal Expansion of GaN and AlN, Mat. Res. Soc. Symp. Proc. 482, Nitride Semiconductors, Boston, Massachusetts, USA, December 1 - 5 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S. Nakamura and S. Strite, (Materials Research Society, Warrendale, Pennsylvania, 1998) 863. 36. V.I. Koshchenko, Ya.Kh. Grinberg and A.F. Demidenko, Thermodynamic Properties of AlN (5 - 2700 ºK), GaP (5 - 1500 ºK), and BP (5 - 800 ºK), Inorg. Mater. 20 (1984) 1550 (Izv. Akad. Nauk SSSR, Neorg. Mater. 20 (1984) 1787). 37. T.B. Jackson, K.Y. Donaldson and D.P.H. Hasselman, Temperature Dependence of the Thermal Diffusivity/Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 73 (1990) 2511. 38. P. S. de Baranda, A.K. Knudsen and E. Ruh, Effect of CaO on the Thermal Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 76 (1993) 1751. 39. S.N. Ivanov, P.A. Popov, G.V. Egorov, A.A. Sidorov, B.I. Kornev, L.M. Zhukova and V.P. Ryabov, Thermophysical properties of aluminum nitride ceramics, Phys. Solid State 39 (1997) 81. 40. I. Barin, Thermochemical Data of Pure Substances, second edition, VCH, Weinheim, FRG (1993). 41. M.W. Chase Jr., C.A. Davies, J.R. Downey Jr., D.J. Frurip, R.A. McDonald and A.N. Syverud, JANAF Thermochemical Tables, third edition, American Chemical Society and American Institute of Physics, Washington, USA (1985). 42. R.J. Bruls and J.C. van Miltenburg, unpublished results.
174 The Grüneisen parameters of MgSiN2, AlN and β-Si3N4
43. I. Tomeno, High Temperature Elastic Moduli of Si3N4 Ceramics, Jpn. J. Appl. Phys. 20 (1981) 1751. 44. C.M.B. Henderson and D. Taylor, Thermal Expansion of the Nitrides and Oxynitride of Silicon in Relation to their Structures, Trans. J. Br. Ceram. Soc. 74 (1975) 49.
45. I.C. Huseby, G.A. Slack and R.H. Arendt, Thermal Expansion of CdAl2O4,
β -Si3N4, and Other Phenacite-Type Compounds, Bull. Am. Ceram. Soc. 60 (1981) 919. 46. I. Ya. Guzman, A.F. Demidenko, V.I. Koshchenko, M.S. Fraifel'd and Yu.V.
Egner, Specific Heats and Thermodynamic Functions of Si3N4 and Si2ON2, Inorg. Mater. 12 (1976) 1546. (Izv. Akad. Nauk SSSR, Neorg. Mater. 12 (1976) 1879). 47. K. Watari, Y. Seki and K. Ishizaki, Temperature Dependence of Thermal Coefficients for HIPped Sintered Silicon Nitride, J. Ceram. Soc. Jpn. Inter. Ed. 97 (1989) 170. 48. P. Rocabois, C. Chatillon and C. Bernard, Thermodynamics of the Si-O-N System: High-Temperature Study of the Vaporization Behaviour of Silicon Nitride by Mass Spectrometry, J. Am. Ceram. Soc. 79 (1996) 1351. 49. N.F. Mott and R.W. Gurney, Electronic processes in ionic crystals, edited by R.H. Fowler and P. Kapitza, Chapter II, Lattice defects in thermal equilibrium (Oxford, UK, Clarendon Press, 1940), pp. 27 - 33. 50. J.H. Crawford, Jr. and L.M. Slifkin, Point Defects in Solids 1, General and Ionic Crystals (Plenum Press, New-York, USA, 1972), Chapter 1, Statistical Thermodynamics of Point Defects in Crystals by A.D. Franklin. 51. K. Heritage, C. Frisby and A. Wolfenden, Impulse excitation technique for dynamic flexural measurements at moderate temperatures, Rev. Sci. Instr. 59 (1988) 973. 52. J.B. Wachtman, Jr., W.E. Tefft, D.G. Lam, Jr., and C.S. Apstein, Exponential Temperature Dependence of Young's Modulus for Several Oxides, Phys. Rev. 122 (1961) 1754.
175 Chapter 7.
53. P. Boch, J.C. Glandus, J.C. Jarrige, J.P. Lecompte and J. Mexmain, Sintering, Oxidation and Mechanical Properties of Hot Pressed Aluminium Nitride, Ceram. Int. 8 (1982) 34. 54. J.C. van Miltenburg, G.J.K. van den Berg and M.J. van Bommel, Construction of an adiabatic calorimeter. Measurements on the molar heat capacity of synthetic sapphire and n-heptane. J. Chem. Thermodyn. 19 (1987) 1129. 55. P. Debije, Zur Theorie der spezifischen Wärmen, Ann. Physik 39 (1912) 789. 56. See for numerical calculation W.M. Rogers and R.L. Powell, Tables of Transport Integrals, Natl. Bur. Stand. Circ. 595 (1958) 1. 57. A.A. Kahn and J.C. Labbe, Aluminium Nitride - Molybdenum CeramicMatrix Composites. Influence of Molybdenum Addition on Electrical, Mechanical and Thermal Properties, J. Eur. Ceram. Soc. 17 (1997) 1885. 58. O.G. Peterson, D.N. Batchelder and R.O. Simmons, Measurements of X-ray Lattice Constant, Thermal Expansivity, and Isothermal Compressibility of Argon Crystals, Phys. Rev. 150 (1966) 703. 59. R.R. Reeber, Thermal Expansion of Some Group IV Elements and ZnS, Phys. Stat. Sol. a 32 (1975) 321.
176 Chapter 8.
Theoretical thermal conductivity of MgSiN2, AlN and
β-Si3N4 using Slack's equation
Abstract
The maximum achievable thermal conductivity of MgSiN2, AlN and β-Si3N4 ceramics is estimated based on the theory of Slack. Using this procedure the estimate obtained at the Debye temperature θ for MgSiN2 and β-Si3N4 appears to be too high, whereas the value for AlN is in good agreement with the highest experimentally observed value. Using better input parameters (especially the Debye temperature) resulted in better estimates. In order to increase the validity of Slack’s equation below the Debye temperature the temperature dependence of this equation was modified, resulting in a fair agreement between the predicted and experimentally observed values at 300 K for MgSiN2 and β-Si3N4. However, for AlN the discrepancy between the predicted and calculated value is considerable. Nevertheless, the modified Slack equation in combination with reliable input parameters seems to result in an accurate (somewhat conservative) estimate of the maximum achievable value making it suitable for materials selection.
1. Introduction
The nitride materials MgSiN2 [1, 2], AlN [3 - 5] and β-Si3N4 [6, 7] are (potentially) interesting as high performance materials, able to resist severe thermal loads, e.g. for substrates in microelectronics, engine parts, gas turbines, etc. One of the most important physical properties for these ceramics is the thermal conductivity. In view of guiding the optimisation of the processing of these materials, an estimate
177 Chapter 8.
of the maximum achievable thermal conductivity (especially at room temperature) is highly desirable. The maximum achievable thermal conductivity κ [W m-1 K-1] for these non-metallic materials is often discussed [6, 8 - 10] in view of Slack's equation [11]: BMδθ 2 θ κ = for T ≥ θ (1) 2 γ 2 n 3 T in which B [W mol kg-1 m-2 K-3] is a constant 3.04 107, M [kg mol-1] the mean atomic mass, δ 3 [m3] the average volume of one atom in the primitive unit cell, θ [K] the Debye temperature, γ [-] the Grüneisen parameter, n [-] the number of atoms per primitive unit cell and T [K] the absolute temperature.
For MgSiN2 (with θ ≈ 900 K [12]) the above mentioned equation was used to calculate the maximum achievable thermal conductivity at 300 K (i.e. below the Debye temperature) resulting in 75 W m-1 K-1 [10]. However, the highest experimental value at 300 K, observed in several studies concerning the influence of the processing conditions on the thermal conductivity [2, 13 - 17], does not exceed about 25 W m-1 K-1. For AlN (with θ ≈ 940 K [12]) theoretical estimates of 157 W m-1 K-1 at 621 K [10] and 97 W m-1 K-1 at 516 K [8] have been reported, based on Slack's equation. Beside the relative large difference between the two estimates in view of the temperatures for which these estimates are reported, both values considerably deviate from the values measured on pure AlN single crystals (about 91 W m-1 K-1 and 125 W m-1 K-1, respectively [8]).
For β-Si3N4 a first crude estimate of the intrinsic value at 300 K was made based on Slack's equation at T = θ (≈ 955 K [12]), and subsequently extrapolating this value to 300 K using the measured temperature dependence of the thermal conductivity of SiC. This results in a value of 200 - 320 W m-1 K-1 [18]. This value is considerably higher than the highest experimental value at 300 K of
-1 -1 122 W m K for isotropic β-Si3N4 reported by Hirosaki [9] or the averaged value of 106 W m-1 K-1 measured on a single grain [19] (thermal conductivity along the c-axis 180 W m-1 K-1 and along the a-axis 69 W m-1 K-1).
178 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
For MgSiN2, AlN and β-Si3N4 a reliable prediction of the thermal conductivity at the Debye temperature using Slack's formula is not reported although some estimates at lower temperatures using this formula can be found in the literature [10, 18]. These estimates at lower temperatures deviate all substantially from the best experimentally observed values, being in most cases higher than the experimentally observed value. This puts the question forward whether reliable estimates at the Debye temperature and 300 K can be obtained using Slack's equation as a starting point. For calculation of the thermal conductivity at 300 K, i.e. below the Debye temperature the equation is used outside its range of validity. This triggers a systematic evaluation of the temperature dependence below the Debye temperature. In this chapter the estimation of the maximum achievable thermal conductivity will be described of MgSiN2, AlN and β-Si3N4 at the Debye temperature θ and 300 K based on a modified formula of Slack, combined with the use of more accurate input parameters. First the history of Slack's formula will be briefly discussed, subsequently the theoretical values at the Debye temperature will be calculated and the relevance of the accuracy of input parameters discussed. Finally, an improved temperature dependence of the thermal conductivity as calculated from Slack's formula is presented based on the temperature dependence of the heat capacity and thermal diffusivity. The values calculated at 300 K are compared with those obtained using the classical Slack theory and experimental values. Some preliminary results concerning MgSiN2 have already been reported [15, 20].
2. The Slack equation
The subject of the absolute magnitude of the thermal conductivity by lattice vibrations (for T ≥ θ ) was first treated by Liebfried et al. [21] for a face centred cubic (FCC) lattice having 1 atom per primitive unit cell (rare-gas crystals). Their result was adjusted by Julian [22] by a factor 2 correcting a mistake in counting. Slack [11] generalised Julian's equation making it suitable for complex lattices
179 Chapter 8.
(n > 1) for T ≥ θ assuming that the heat transport takes mainly place by acoustic vibrations, resulting in the expression for the thermal conductivity as given in equation (1). In this equation B is taken as 3.04 107 W mol kg-1 m-2 K-3 [11], the mean atomic mass M can be calculated from the stoichiometry of MgSiN2, AlN and 3 β-Si3N4 and the average volume of the atoms δ and the number of atoms per primitive cell n can be obtained from crystallographic data (see Table 8-1). Slack [11] proposed to use for θ the Debye temperature evaluated from elastic constants or heat capacity data near 0 K (θ 0) and for γ the value of the thermodynamic
Grüneisen parameter at T = θ = θ 0 (γθ). For MgSiN2, AlN and β-Si3N4 the Debye temperature θ 0 and Grüneisen parameter γ as a function of the absolute temperature have been recently reported (θ 0 [12] and γ [23]) (see Table 8-1). This makes it possible to calculate the theoretical thermal conductivity (κThe) at T = θ (see Table 8-1).
Table 8-1: The mean atomic mass M , the average volume occupied by one atom δ 3, the Debye temperature θ, the Grüneisen parameter γ, the number of atoms per primitive unit cell n
and theoretical and experimental thermal conductivity at T = θ, κThe(θ ) and κExp(θ ) for
MgSiN2, AlN and β-Si3N4.
3 Material M δ θ (= θ 0) γ (= γθ) n κThe(θ ) κExp(θ ) [kg mol-1] [m3] [K] [-] [-] [W m-1 K-1][W m-1 K-1] -29 MgSiN2 0.0201 1.064 10 [24] 900 0.98 16 [27] 18 10 [20, 30] AlN 0.0205 1.045 10-29 [25] 940 0.95 4 [28] 53 51 [8] -29 β-Si3N4 0.0200 1.039 10 [26] 955 0.62 14 [29] 54 38 [9]
For MgSiN2 own data for the thermal diffusivity [20] and heat capacity [30] were used to obtain the experimental thermal conductivity (κExp) at T = θ 0
(Table 8-1). For AlN [8] and β-Si3N4 [9] literature data of the thermal conductivity as a function of the temperature were used to obtain the value at T = θ 0 (Table 8-1). For AlN the thermal conductivity was measured on almost pure AlN
180 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
single crystals (0.4 - 1800 K) [8] and for β-Si3N4 the highest experimental values for isotropic material (300 - 1700 K) reported by Hirosaki [9] were used. As for high temperatures the thermal resistance is largely determined by intrinsic thermal phonon-phonon scattering processes, it can be assumed that the experimentally obtained values (at T = θ ) equal about the intrinsic values. Except for AlN, the predictions at T = θ are too optimistic (Table 8-1), triggering a detailed evaluation of the input parameters.
3. Influence of input parameters
In order to understand the choice of the input parameters used for the Slack equation, the 'exact' nature of B, θ and γ in the Slack equation should be discussed. Actually the 'constant' B depends on the Grüneisen parameter γ, so B = B(γ ) [W mol kg-1 m-2 K-3]. The dependence of B on γ is given by [11, 22]:
5.720 107 * 0.849 B(γ) = (2) 2(1- 0.514γ -1 + 0.228γ -2 )
4.0E+07 B(γ ) B(γ = 2) 3.0E+07 ) γ
( 2.0E+07 B
1.0E+07
0.0E+00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 γ [-]
Fig. 8-1: B(γ ) and B(γ = 2) plotted versus the Grüneisen parameter (γ ).
For γ = 2, which is approximately the case for many solids, the originally used B value of 3.04 107 [W mol kg-1 m-2 K-3] results. As the Grüneisen parameter γ
181 Chapter 8.
for solid substances can vary between 0.5 and 3, the true B value can vary with about 10% (Fig. 8-1). Only for γ < 0.5, the B value drops significantly. As it is assumed that only acoustic phonons contribute to the heat transport, the Debye temperature θ and Grüneisen parameter γ should be based on the acoustic vibration modes only. For the Debye temperature and Grüneisen A parameter this results in θ∞ [K] (the high temperature limit of the Debye temperature based on the acoustic vibration modes only, as discussed by Domb et al. [31]) and (γ A)2 [-] the value of γ 2 based on the acoustic branches [11], respectively. Furthermore (γ A)2 is obtained from the average of γ 2 for all individual modes. Because B(γ ) is a function of the Grüneisen parameter γ and κ ∼ θ 3/γ 2 it is important to obtain reliable estimates for the Debye temperature θ and the Grüneisen parameter γ.
1300
MgSiN2 1200 AlN
β -Si3N4 1100
[K] 1000 θ
900
800
700 0 50 100 150 200 250 300 350 400 T [K]
Fig. 8-2: The Debye temperature (θ ) versus the absolute temperature
(T ) for MgSiN2, AlN and β-Si3N4 as obtained from heat capacity data.
The high temperature limit of the Debye temperature based on the acoustic vibration modes only can be estimated from heat capacity data as will be discussed below. In Fig. 8-2 the Debye temperature versus temperature plot is presented as obtained from heat capacity measurements for MgSiN2 [30], and AlN and β-Si3N4 [23], respectively. For all three materials a similar temperature dependence is
182 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
observed. When extrapolated to 0 K the Debye temperature is approximately equal to the value obtained from elastic constants [12] (Table 8-2) as expected from the literature [32]. This value equals the Debye temperature at 0 K related to the acoustic modes only whereas the high temperature limit of this Debye temperature is needed. With increasing temperature the Debye temperature first decreases. Subsequently a minimum is observed and the Debye temperature starts to increase with temperature approaching a constant value at high temperatures. As at low temperatures only the acoustic vibration modes are excited these determine the heat capacity. So the low temperature region of the Debye temperature versus temperature plot is determined by the acoustic phonons (decrease of θ ). As the temperature increases the optic phonons are also excited resulting in an increase of the Debye temperature. Therefore the minimum in the
Debye temperature versus temperature plot θ m is a more realistic estimate for the high temperature limit of the Debye temperature related to the acoustic phonons A A θ∞ than θ 0, resulting in a lower value for θ∞ (Table 8-2).
Table 8-2: The Debye temperature evaluated from elastic constants θ E, the minimum Debye temperature
θm and the high temperature limit of the Debye C temperature θ∞ evaluated from heat capacity
measurements for MgSiN2, AlN and β-Si3N4.
E A C Material θ (=θ 0) θm (≈θ∞ ) θ∞ [K] [K] [K]
MgSiN2 900 741 996 AlN 940 818 989
β-Si3N4 955 837 1200
As no better estimates of (γ A)2 can be made, it was assumed that for all temperatures (γ A)2 equals the square value of the thermodynamic Grüneisen
2 parameter at the Debye temperature γθ with θ = θ m.
183 Chapter 8.
Using the exact value of B(γ ) instead of the constant B-value of 3.04 107 results in some increase of the resulting thermal conductivity (Table 8-3), whereas the use of θ m instead of the higher θ 0 results in a significant decrease of the resulting thermal conductivity (Table 8-3).
Table 8-3: The most appropriate Debye temperature θ = θm, the Grüneisen
parameter γ at T = θm, the resulting B(γ ) value, and the
theoretical and experimental thermal conductivity at T = θm,
κThe(θ ) and κExp(θ ) for MgSiN2, AlN and β-Si3N4.
Material θ (= θm) γ (= γθ ) B(γ ) κThe(θ ) κExp(θ ) [K] [-] [-] [W m-1 K-1][W m-1 K-1] 7 MgSiN2 741 0.97 3.408 10 13.9 13.4 AlN 818 0.93 3.415 107 47.0 62.0 7 β-Si3N4 837 0.61 3.153 10 44.7 44.5
As compared to the conventional input parameters, the agreement between the best experimentally observed value and the calculated theoretical thermal conductivity at T = θ = θ m for MgSiN2 and β-Si3N4 considerably improves whereas the estimate for AlN becomes worse (Table 8-3). The relatively low predicted value for AlN is most probably caused by the fact that also optic phonons contribute to the heat transport whereas only acoustic modes are considered. Some experimental confirmation concerning this point can be found in Ref. 33. Furthermore, molecular dynamic calculations indicated that AlN has some low energy optic modes with a relatively large dispersion [28]. These modes have thus a relative large group velocity and therefore can contribute substantially to the heat transport. Considering the improvement of the values for MgSiN2 and β-Si3N4 it can be concluded that the use of more appropriate input parameters results in better estimates than the use of the conventional input parameters.
184 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
4. The modification of the Slack equation
The question arises whether the estimates at T = θ can be extrapolated beyond their validity range to room temperature (300 K). As the simple T -1 dependence of the traditional Slack equation results in questionable estimates at 300 K for
MgSiN2 and AlN (Table 8-4), a more appropriate description of the temperature dependence is needed.
Table 8-4: The theoretical thermal conductivity (at the Debye temperature θm and 300 K) estimated with the traditional Slack equation using more appropriate input parameters, as compared
to the highest experimentally obtained values for MgSiN2, AlN and β-Si3N4.
Slack Experimental κThe/κExp
Material κThe(θ ) κThe(300 K) κExp(θ ) κExp(300 K) θ = θ m 300 K [W m-1 K-1][W m-1 K-1][W m-1 K-1][W m-1 K-1][-][-]
MgSiN2 14 34 13 23 1.08 1.48 AlN 47 128 62 285 0.76 0.45
β-Si3N4 45 124 45 122 1.00 1.02
The temperature dependence of the thermal conductivity (κ [W m-1 K-1]) is given by the temperature dependence of thermal diffusivity (a [m2 s-1]), molar
-3 -1 -1 density (ρ m [mol m ]) and heat capacity at constant volume (CV [J mol K ]) as [34]: κ = ρ a mCV (3) For T ≥ θ the T -1 dependence of the Slack equation results as in this temperature
-1 region ρm and CV are about constant and a ∼ T as will be explained below. Combining the above mentioned equation with the classical formula for the thermal conductivity of a phonon conductor (i.e. heat transport predominantly takes place by lattice vibrations) κ = 1 v l ρ C [21, 35] results in the following 3 s tot m V expression for the thermal diffusivity a [36]:
a = 1v l (4) 3 s tot
185 Chapter 8.
-1 in which vs [m s ] is the average phonon velocity (i.e. essentially the velocity of sound) and ltot [m] the total mean free path of the phonons. As the average phonon velocity vs is almost temperature independent [21, 35], the temperature dependence of the thermal diffusivity is mainly determined by that of the total phonon mean -1 free path: a ∼ ltot. So for T ≥ θ the thermal conductivity κ ~ a ∼ ltot ~ T as ρ m, vs -1 and CV are about constant, and ltot ~ T [35, 37, 38]. However, for T < θ the thermal diffusivity a and the heat capacity CV are both strong functions of the temperature (in contrast to the density ρ m) and thus both determine the temperature dependence of the thermal conductivity. The temperature dependence of the thermal diffusivity can be deduced by considering that of the total phonon mean free path. The total phonon mean free path (ltot [m]) is determined by the lattice characteristics (intrinsic properties) as well as defects and grain boundaries present (extrinsic properties), and can be written as [39 - 41]:
1 1 1 = + ∑ (5) ltot lpp i li in which lpp [m] is the mean free path due to thermal phonon-phonon scattering and li [m] the mean free path due to other phonon scattering mechanisms e.g. phonon- defect scattering, phonon-grain boundary scattering, etc. For pure, defect free single crystalline materials ltot equals lpp, and in that case the temperature dependence of the thermal diffusivity is determined by the temperature dependence of lpp only.
For the phonon mean free path due to thermal phonon-phonon scattering, lpp, of pure crystalline materials it is known that approximately [35, 37]:
θ~ = lpp l0 exp -1 (6) bT ~ in which l0 [m] is a pre-exponential factor, θ [K] a characteristic temperature below which Umklapp processes start to disappear given by θ /n1/3 [11, 42], b [-] a constant about equal to 2 [35, 37, 43] and T [K] the absolute temperature. So the
186 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
temperature dependence of the thermal diffusivity is given by: θ a ~ exp -1 (7) bTn1 3
From the discussion of the input parameters it is clear that within the framework of the Slack equation the Debye temperature in the above equation is A based on the high temperature limit of the acoustic phonons only (θ = θ∞ ). The value for b for describing the temperature dependence of the thermal diffusivity is not exactly known. Based on the simple Debye theory it can be argued that b = 2 [37, 43], but larger and smaller values down to 1 are also reported [21, 35]. Leibfreid et al. [21] suggest that for a FCC lattice b = (5/3)1/2. However, the scarce experimental results confirm the value of b ≈ 2 (2.3, 2.7 and 2.1 for solid helium, diamond and sapphire, respectively [35]).
3.00
FD(T /θ ) 2.50
2.00 ] R [
V 1.50 C
1.00
0.50
0.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 T/θ [-]
Fig. 8-3: The heat capacity at constant volume (CV) versus the reduced temperature (T /θ ) using the Debye function
(FD(θ /T )).
-1 -1 The heat capacity at constant volume CV is 0 J mol K at 0 K and with increasing temperature it increases having a maximum value at high temperatures of 3R per atom mole (R is the gas constant 8.314 J mol-1 K-1) (Fig. 8-3).
Theoretically the temperature dependence of CV is given by the Debye function
187 Chapter 8.
FD(θ /T) [44 - 46] (Fig. 8-3). For 0.1 ≤ T /θ ≤ 0.4 the increase of FD(θ /T ) is the largest and for T ≥ θ the heat capacity is almost constant. If the temperature is sufficiently high (T /θ / 0.3) FD(θ /T ) describes very well the experimental heat C capacity with θ = θ∞ [K], the high temperature limit of the characteristic Debye temperature obtained from heat capacity measurements (see Fig. 8-2).
From the temperature dependence of a, ρ m and CV it may be expected that:
A θ ∞ C κ ~ exp -1 F (θ ∞ / T) (8) 1 3 D bTn
A C in which b, θ∞ and θ∞ are constants if the temperature is not too low (T /θ / 0.3).
For T ≥ θ, FD(θ /T) is almost constant, and the exponential function can be approximated by θ /bTn1/3 as for θ /bTn1/3 = x it can be written for 2x < 1 that exp(x) - 1 = (1 + x + x2/2 +…) - 1 ≈ x, so κ ∼ T -1 as suggested by Slack's equation. In general for bTn1/3 ≥ 2θ the temperature dependence can be calculated accurately within 23 % ({½ [exp(½) - 1]-1 × 100 %} - 100%) as the thermal diffusivity is about inversely proportional with the absolute temperature, whereas for Tn1/3 ≥ 2θ the exact value of b is of minor importance and the maximum error in the calculated temperature dependence equals about 15 - 20 %. Due to the large number of atoms per primitive unit cell for MgSiN2 and Si3N4 the error made at 300 K by this assumption does most probably not exceed 15 - 20 %. However, for AlN with n = 4 the error at 300 K can be considerable. It is clear that the complexity of the crystal structure (n) is of importance for the temperature dependence of the thermal conductivity below the Debye temperature. This can be visualised by dividing the temperature dependence of κ as ~ function of the reduced temperature T /θ by its high temperature limit ((θ /b) T -1 × 3R) assuming that b = 2 and both Debye temperatures for describing the thermal diffusivity and the heat capacity are the same (Fig. 8-4). For rare-gas crystals (n = 1) Julian found that κ ∼ T -1 for T ≥ θ /4 [22]. This result is somewhat different from the one given in Fig. 8-4 which shows a faster than T -1 dependence. From Fig. 8-4 it can be seen that for simple crystals
188 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
(2 ≤ n ≤ 4) with decreasing temperature the drop in the heat capacity is compensated by the exponential increase of the thermal diffusivity resulting in a pseudo T -1 dependence for T ≥ θ /4. For large n (n / 8) the heat capacity decreases for T . θ whereas the thermal diffusivity is still inversely proportional with the absolute temperature for T / 2θ /bn1/3. So, the decrease of the heat capacity is only partially compensated by the exponential increase of the thermal diffusivity resulting in a considerable deviation of the T -1 dependence for T/θ . 0.5 (Fig. 8-4). Based on Julian's result De With and Groen [10] assumed that the T -1 dependence of Slack's equation is valid down to T ≅ θ /4. They used the Slack equation to predict the intrinsic thermal conductivity of several new (oxy-)nitride materials with large n. From the above discussion it is clear that the T -1 dependence of the Slack formula below the Debye temperature is in general too simple and a more general 'all temperature' formula should be used.
[-] 2.00 θ
/ n = 1 n = 2 n = 4
1/3 1.75 n = 8 n = 16 n = 64 Tn 1.50 )-1] 2
1/3 1.25 Tn
/2 1.00 θ 0.75
× [exp( 0.50 R
)/3 0.25 T / θ (
D 0.00 F 0.00 0.25 0.50 0.75 1.00 1.25 T/θ [-]
Fig. 8-4: Theoretical deviation from the T -1 temperature dependence of the thermal conductivity versus the reduced temperature (T/θ) as calculated from the temperature dependence of the thermal diffusivity and the heat capacity with b = 2 (note 1/3 1/3 that FD(θ/T)/3R × [exp(θ/2Tn )-1] 2Tn /θ = 1 represents the T -1 dependence of the Slack equation).
189 Chapter 8.
For describing the absolute value of the thermal conductivity the (above mentioned) temperature dependence of the thermal diffusivity and the heat capacity (equation (8)) is combined with the Slack equation (equation (1)) resulting in a modified Slack equation. It is assumed that the Slack and modified Slack equation give the same result for T >> θ . This results in the following modified Slack equation for the theoretical thermal conductivity:
δ ()θ A 2 θ C θ A BM ∞ F D ( ∞ T) ∞ 1 κ (T) = * exp - 1 bn 3 (9) 2 2 1 γ n 3 (3R) bTn 3
C in which θ∞ [K] is the high temperature limit of the Debye temperature as
A evaluated from heat capacity measurements and θ∞ [K] the high temperature limit of the acoustic phonons. The expression before the square brackets (Slack part)
A represents the thermal conductivity at T = θ∞ (= θ m) for the traditional Slack equation and between square brackets the temperature dependence of the thermal conductivity is given. As for high temperatures (T → ∞) the θ /T temperature dependence should result, the temperature dependence of the specific heat and the thermal diffusivity in equation (9) are normalised by a factor 3R and bn1/3, respectively. It is noted that at T = θ the expression between square brackets differs from unity resulting in different estimates for the thermal conductivity at T = θ as compared to the traditional Slack equation (see Fig. 8-4).
Table 8-5: The temperature independent Slack part, temperature dependent heat capacity and thermal diffusivity part of the modified Slack equation, and the resulting estimated thermal
conductivity for MgSiN2, AlN and β-Si3N4 at T = θ = θm with various values for b.
2 C θ 1 B()γ Mδ θ (θ ∞ T) m m F D exp -1 bn 3 κ θ Material 1 The(T = m) 2 2 3 γ θ n 3 3R bTn [W m-1 K-1][-] [-] [W m-1 K-1]
b = 2 b = √5/3 b = 1 b = 2 b = √5/3 b = 1
MgSiN2 13.85 0.915 1.106 1.171 1.227 14.0 14.8 15.6 AlN 46.99 0.930 1.175 1.289 1.393 51.4 56.4 60.9
β-Si3N4 44.69 0.904 1.111 1.179 1.239 44.9 47.6 50.1
190 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
A C Using the modified Slack equation and the values for θ∞ (= θ m) and θ∞ (Table 8-2), and b = 2, b = (5/3)1/2 and b = 1 the theoretical thermal conductivity at
T = θ = θ m and 300 K was calculated (Table 8-5 and Table 8-6).
The results at T = θ = θ m using the modified Slack equation (Table 8-5) show a better agreement for AlN with the experimental values than the conventional Slack equation (Table 8-4), whereas the estimates for MgSiN2 and
β-Si3N4 as compared to the conventional Slack equation (Table 8-4) remain about the same being in good agreement with the experimentally observed values.
Table 8-6: The temperature independent Slack part, temperature dependent heat capacity and thermal diffusivity part of the modified Slack equation, and the resulting estimated thermal
conductivity for MgSiN2, AlN and β-Si3N4 at T = 300 K with various values for b.
2 θ C θ 1 B()γ Mδ θ F D ( ∞ T) m Material m exp -1 bn 3 κ (T =300) 2 1 The 2 3 γ θ n 3 3R bTn [W m-1 K-1][-] [-] [W m-1 K-1] b = 2 b = √5/3 b = 1 b = 2 b = √5/3 b = 1
MgSiN2 13.85 0.609 3.187 3.698 4.196 26.9 31.2 35.4 AlN 46.99 0.613 4.319 5.703 7.257 124.3 164.2 208.9
β-Si3N4 44.69 0.495 3.779 4.516 5.260 83.6 99.9 116.4
-1 -1 The estimate at 300 K for MgSiN2 (27 - 35 W m K ) is in reasonable agreement with the best experimentally observed values (23 W m-1 K-1 [20]). This indicates that the former estimate based on the conventional Slack equation of 75 W m-1 K-1 [10] is too optimistic mainly due to the use of inappropriate input parameters. The estimate for the thermal conductivity at 300 K of β-Si3N4 (84 - 116 W m-1 K-1) is also in favourable agreement with the best experimentally -1 -1 observed value (122 W m K [9]). It is noted that for β-Si3N4 at room C temperature T/θ∞ = 300/1200 = 0.25 indicating that the Debye function used to calculate the heat capacity is not totally correct (see also Fig. 8-2) resulting in an underestimation of the heat capacity (about 5 %). For AlN the estimate (124 - 209 W m-1 K-1) is substantially lower than the experimentally observed value
191 Chapter 8.
(285 W m-1 K-1 [8]) providing only a rough indication for the true intrinsic value. As pointed out before, this is caused by the used assumption that only acoustic phonons contribute to the heat conduction, resulting in an underestimation of the thermal conductivity. Introduction of a more realistic temperature dependence shows that AlN has a higher intrinsic thermal conductivity than β-Si3N4 whereas the conventional Slack equation predicted that both materials have about the same thermal conductivity. Moreover, the calculations indicate that for applications where a high thermal conductivity is required β-Si3N4 is a much more interesting compound than MgSiN2. From a first approximate based on the conventional Slack equation it was concluded that MgSiN2 might be interesting as a potential substrate material [10]. However, this expectation is not supported by the presently discussed refinement based on the modified Slack equation, which provides some additional theoretical evidence for the experimentally observed limited thermal conductivity of MgSiN2. Furthermore, by varying the value of b an impression of the reliability of the estimate is obtained indicating that the estimate for AlN is less reliable than the estimate for MgSiN2 and β-Si3N4.
Table 8-7: Comparison of the modified Slack estimates for b = 1 - 2 based on reliable input
parameters (κThe) and experimentally observed (κExp) thermal conductivity at T = θ 0 and
300 K for MgSiN2, AlN and β-Si3N4. Between brackets the theoretical thermal conductivity based on the conventional input parameters using the traditional Slack equation is presented.
Material Slack Experimental κThe/κExp
κThe(θ 0) κThe(300 K) κExp(θ0) κExp(300 K) θ 0 300 K [W m-1 K-1] [W m-1 K-1][W m-1 K-1] [W m-1 K-1][-][-]
MgSiN2 12 - 13 (18) 27 - 36 (54) 10 23 1.2 - 1.3 1.2 - 1.6 AlN 45 - 52 (53) 124 - 209 (166) 51 285 0.9 - 1.0 0.4 - 0.7
β-Si3N4 40 - 44 (54) 84 - 116 (173) 38 122 1.1 - 1.2 0.7 - 1.0
Generally, the estimates using the modified Slack equation (Table 8-5 and Table 8-6) somewhat improved as compared to the estimates obtained using the
192 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
traditional Slack equation (T -1 dependence) with more appropriate input parameters (Table 8-4). However, especially the choice of reliable values for θ and γ are important to obtain an accurate estimate of the thermal conductivity (Table 8-7). An error of 10 % in θ results in an error of 30 - 40 % and an error of 10 % in γ results in an error of ~ 20 % in the resulting thermal conductivity estimate. So, the reliability of the calculated thermal conductivity is more dependent on the accuracy of the input parameters rather than a correct description of the temperature dependence when the temperature is not too low. However, when also optic phonons contribute substantially to the heat transport, like in the case of AlN, the maximum achievable thermal conductivity can be substantially underestimated providing a conservative estimate of the maximum achievable thermal conductivity. This provides only a rough indication of the true intrinsic thermal conductivity. Nevertheless, the modified Slack equation results in a good impression whether or not materials have potentially desirable thermal properties.
5. Applicability, reliability and limitations of Slack modified
From the discussion about the influence of n on the temperature dependence of the thermal conductivity a temperature above which the modified Slack equation gives reasonable estimates for the intrinsic thermal conductivity was estimated to be
1/3 Table 8-8: The thermal conductivity for MgSiN2, AlN and β-Si3N4 at T = θ m/n with various values for b using the modified Slack equation.
θ C F D ( ∞ T) θ 1 κThe κExp θ 1/3 exp m -1bn 3 Material m/n 1 3R 3 1/3 1/3 bTn (T = θm/n ) (T = θm/n ) [K] [-] [-] [W m-1 K-1][W m-1 K-1] b = 2 b = 1 b = 2 b = 1
MgSiN2 294 0.597 3.269 4.330 27.0 35.8 25 AlN 515 0.837 2.060 2.728 81.0 107.3 120
β-Si3N4 347 0.586 3.127 4.141 81.9 108.4 110
193 Chapter 8.
C 1/3 T ≅ θ∞ /3 and higher based on the specific heat part and T / θ m/n for the thermal 1/3 C diffusivity part. The estimates obtained at T = θ m/n (/ θ∞ /3) are indeed in reasonable agreement with the experimentally observed values (Table 8-8). For
1/3 T = 300 K the criterion of T / θ m/n is not fulfilled by AlN (Table 8-8) indicating that, besides the already discussed contribution of optic phonons to the heat conduction, the estimate of the acoustic phonons to the heat conduction at 300 K might be less reliable. An important complication encountered using the (modified) Slack equation is the estimation of γ. By using the thermodynamic Grüneisen parameter no difference can be made between the Grüneisen parameter of acoustic and optic phonons. Furthermore acoustic modes might have a negative gamma decreasing the value of the thermodynamic Grüneisen parameter, however, contributing substantially to the thermal resistance [47]. Also it was assumed that the Grüneisen parameter is temperature independent. For temperatures near the Debye temperature and above this assumption is allowed [23]. However, for materials with a high Debye temperature this assumption might be incorrect at room temperature [23, 48]. In view of the discussion about the temperature dependence of the thermal conductivity this could be interpreted as l0, the pre-exponential factor for describing the phonon mean free path (equation (6)), being not constant 2 Α but l0 ~ B(γ )/γ . Assuming that γ is temperature dependent and can be approximated by the (temperature dependent) thermodynamic Grüneisen parameter
C (when the temperature is not too low (T / θ∞ /3)), the maximum achievable thermal conductivity at 300 K was calculated using the modified Slack equation. For b = 2 a value of 33 W m-1 K-1, 215 W m-1 K-1 and 133 W m-1 K-1 is obtained for the intrinsic thermal conductivity at 300 K of MgSiN2, AlN and β-Si3N4, respectively. These estimates are in very good agreement with the experimentally observed values (Table 8-7). However, this result might be a coincidence / fortuitous. Although, the κ ∼ n-2/3 of the Slack formula (equation (1)) works well some discussion about this point can be found in the literature (κ ∼ n-1/3 [42] and κ ∼ n-1/2
194 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
[49]). Considering this point it would be interesting, as suggested previously by Spitzer [50], to study the thermal conductivity of materials with different modifications, having only a slightly different atomic orientation, resulting in about the same density, Debye temperature and Grüneisen parameter, yet different number of atoms per primitive unit cell like α-Si3N4 (n = 28) and β-Si3N4 (n = 14).
6. Conclusions
In order to obtain reliable estimates of the intrinsic thermal conductivity at T = θ using the Slack equation the input parameters should be carefully chosen. The T -1 dependence of the Slack equation can generally not be used at T < θ and should be modified in order to obtain a more realistic description of the temperature dependence of the thermal conductivity. The presented modified Slack equation is considered to describe the temperature dependence of the intrinsic thermal conductivity for T ≅ θ /3, and higher. Furthermore, the choice of the appropriate input parameters, especially the Debye temperature and Grüneisen parameter, is of crucial importance. Considering the accuracy and boundary conditions using the modified Slack equation it can be stated that the Slack equation provides a reasonably good impression of the thermal conductivity when T / θ /n1/3 as for this temperature the exact value of b is of minor importance. Estimates of the maximum thermal conductivity at 300 K resulted in a value of 27 - 35 W m-1 K-1, 124 - 209 W m-1 K-1 and 84 - 116 W m-1 K-1 as compared to the highest experimentally observed values of 23 W m-1 K-1, 285 W m-1 K-1 and -1 -1 122 W m K for MgSiN2, AlN and β-Si3N4, respectively. The calculated values for MgSiN2 and β-Si3N4 are in good agreement with the measured thermal conductivity whereas the calculated value for AlN is considerably below the experimentally observed thermal conductivity as the (modified) Slack equation neglects the contribution of optic phonons to the heat conduction.
195 Chapter 8.
Although the match with the true intrinsic thermal conductivity can be disappointing due to the contribution of optic phonons to the heat conduction, the modified Slack equation is a useful tool for estimating the intrinsic thermal conductivity and understanding the differences in thermal conductivity between several materials. Considering the accuracy of the modified Slack equation at lower temperatures it would be desirable to have an alternative method for estimating the intrinsic thermal conductivity in order to get a better impression of the true maximum achievable thermal conductivity of (new) potentially interesting materials.
References
1. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent
Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non- Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh, 1994) 239. 2. I.J. Davies, T. Shimazaki, M. Aizawa, H. Suemasu, A. Nozue and K. Itatani, Physical Properties of Hot-Pressed Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 276. 3. M.P. Borom, G.A. Slack and J.W. Szymaszek, Thermal Conductivity of Commercial Aluminum Nitride, Bull. Am. Ceram. Soc. 51 (1972) 852. 4. W. Werdecker and F. Aldinger, Aluminum Nitride - An Alternative Ceramic Substrate for High Power Applications in Microcircuits, IEEE Trans. Compon., Hybrids, Manuf. Technol., CHMT-7 (1984) 399. 5. T.B. Jackson, A.V. Virkar, K.L. More, R.B. Dinwiddie, Jr. and R.A. Cutler, High-Thermal-Conductivity Aluminum Nitride Ceramics: The Effect of Thermodynamic, Kinetic, and Microstructural Factors, J. Am. Ceram. Soc. 80 (1997) 1421.
196 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
6. K. Watari, K. Hirao, M.E. Brito, M. Toriyama and S. Kanzaki, Hot Isostatic
Pressing to Increase Thermal Conductivity of Si3N4 Ceramics, J. Mater. Res. 14 (1999) 1538. 7. K. Hirao, K. Watari, M.E. Brito, M. Toriyama and S. Kanzaki, High Thermal Conductivity of Silicon Nitride with Anisotropic Microstructure, J. Am. Ceram. Soc. 79 (1996) 2485. 8. G.A. Slack, R.A. Tanzilli, R.O. Pohl and J.W. Vandersande, The Intrinsic Thermal Conductivity of AlN, J. Phys. Chem. Solids 48 (1987) 641. 9. N. Hirosaki, Y. Okamoto, M. Ando, F. Munakata and Y. Akimune, Thermal Conductivity of Gas-Pressure-Sintered Silicon Nitride, J. Am. Ceram. Soc. 79 (1996) 2878. 10. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)- nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of Properties and Performances by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405. 11. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1979) 1. 12. Chapter 6; R.J. Bruls, H.T. Hintzen, R. Metselaar and G. de With, The
temperature dependence of the Young's modulus of MgSiN2, AlN and Si3N4, accepted for publication in J. Eur. Ceram. Soc. 13. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan,
Hot-pressing of MgSiN2 ceramics, J. Mat. Sci. Lett. 13 (1994) 1314. 14. H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Densification of MgSiN2, Ceram. Trans. 51, Int. Conf. Cer. Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995) 585.
15. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics 2, Basic Science - Developments in
197 Chapter 8.
Processing of Advanced Ceramics - Part II, Faenza, Italy, October 1995, edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 289. 16. I.J. Davies, H. Uchida, M. Aizawa, and K. Itatani, Physical and Mechanical Properties of Sintered Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 40. 17. Chapter 3; R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar,
Preparation, Characterisation and Properties of MgSiN2 Ceramics, to be published 18. J.S. Haggerty and A. Lightfoot, Oppertunities for enhancing the thermal
conductivities of SiC and Si3N4 ceramics through improved processing, Ceram. Eng. Sci. Proc. 16, 19th Annual Conference on Composites, Advanced Ceramics, Materials, and Structures-A, Cocoa Beach, Florida, USA, January 8 - 12, 1995, edited by J.B. Wachtman (The American Ceramic Society, Westerville, OH, 1995) 475. 19. B. Li, L. Pottier, J.P. Roger, D. Fournier, K. Watari and K. Hirao, Measuring the Anisotropic Thermal Diffusivity of Silicon Nitride Grains by Thermoreflectance Microscopy, J. Eur. Ceram. Soc. 19 (1999) 1631. 20. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modeling of the Thermal th Diffusivity/Conductivity of MgSiN2 Ceramics, ITCC 24 and ITES 12, 24 International Thermal Conductivity Conference and 12th International Thermal Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 26-29, 1997, edited by P.S. Gaal and D.E. Apostolescu (Technomic Publishing Co., Inc., Lancaster, 1999) 3. 21. G. Leibfried and E. Schlömann, Wärmeleitung in elektrisch isolierenden Kristallen, Nachr. Akad. Wiss. Göttingen, Math. Physik. Kl. IIa (1954) 71. 22. C.L. Julian, Theory of Heat Conduction in Rare-Gas Crystals, Phys. Rev. 139 (1965) A128. 23. Chapter 7; R.J. Bruls, H.T. Hintzen, R. Metselaar, G. de With and J.C. van Miltenburg, The temperature dependence of the Grüneisen parameter of
MgSiN2, AlN and β-Si3N4, submitted to J. Phys. Chem. Solids
198 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
24. Chapter 4; R.J. Bruls, H.T. Hintzen, R. Metselaar and C.-K. Loong,
Anisotropic thermal expansion of MgSiN2 from 10 to 300 K as measured by neutron diffraction, J. Phys. Chem. Solids 61 (2000) 1285. 25. K. Wang and R.R. Reeber, Thermal Expansion of GaN and AlN, Mat. Res. Soc. Symp. Proc. 482, Nitride Semiconductors, Boston, Massachusetts, USA, December 1 - 5 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S. Nakamura and S. Strite, (Materials Research Society, 1998) 863. 26. C.M.B. Henderson and D. Taylor, Thermal Expansion of the Nitrides and Oxynitride of Silicon in Relation to their Structure, Trans. J. Br. Ceram. Soc. 74 (1975) 49.
27. J. David, Y. Laurent and J. Lang, Structure de MgSiN2 et MgGeN2, Bull. Soc. Fr. Minéral. Cristallogr. 93 (1970) 153. 28. J.C. Nipko and C.-K. Loong, Phonon Excitations and Related Thermal Properties of Aluminum Nitride, Phys. Rev. B. 57 (1998) 10550.
29. R. Grün, The Crystal Structure of β-Si3N4; Structural and Stability
Considerations Between α- and β-Si3N4, Acta Cryst. B35 (1979) 800. 30. Chapter 5; R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,
Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998) 7871. 31. C. Domb and L. Salter, The Zero Point Energy and Θ Values of Crystals, Phil. Mag. 43 (1952) 1083. 32. P. Debye, Zur Theorie der spezifischen Wärmen, Ann. Phys. 39 (1912) 789. 33. Chapter 9; R.J. Bruls, H.T. Hintzen, R. Metselaar, A new Estimation Method for the Thermal Diffusivity/Conductivity of Non-Metallic Compounds: A case
study for MgSiN2, AlN and β-Si3N4 ceramics, to be published. 34. R. Berman, Thermal Conduction in Solids (Clarendon Press, Oxford, 1976), p. 7. 35. R. Berman, The Thermal Conductivity of Dielectric Solids at Low Temperatures, Advances in Phys. 2 (1953) 103.
199 Chapter 8.
36. K. Watari, Y. Seki and K. Ishizaki, Temperature Dependence of Thermal Coefficients for HIPped Sintered Silicon Nitride, J. Ceram. Soc. Jpn. Inter. Ed. 91 (1989) 170. 37. P. Debye, Zustandsgleichung und Quantenhypothese mit einem Anhang über Wärmeleitung, in: Vorträge über die Kinetische Theorie der Materie und der Elektrizität (Teubner, Berlin, 1914), pp. 19 - 60. 38. R. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen, Ann. Phys. 3 (1929) 1055. 39. F.R. Chavat and W.D. Kingery, Thermal Conductivity: XIII, Effect of Microstructure on Conductivity of Single-Phase Ceramics, J. Am. Ceram. Soc. 40 (1957) 306. 40. P.G. Klemens, The thermal conductivity of dielectric solids at low temperatures, Proc. Roy. Soc. (London), A208 (1951) 108. 41. K. Watari, K. Ishazaki and F. Tsuchiya, Phonon Scattering and Thermal Conduction Mechanisms of Sintered Aluminium Nitride Ceramics, J. Mater. Sci. 28 (1993) 3709. 42. M. Roufosse and P.G. Klemens, Thermal Conductivity of Complex Dielectric Crystals, Phys. Rev. B. 7 (1973) 5379. 43. J.R. Drabble and H.J. Goldsmid, Thermal Conduction in Semiconductors, International Series of Monographs on Semiconductors 4, edited by H.K. Henisch (Pergamon Press, Oxford, 1961), p. 141. 44. P. Debije, Zur Theorie der spezifischen Wärmen, Ann. Physik 39 (1912) 789. 45. See for numerical calculation W.M. Rogers and R.L. Powell, Tables of Transport Integrals, Natl. Bur. Stand. Circ. 595 (1958) 1. 46. See for example C. Kittel, Introduction to Solid State Physics, fifth edition (John Wiley & Sons, Inc., New York, 1976), pp. 136 - 140. 47. W.B. Daniels, The anomalous thermal expansion of germanium, silicon and compounds crystallizing in the zinc blende structure, International Conference on the Physics of Semiconductors, Exeter, UK, July 1962, edited by A.C. Stickland (London Institute of Physics, 1962) 482.
200 Theoretical thermal conductivity of MgSiN2, AlN and β-Si3N4 using Slack's equation
48. G.A. Slack and I.C. Huseby, Thermal Grüneisen parameters of CdAl2O4,
β-Si3N4, and other phenacite-type compounds, J. Appl. Phys. 53 (1982) 6817. 49. A. Missenard, Conductivité Thermique des Solides, Liquides, Gaz et leurs Mélanges, ch.1.II, Eyrolles, Paris, 1965. 50. D.P. Spitzer, Lattice Thermal Conductivity of Semi-Conductors: a Chemical Bond Approach, J. Phys. Chem. Solids 31 (1970) 19.
201 Chapter 8.
202 Chapter 9. A new method for estimation of the intrinsic thermal conductivity
A case study for MgSiN2, AlN and β-Si3N4
Abstract A new method for estimating the maximum achievable thermal conductivity of non-electrically conducting materials is presented. The method is based on temperature dependent thermal diffusivity data using a linear extrapolation method enabling to distinguish between phonon-phonon and phonon-defect scattering. The thermal conductivity estimated in this way for MgSiN2, AlN and β-Si3N4 ceramics at 300 K equals 26 - 28 W m-1 K-1, 178 - 200 W m-1 K-1 and 79 - 94 W m-1 K-1, respectively in favourable agreement with the highest experimental values of 23 W m-1 K-1, 246 - 266 W m-1 K-1 and 106 - 122 W m-1 K-1. The difference between the estimated and experimentally observed value for AlN can be understood in view of optic phonons that are substantially contributing to the heat conduction. The reliability, accuracy and limitations of this method are discussed.
1. Introduction
Several ceramic materials have been investigated intensively for substrate applications [1] because of their potentially high thermal conductivity in combination with a high electrical resistivity. Especially AlN has drawn a lot of attention [2 - 4], but recently also the nitride materials β-Si3N4 [5 - 7] and MgSiN2 [8, 9] are considered to be potentially interesting. In a previous paper [10] the
203 Chapter 9.
intrinsic thermal conductivity of MgSiN2, AlN and Si3N4 was (theoretically) estimated based on a modification of Slack’s equation for non-metallic materials [11]. Based on the results obtained it can be concluded that this equation only provides a rough indication of the maximum achievable thermal conductivity, and that a more accurate and simpler estimation method would be useful. Another (experimental) method reported in the literature to estimate the maximum achievable thermal conductivity is by linear extrapolation of the measured inverse thermal conductivity (thermal resistivity) values [12] versus the absolute temperature. Usually, it is assumed that the slope is determined by the lattice characteristics (intrinsic properties) and the intercept at 0 K by defects (impurities, grain boundaries, etc.) [12 - 14]. It will be shown that this last assumption is only partially correct. So, also this method is not generally applicable. However, by combining some of the concepts of both approaches a new estimation method was developed. In this chapter a new method will be described for the estimation of the maximum achievable thermal conductivity of non-metallic crystals (i.e. heat transport takes place by lattice vibrations) based on temperature dependent thermal diffusivity data. With this method the maximum achievable thermal conductivity of
MgSiN2, AlN and β-Si3N4 was calculated at 300, 600 and 900 K. The results were compared with experimental values, values obtained using the (modified) Slack theory and other (theoretical) estimates. Some preliminary results considering
MgSiN2 have already been reported elsewhere [15].
2. The temperature dependence of the thermal diffusivity and conductivity
The thermal conductivity (κ [W m-1 K-1]) of a material can be calculated using [16]: κ = ρ a mCV (1)
2 -1 -3 in which a [m s ] is the thermal diffusivity, ρm [mol m ] the molar density and CV [J mol-1 K-1] the heat capacity at constant volume. The density is only a weak
204 A new method for estimation of the intrinsic thermal conductivity
function of the temperature, so the temperature dependence of the thermal conductivity is determined by that of the thermal diffusivity and the heat capacity. For a phonon conductor (i.e. heat transport predominantly takes place by lattice vibrations) the thermal diffusivity a equals [16 - 19]:
= 1 a 3vsltot (2)
-1 in which vs [m s ] is the average phonon velocity (i.e. essentially the velocity of sound) and ltot [m] the total mean free path of the phonons. The average phonon velocity vs is almost temperature independent [20], so that a ∼ ltot. If secondary phases are not taken into account then the total phonon mean free path is determined by the lattice characteristics (intrinsic properties) as well as defects and grain boundaries present in the material (extrinsic properties), and can be written as [12, 21 - 23]:
1 1 1 1 1 = + + + ∑ (3) ltot lpp lpd lgb x lx in which lpp [m] is the mean free path due to thermal phonon-phonon scattering, lpd [m] the mean free path due to phonon-defect (vacancies, impurities, isotopes) scattering, lgb [m] the mean free path due to phonon-grain boundary scattering and lx [m] the mean free path due to other scattering mechanisms induced by e.g. stacking faults, dislocations, etc. For the temperature dependence of the phonon mean free path due to thermal phonon-phonon scattering, lpp, of pure crystalline materials it is known that approximately [17, 24]:
~ θ ~ θ = θ = lpp l0 exp -1 with 1 (4) bT n 3
~ in which l0 [m] is a pre-exponential factor, θ [K] a characteristic temperature (so-called reduced Debye temperature) below which Umklapp processes start to disappear [11, 25], b [-] a constant ≈ 2 [17, 18, 24, 26], T [K] the absolute
205 Chapter 9.
temperature, θ [K] the Debye temperature and n [-] the number of atoms per primitive unit cell. For most materials only the first three terms of equation 3 are considered to be of importance [12, 22, 27]. However, for the present discussion it is sufficient to assume that lx is temperature independent. The temperature dependence of phonon- defect scattering lpd has been studied by Klemens [21, 28] and Ambegaokar [29]. It was shown that this term for low defect concentrations is (almost) temperature independent. The phonon-grain boundary scattering term lgb is temperature independent if the influence of the thermal expansion is neglected. So, the temperature dependence of ltot is dominated by the lpp term, whereas the other terms can be assumed to be negligibly temperature dependent [12, 21, 30]. This implies that in general at low temperature ltot is determined by temperature independent extrinsic scattering processes (at defects and grain boundaries), whereas at high temperatures it is determined by the temperature dependent intrinsic phonon- phonon scattering process. ~ If the temperature is sufficiently high (T > θ /b) we can write more generally (i.e. including all above mentioned phonon scattering mechanisms) for the inverse of the thermal diffusivity:
1 1 A ∼ ∼ + B = a l θ~ tot exp -1 bT
~ ~ 2 bT 1 θ 1 θ A 1- + + ... + B ≈ (5) θ~ 2 bT 12 bT
bT 1 bA + + ()1 A ~ - B = ~ T B - 2 A θ 2 θ
The constant A in equation (5) is related to the temperature dependent phonon- phonon scattering processes (intrinsic lattice diffusivity) and B to the temperature independent phonon scattering processes (impurities, defects, grain boundaries, etc.). It is obvious that equation (5) shows a linear relation between a -1 and T:
206 A new method for estimation of the intrinsic thermal conductivity
− ~ a 1 = A'T + B' (for T / θ /b)(6)
~ in which the slope A' (= bA/θ ) [m-2 s K-1] is determined by the intrinsic lattice characteristics (phonon-phonon scattering mechanisms), and the intercept B' (= B - ½ A) [m-2 s] by the impurities and microstructure (B: temperature independent scattering processes) as well as the intrinsic lattice characteristics (A). From equation (6) it can be concluded that for pure defect free single crystalline materials (B = 0) a plot of the inverse of the thermal diffusivity versus the absolute ~ temperature for measurements at T > θ /b extrapolated to 0 K should result in a straight line with (negative) intercept -½ A and which intercepts the temperature ~ axis at T = θ /2b (= ½ A/A'). If the temperature is sufficiently high so that the heat capacity is temperature independent (T / θ [31]) then κ ~ a (~ ltot) and the well known linear relation for the thermal resistivity results [12 - 14]:
κ -1 = A''T + B'' (for T / θ )(7) ~ This equation is often interpreted as being A'' (~ bA/θ ) determined by the intrinsic lattice diffusivity, which is correct, and the intercept value B'' (~ (B - ½ A)) as being only determined by the microstructure and impurities, which is incorrect. This results in the erroneous conclusion that for a pure defect free single crystalline material (for T > θ ) the thermal resistivity versus the absolute temperature plot gives a straight line through the origin [5, 13, 14, 32] as B = 0 instead of B'' = 0. It is noted that at very high temperatures (T ≥ 2θ ), where the phonon mean free path is limited by the inter-atomic distances, equations (5) and (6) are no longer valid [33] because they predict a decrease of the phonon mean free path to ~ zero. For most materials n > 1 so that θ < θ. Considering the above discussion it is clear that the linear temperature dependence for the inverse thermal diffusivity ~ a -1 can be observed at much lower temperatures (T > θ /b) than for the thermal resistivity κ -1 as for T . θ the heat capacity is still temperature dependent. Furthermore the thermal diffusivity is directly related to the total phonon mean free
207 Chapter 9.
path which has to be maximised in order to optimise the thermal conductivity. So for identifying the dominant scattering mechanisms it is much more interesting to study the temperature dependence of the thermal diffusivity rather than that of the thermal conductivity. So, temperature dependent thermal diffusivity measurements when ~ performed in a suitable temperature region (θ /b (= θ /bn1/3 ) ≤ T ≤ 2θ ) can be a powerful tool in understanding and optimising the thermal conductivity of promising materials.
3. Experimental
For MgSiN2 the thermal diffusivity a as a function of the temperature T (300 - 900 K) was measured on small ceramic samples (∅ 11 mm × 1 mm) cut from several large fully dense ceramic pellets processed under different conditions (for details see [34]) using the photo/laser flash method [35] (laser flash equipment, Compotherm Messtechnik GmbH). The method used to prepare the ceramic pellets is described elsewhere [34, 36, 37]. By carefully grinding and polishing, samples with a uniform thickness and a low roughness were obtained. Samples varying in microstructure, oxygen content and processed with and without additive were investigated (Table 9-1). The accuracy of the measurement was estimated to be within 5%. Some samples were coated with a thin layer of gold and/or carbon before measuring the thermal diffusivity. The thin gold layer prevents direct transmission of the laser beam and aids the energy transfer to the sample. Carbon was used to increase the absorptivity of the front surface, and the emissivity of the back surface. These additional layers reduce the measured thermal diffusivity only slightly. A gold layer is always coated with a carbon layer because the gold layer reflects the laser flash. The radiative heat losses were minimised by measuring the samples in vacuum. The molar density ρm and heat capacity at constant volume CV required for calculating the thermal conductivity were obtained from the literature
(density [38] and heat capacity [38, 39] assuming that CV = Cp resulting in a maximum relative error of approximately 10 % [40]).
208 A new method for estimation of the intrinsic thermal conductivity
Table 9-1: Preparation characteristics as reported for several MgSiN2 ceramic samples [34]. Sample Densification method and Additives Oxygen Grain reaction conditions content size [wt. %] [µm] RB02 hot-pressing None 3.8 —
1823 K, N2, 75 MPa, 2 h RB11 hot-pressing None 1.8 —
1823 K, N2, 75 MPa, 2 h RB13 reaction hot-pressing None 1.0 ~ 0.5
1873 K, N2, 75 MPa, 2 h
RB32 reaction hot-pressing 4.2 wt. % Mg3N2 1.0 —
1873 K, N2, 75 MPa, 2 h RB34 reaction hot-pressing none 1.0 ~ 1.5
1973 K, N2, 75 MPa, 2 h
RB37 reaction hot-pressing 6.0 wt. % Y2O3 — —
1873 K, N2, 75 MPa, 2 h
In the literature many temperature dependent thermal diffusivity/ conductivity data are reported for several AlN [4, 32, 41 - 47] and Si3N4 [6, 48 - 52] ceramics having different thermal properties. When necessary, the thermal diffusivity as a function of the temperature was calculated from the temperature dependence of the thermal conductivity, the density and heat capacity reported in the corresponding reference or literature [38].
4. Results for MgSiN2, AlN and β-Si3N4
4.1. The temperature dependence of the thermal diffusivity a
As expected, the thermal diffusivity for the MgSiN2 samples processed in different ways decreases for higher temperatures (Fig. 9-1). The same is true for the AlN
(Table 9-2) and (β-)Si3N4 (Table 9-3) samples prepared in different ways (for details concerning the processing see the corresponding references). For all three materials the observed thermal diffusivity/conductivity at 300 K varied over a
209 Chapter 9.
Table 9-2: Preparation characteristics as reported in the literature for several AlN ceramic materials.
Sample Densification method and Additives reaction conditions Single crystal W-201 [41] Sublimation-recondensation none
2523 K, 95 % N2/ 5 % H2 [53] Shapal [42, 54]Not reported □ AlN without additive [4] Hot-pressing, 2123 K, 10 min. none annealing, 2123 K, 100 min. BP research AlN [32, 54] Not reported Shapal SH-04 [32, 54] Not reported Shapal SH-15 Super Shapal Toshiba TAN-170 [32, 54] Not reported Carborundum AlN [32, 54] Not reported
AlN [43] Pressureless sintering 4 wt. % Y2O3
2023 K, N2, 10 h
B(N2) [44] Pressureless sintering 1 wt. % Y2O3
2133 K, N2, 1 h
H(N2) 3 wt. % Y2O3
G(N2) 10 wt. % Y2O3
C1 [44, 45] Pressureless sintering 3 wt. % Y2O3 + 0 wt. % CaO
2098 K, N2, 1 h
I1 [44] 3 wt. % Y2O3 + 1 wt. % CaO
B1 [44, 45] 3 wt. % Y2O3 + 2 wt. % CaO
H(N2) [44, 46] Pressureless sintering 3 wt. % Y2O3 + 0 wt. % SiO2
2133 K, N2, 1 h
N(N2) [44] 3 wt. % Y2O3 + 0.3 wt. % SiO2
O(N2) [44, 46] 3 wt. % Y2O3 + 1 wt. % SiO2
Q(N2) [44, 46] 3 wt. % Y2O3 + 2 wt. % SiO2
S(N2) [44, 46] 3 wt. % Y2O3 + 5 wt. % SiO2
-1 -1 relatively broad range (MgSiN2: 16 - 23 W m K (Table 9-4); AlN: -1 -1 -1 -1 24 - 285 W m K (Table 9-5); β-Si3N4: 14 - 122 W m K (Table 9-6)), indicating large differences in impurity content and microstructure for the different samples. The difference in thermal diffusivity between the samples is less
210 A new method for estimation of the intrinsic thermal conductivity
Table 9-3: Preparation characteristics and resulting β-fraction as reported in the literature for several
β-Si3N4 and α/β-Si3N4 composite ceramic materials.
Sample Densification method Additives β-fraction and reaction conditions
SN5 [51] gas-pressure sintered 0.5 mol % Y2O3 + 0.5 mol % 100
2473 K, 30 MPa (N2), 4 h Nd2O3 A [48] high pressure hot-pressing None 100 2173 K, 3 GPa, 1 h B 4 wt. % MgO 100
D4 wt. % Al2O3 100
C high pressure hot-pressing 4 wt. % Y2O3 100 2073 K, 3 GPa, 1 h
□ [49] capsule-HIPped 3 mol % Y2O3 + 3 mol % Al2O3 ~ 25 1973 K, 60 MPa (Ar), 1 h
○ 2 mol % Y2O3 + 4 mol % Al2O3 ~ 34
▲ 4 mol % Y2O3 + 2 mol % Al2O3 ~ 67
■-100 capsule-HIPped 3 mol % Y2O3 + 3 mol % Al2O3 100 2023 K, 60 MPa (Ar), 1 h
■- 90 capsule-HIPped 3 mol % Y2O3 + 3 mol % Al2O3 ~ 90 1973 K, 60 MPa (Ar), 1 h
■- 34 capsule-HIPped 3 mol % Y2O3 + 3 mol % Al2O3 ~ 34 1823 K, 60 MPa (Ar), 1 h
+ 6/0 capsule-HIPped 6 mol % Y2O3 + 0 mol % Al2O3 100 2023 K, 60 MPa (Ar), 1 h
+ 5/1 5 mol % Y2O3 + 1 mol % Al2O3 100
+ 4/2 4 mol % Y2O3 + 2 mol % Al2O3 100
+ 3/3 3 mol % Y2O3 + 3 mol % Al2O3 100
+ 2/4 2 mol % Y2O3 + 4 mol % Al2O3 100
+ 1/5 1 mol % Y2O3 + 5 mol % Al2O3 100
+ 0/6 0 mol % Y2O3 + 6 mol % Al2O3 100
Tape cast [6] hot-pressed, 2073 K, 5 wt. % Y2O3 + 5 vol. % — 40 MPa, 2 h and rod-like β-Si3N4 seeds Subsequently HIPped,
2773 K, 200 MPa (N2), 2 h pronounced at higher temperatures (see e.g. Fig. 9-1) as then intrinsic phonon scattering processes are dominating the thermal diffusivity/conductivity (because
1 1 1 > + + ∑ 1 so A'T > B'). l l l pp pd gb x lx
211 Chapter 9.
1.0E-05 RB02 RB11 8.0E-06 RB13
] RB32 -1
s RB34 2 6.0E-06 RB37 [m a
4.0E-06
2.0E-06 0 200 400 600 800 1000 T [K]
Fig. 9-1: The thermal diffusivity (a) plotted versus the absolute
temperature (T ) for several MgSiN2 samples.
4.2. Inverse thermal diffusivity a -1 versus temperature T plots
For all three compounds the inverse of the thermal diffusivity plotted against the absolute temperature can be described with a linear fit (Fig. 9-2 - Fig. 9-8) resulting
4.5E+05
4.0E+05 RB02 RB11 RB13 RB32 3.5E+05 RB34 RB37 3.0E+05 ] -2 2.5E+05
[sm 2.0E+05 -1 a 1.5E+05
1.0E+05
0.5E+05
0.0E+00 0 200 400 600 800 1000 T [K]
Fig. 9-2: The inverse thermal diffusivity (a -1) versus the absolute
temperature (T ) plot for MgSiN2 ceramic samples processed in different ways.
212 A new method for estimation of the intrinsic thermal conductivity
in a good description of the temperature dependence (R > 0.99) (Table 9-4 - Table 9-6). The indicated uncertainties for the slope and the intercept correspond with the 95% confidence interval. -2 -1 For all MgSiN2 samples about the same slope A' of 400 - 430 m s K is observed (Table 9-4 and Fig. 9-2) indicating that the lattice characteristics are not influenced by the processing conditions used. On the contrary, the intercept B' shows a relative large variation, as the samples differ in impurity content and grain size [34]. As expected, the samples with the highest purity and grain size have in general the lowest intercept value.
Table 9-4: Slope and intercept values (with R-value) of linearly fitted inverse thermal diffusivity (a -1) versus the absolute temperature (T ), and
measured room temperature (~ 300 K) thermal conductivity κ300 for
MgSiN2 ceramic samples proceed in different ways.
Sample Slope A' Intercept B' R-value κ300 [m-2 s K-1][m-2 s] [-] [W m-1 K-1] RB02 424.6 ± 7.5 5.3 ± 4.6 103 0.9992 19 RB11 409.8 ± 8.0 27.7 ± 4.9 103 0.9990 16 RB13 411.8 ± 9.5 4.0 ± 5.8 103 0.9987 20 RB32 394.5 ± 6.6 - 1.7 ± 4.0 103 0.9993 21 RB34 402.8 ± 4.4 -13.6 ± 2.7 103 0.9997 23 RB37 437.2 ± 5.1 -19.7 ± 3.1 103 0.9997 22 Mean 413.5 ± 14.0 — — —
The AlN ceramics processed with several additives have a typical slope value of 80 - 90 m-2 s K-1 (Table 9-5). These values are somewhat smaller than the value observed for the hot-pressed AlN sample without sintering additive (104.8 ± 3.0 m-2 s K-1). Also for the best heat conducting sample (single crystal W-201 [41]) a somewhat larger slope is observed (100.0 ± 1.0 m-2 s K-1) as compared to the typical value (Fig. 9-3). This observation is related to the fact that this sample is a single crystal for which the thermal conductivity was determined
213 Chapter 9.
along the c-axis resulting for the a -1 versus T plot in the anisotropic slope value of the c-axis.
Table 9-5: Slope and intercept values with R-value of linearly fitted inverse thermal diffusivity (a -1) versus the absolute temperature (T ) and measured room temperature (~ 300 K) thermal
conductivity κ300 for AlN ceramic samples processed in different ways.
Sample Slope A' Intercept B'R-value κ300 [m-2 s K-1][m-2 s] [-] [W m-1 K-1] single crystal W-201 [41] 100.0 ± 1.0 - 22.79 ± 0.86 103 0.9999 285 Shapal [41, 54] 83.5 ± 1.5 - 8.08 ± 1.10 103 0.9986 141 □ AlN without additive [4] 104.8 ± 3.0 3.38 ± 1.75 103 0.9988 70 BP research AlN [32, 54] 81.6 ± 1.0 - 14.33 ± 0.45 103 0.9996 228 Shapal SH-04 [32, 54] 85.6 ± 1.6 - 11.66 ± 0.77 103 0.9995 167 Shapal SH-15 91.9 ± 1.5 - 14.62 ± 0.77 103 0.9995 144 Super Shapal 84.4 ± 1.9 - 14.79 ± 0.94 103 0.9987 212 Toshiba TAN-170 [32, 54] 85.0 ± 1.5 - 11.69 ± 0.69 103 0.9995 170 Carborundum AlN [32, 54] 82.4 ± 1.1 - 13.81 ± 0.51 103 0.9995 212 3 AlN [43] (4 wt. % Y2O3) 89.3 ± 9.0 - 16.11 ± 4.21 10 0.9850 208 3 B(N2) [44] (1 wt. % Y2O3) 87.0 ± 1.9 - 4.72 ± 1.05 10 0.9990 119 3 H(N2) (3 wt. % Y2O3) 91.6 ± 2.4 - 13.41 ± 1.35 10 0.9986 159 3 G(N2) (10 wt. % Y2O3) 94.3 ± 1.1 - 12.74 ± 0.58 10 0.9998 148 C1 [44, 45] (0 wt. % CaO) 89.2 ± 1.0 - 10.50 ± 0.56 103 0.9997 144 I1 [44] (1 wt. % CaO) 91.5 ± 0.5 - 9.11 ± 0.30 103 0.9999 129 B1 [44, 45] (2 wt. % CaO) 100.0 ± 1.0 - 10.97 ± 0.52 103 0.9998 124 3 H(N2) [44, 46] (0 wt. % SiO2) 91.6 ± 2.4 - 13.41 ± 1.35 10 0.9986 159 3 N(N2) [44] (0.3 wt. % SiO2) 88.5 ± 0.6 - 8.19 ± 0.32 10 0.9999 129 3 O(N2) [44, 46] (1 wt. % SiO2) 96.0 ± 1.5 - 6.62 ± 0.83 10 0.9995 106 3 Q(N2) [44, 46] (2 wt. % SiO2) 148.9 ± 5.3 9.10 ± 2.93 10 0.9935 46 3 S(N2) [44, 46] (5 wt. % SiO2) 200.4 ± 11.4 45.83 ± 6.30 10 0.9975 24
The resulting slope value is not much influenced by the addition of small amounts of Y2O3 (≤ 10 wt. %) (Fig. 9-4) and CaO (≤ 2 wt. % together with 3 wt. %
Y2O3 (Table 9-5)), whereas the slope changes drastically for larger amounts SiO2
214 A new method for estimation of the intrinsic thermal conductivity
addition (≥ 2 wt. % together with 3 wt. % Y2O3) (Fig. 9-5). From these observations it can be concluded that Y2O3 and CaO additions mainly influence the defect chemistry and microstructure of the AlN ceramics (phonon-defect and phonon-grain boundary scattering), whereas SiO2 addition also results in a change of the lattice characteristics (phonon-phonon scattering). In complete agreement with this conclusion, De Baranda et al. [46] reported that for an SiO2 addition of
2 wt. % together with 3 wt. % Y2O3 and above sialon polytypoids with an AlN like structure are formed, resulting in the formation of a different lattice and thus a different slope value (Table 9-5 and Fig. 9-5).
1.0E+05 without additives Shapal, non-irradiated 0.8E+05 single crystal W-201 ]
-2 0.6E+05 [s m
-1 0.4E+05 a
0.2E+05
0.0E+00 0 200 400 600 800 1000 1200 T [K]
Fig. 9-3: The inverse thermal diffusivity (a -1) versus temperature (T ) plot for AlN samples without additive (×), a typical sample () and a single crystal (▲).
The intercept value B' is the smallest for the (almost) defect free single crystal and largest for hot-pressed ceramics processed without additives containing many defects due to the oxygen impurities dissolved into the AlN lattice (Fig. 9-3 and Table 9-5). By suitable processing (typical sample) the defect concentration in the AlN lattice is reduced resulting in a decrease of the intercept approaching the value for the (almost) defect free single crystal.
215 Chapter 9.
1.0E+05 AlN without additives
1 wt. % Y2O3 0.8E+05 3 wt. % Y2O3
AlN 4 wt. % Y2O3
] 10 wt. % Y2O3
-2 0.6E+05 [s m
-1 0.4E+05 a
0.2E+05
0.0E+00 0 100 200 300 400 500 600 700 800 T [K]
Fig. 9-4: The inverse thermal diffusivity (a -1) versus temperature (T ) plot for AlN ceramics, processed with different amounts of
Y2O3 as a sintering additive (data from several references).
2.5E+05 3 wt. % Y2O3 + 0 wt. % SiO2
3 wt. % Y2O3 + 0.3 wt. % SiO2
2.0E+05 3 wt. % Y2O3 + 1 wt.% SiO2
3 wt. % Y2O3 + 2 wt. % SiO2
3 wt. % Y2O3 + 5 wt. % SiO2 ]
-2 1.5E+05 [s m
-1 1.0E+05 a
0.5E+05
0.0E+00 0 100 200 300 400 500 600 700 800 T [K]
Fig. 9-5: The inverse thermal diffusivity (a -1) versus temperature (T )
plot for AlN ceramics, processed with 3 wt. % Y2O3 and
different amounts of SiO2 as sintering additives.
With increasing Y2O3 addition the intercept value B' first decreases and subsequently increases again (Table 9-5 and Fig. 9-4) in agreement with other observations [55, 56] that with increasing Y2O3 addition the thermal conductivity first increases (till about 4 - 6 wt. % addition [56]) and subsequently decreases.
This indicates that (as expected) Y2O3 is an effective sintering aid for sintering of
216 A new method for estimation of the intrinsic thermal conductivity
AlN by reducing the defect concentration (Al vacancies) in the AlN lattice. For higher dopant levels the thermal conductivity decreases as the thermal conductivity of yttrium aluminates (and Y2O3) is much lower than that for AlN [4, 55] resulting in an increase of the observed slope value too.
4.0E+05 D 4wt% Al2O3
3.5E+05 C 4wt% Y2O3 B 4wt% MgO 3.0E+05 A no additives
] 2.5E+05 SN5 β-Si3N4 -2 2.0E+05 [s m -1
a 1.5E+05
1.0E+05
0.5E+05
0.0E+00 0 500 1000 1500 2000 T [K]
Fig. 9-6: The inverse thermal diffusivity (a -1) versus temperature (T )
plot for β-Si3N4 ceramics processed in different ways.
The lowest slope value A' for the isotropic β-Si3N4 samples equals 110 - 130 m-2 s K-1 (Table 9-6). The slope observed for the sample with the highest thermal diffusivity/conductivity (SN5) equals (129.1 ± 2.9 m-2 s K-1). The addition of MgO and Y2O3 has only a limited influence on the slope, whereas in contrast the addition of Al2O3 has a strong effect (Fig. 9-6). The reason for this different behaviour is that the Al2O3 addition can dissolve into the β-Si3N4 lattice resulting in the formation of a β-sialon (Si6-zAlzOzN8-z), whereas Y2O3 and MgO can only react with SiO2 on the surface of the Si3N4 grains to form a separate secondary phase. The relatively large scattering in the data points for the samples A to D, especially at higher temperatures (Fig. 9-6), can be partially ascribed to the inaccuracy introduced when obtaining the data from a plot of ref. 48.
For a lower β content of the α /β-Si3N4 composite ceramics (Table 9-3) the observed slope increases (Table 9-6). This observation can be explained in view of
217 Chapter 9.
the difference between the crystal structure of the α- and β-modifications of Si3N4.
As the α-modification is more complex than the β-modification (α-Si3N4: n = 28;
β-Si3N4: n = 14) it is expected for α-Si3N4 to have a higher value for the slope A' ~ (= bA/θ = bAn1/3/θ ~ n, assuming that b, A, and θ are about the same for both modifications) and thus a lower intrinsic thermal diffusivity/conductivity than
β-Si3N4.
Table 9-6: Slope and intercept values with R-value of linearly fitted inverse thermal diffusivity (a -1) versus the absolute temperature (T ) and measured room temperature (~ 300 K) thermal
conductivity κ300 for several (β-)Si3N4 ceramic samples.
Sample Slope A' Intercept B'R-valueκ300 [m-2 s K-1][m-2 s] [-] [W m-1 K-1] SN5 [51] 129.1 ± 2.9 - 22.84 ± 3.14 103 0.9985 122 A [48] (without additive) 125.7 ± 6.3 42.57 ± 4.49 103 0.9838 30 B (4 wt. % MgO) 143.6 ± 5.1 38.46 ± 3.51 103 0.9910 29 3 D (4 wt. % Al2O3) 251.3 ± 19.3 106.38 ± 13.67 10 0.9636 14 3 C (4 wt. % Y2O3) 159.0 ± 9.5 64.24 ± 6.71 10 0.9910 22 □ [49] 210.6 ± 4.3 20.36 ± 3.18 103 0.9990 28 ○ 199.8 ± 6.6 31.35 ± 4.82 103 0.9967 26 ▲ 146.2 ± 5.4 48.01 ± 3.91 103 0.9960 24 ■-100 143.5 ± 4.4 38.24 ± 4.02 103 0.9986 28 ■- 90 143.8 ± 6.6 53.44 ± 4.82 103 0.9937 22 ■- 34 169.2 ± 10.4 83.86 ± 7.56 103 0.9889 16 + 6/0 112.1 ± 1.3 - 3.77 ± 1.20 103 0.9998 73 + 5/1 116.0 ± 1.2 6.40 ± 1.10 103 0.9998 53 + 4/2 127.7 ± 3.6 26.94 ± 3.30 103 0.9988 35 + 3/3 143.5 ± 4.4 38.24 ± 4.02 103 0.9986 28 + 2/4 150.9 ± 4.7 50.57 ± 4.34 103 0.9985 23 + 1/5 137.5 ± 7.4 67.91 ± 6.71 103 0.9957 21 + 0/6 146.2 ± 6.1 92.43 ± 5.53 103 0.9974 17 tape-casting direction [6] 84.5 ± 2.1 -8.96 ± 1.52 103 0.9954 155 Stacking direction 187.0 ± 4.8 -22.29 ± 3.10 103 0.9960 70
218 A new method for estimation of the intrinsic thermal conductivity
3.0E+05 6 mol % Y2O3 / 0 mol % Al2O3
4 mol % Y2O3 / 2 mol % Al2O3 2.5E+05 2 mol % Y2O3 / 4 mol % Al2O3
0 mol % Y2O3 / 6 mol % Al2O3 2.0E+05 ] -2 1.5E+05 [s m -1 a 1.0E+05
0.5E+05
0.0E+00 0 500 1000 1500 T [K]
Fig. 9-7: The inverse thermal diffusivity (a -1) versus temperature (T )
plot for β-Si3N4 ceramics using mixtures of Y2O3 and Al2O3 as sintering additives.
A nice illustration of the influence of the type and amount of additive on the slope and intercept values can be obtained from the data of Watari [49] who studied the influence of in total 6 mol % Y2O3 and Al2O3 addition on the thermal conductivity of β-Si3N4 (Table 9-6 and Fig. 9-7). It can be concluded that Y2O3 without Al2O3 is an effective additive for increasing the thermal conductivity of
β-Si3N4 because it does not dissolve in the lattice (slope A' ≈ constant ≈ 110 m-2 s K-1) and decreases the intercept B' (< 0), whereas with increasing
Al2O3/Y2O3 ratio a sialon is formed resulting in a change of the lattice characteristics (increase of the slope A' (= bAn1/3/θ ) due to lowering θ as a consequence of Si-N → Al-O replacement) and defect concentration (increase of the intercept B' due to Al on Si site and O on N site acting as scattering centres for phonons) (Fig. 9-7 and Table 9-6). Recently it was demonstrated that the thermal diffusivity/conductivity of -1 β-Si3N4 is strongly anisotropic [57]. This observation is fully supported by the a versus T plot of thermal conductivity data of a tape-cast sample (Fig. 9-8) obtained from Ref. 6 showing two different slope values, the one in the casting direction (predominantly along c-axis) below the typically observed value and the one in the stacking direction (predominantly along a-axis) above the typically observed slope
219 Chapter 9.
value (Table 9-6). It is worth noting that the average slope value of the tape-cast sample equals 133 m-2 s K-1 (= 3(1/84.5 + 2/187)-1) which is very close to the value observed (129 m-2 s K-1) for the best isotropic sample SN5 (Table 9-6).
2.5E+05 Tape-casting direction Stacking direction 2.0E+05 isotropic β-Si3N4 (SN5) ]
-2 1.5E+05 [s m
-1 1.0E+05 a
0.5E+05
0.0E+00 0 250 500 750 1000 1250 T [K]
Fig. 9-8: The inverse thermal diffusivity (a -1) versus temperature (T )
plot for β-Si3N4 ceramics along the casting and stacking direction as compared to an isotropic sample.
5. Discussion
From the results of the a -1 versus T plots it is clear that these plots can be very useful for optimisation of the thermal diffusivity/conductivity. The data of a material processed in different ways can be used to study the influence of several additives. Increase of the slope indicates that the additive dissolves into the lattice, whereas a decrease in slope or intercept indicates that the additive improves the thermal conductivity.
5.1. Interpretation of the fitting parameters
In general the observed slopes A' for the three materials have a typical constant -2 -1 -2 -1 value (MgSiN2: 400 - 430 m s K (Table 9-4); AlN: 80 - 90 m s K (Table 9-5);
220 A new method for estimation of the intrinsic thermal conductivity
-2 -1 and β-Si3N4: 110 - 130 m s K (Table 9-6)) and deviations from this constant value can be explained in view of the lattice characteristics. For the samples with the same lattice characteristics (constant A') but with different impurity content and microstructure a relatively large variation in the intercept value B' can be observed. Considering the large variation in thermal conductivity observed for the samples with a approximately constant slope value for the inverse thermal diffusivity versus absolute temperature plot, it can be concluded that all phonon scattering processes, except the intrinsic phonon-phonon scattering, are indeed (almost) temperature independent. As expected from the theory moreover also negative intercept values are found. This indicates that the presented theoretical concept has a sound basis.
5.2. Thermal conductivity estimates for MgSiN2, AlN and β-Si3N4
In order to estimate the maximum achievable theoretical thermal diffusivity/ ~ conductivity (B = 0), besides the slope A' (= bA/θ ) the (theoretical) intercept with ~ the a -1-axis (= -½ A) or the T-axis (= θ /2b) should be known. For the present discussion the intercept with the T-axis was used as this value is only dependent on ~ ~ θ and b. For estimation of this intercept with the T-axis at T = θ /2b (= θ /2bn1/3) reliable values for θ and b are needed. The Debye temperature can be evaluated from elastic constants or heat capacity data near 0 K [11] resulting in θ 0. Recently the θ 0 data obtained from elastic constants for MgSiN2, AlN and Si3N4, have been reported [58] (Table 9-7). The number of atoms per primitive unit cell n can be obtained from θ~ crystallographic data (Table 9-7). This results in a reduced Debye temperature ( 0 ) of 357, 592 and 396 K for MgSiN2, AlN and Si3N4, respectively. As expected, AlN θ~ with the highest 0 shows the lowest slope value A' (Table 9-7). The value of b for describing the temperature dependence of the thermal diffusivity is not exactly known. Based on the simple Debye theory it can be argued that b = 2 [17, 18, 24, 26]. This results in a theoretically calculated intercept of 89, 148 and 99 K for pure, defect free MgSiN2, AlN and Si3N4 ceramics, respectively (Table 9-7).
221 Chapter 9.
~ Table 9-7: The measured slope A' (= bA/θ ), the Debye temperature θ0, the number of atoms per primitive unit cell n, the resulting reduced θ~ θ 1/3 Debye temperature 0 (= 0/n ) and the calculated intercept θ~ β (= 0 /2b with b = 2) for MgSiN2, AlN and -Si3N4.
θ θ~ Material slope A' 0 [58] n 0 intercept [m-2 s K-1] [K] [-] [K] [K] 2 MgSiN2 4.0 - 4.3 10 900 16 [59] 357 89 AlN 0.8 - 0.9 102 940 4 [60] 592 148 2 β-Si3N4 1.1 - 1.3 10 955 14 [61] 396 99
~ For T ≥ θ /2 (T ≥ 179 K, 296 K and 198 K for MgSiN2, AlN and β-Si3N4, respectively) and T ≤ 2θ the maximum achievable thermal diffusivity of MgSiN2,
AlN and β-Si3N4 can be estimated by using the linear extrapolation method of temperature dependent thermal diffusivity measurements. Using the theoretical intercept with the T-axis (Table 9-7) and the experimental slope values for non-optimised (badly-conducting) fully dense -2 -1 -2 -1 MgSiN2 (409.8 m s K for sample RB11 (Table 9-4)), AlN (104.8 m s K for -2 -1 sample □ AlN without additive [4] (Table 9-5)) and β-Si3N4 (125.7 m s K for sample A [48] (without additive) (Table 9-6)) ceramics using no additives during sintering (so the influence on A' is limited) a conservative estimate of the maximum achievable (intrinsic) thermal diffusivity was made. From these data the corresponding thermal conductivity was calculated using the heat capacity and the density data of the corresponding materials. This resulted in the prediction that the room temperature thermal conductivity can be at least improved from 16 to -1 -1 -1 -1 28 W m K for MgSiN2, from 70 to 153 W m K for AlN and 30 to -1 -1 82 W m K for β-Si3N4. Considering the quality of the used samples reasonable -1 -1 estimates as compared to the highest experimental values of MgSiN2 (23 W m K -1 -1 -1 -1 -1 -1 [15]), AlN (246 W m K [4] - 266 W m K [62]) and β-Si3N4 (106 W m K [57] - 122 W m-1 K-1 [51]) for isotropic materials are obtained, indicating that this
222 A new method for estimation of the intrinsic thermal conductivity
method is in general very powerful in providing an indication of the maximum achievable thermal conductivity of optimised ceramics. Taking the typical experimentally observed slope A' and the theoretically calculated intercept (Table 9-7), the maximum achievable thermal diffusivity at
300, 600 and 900 K was calculated (Table 9-8) for MgSiN2, AlN and Si3N4. From these data the corresponding maximum achievable thermal conductivity was calculated using the heat capacity and the density data of the corresponding materials and compared with the highest reported experimental values as a function of the temperature for MgSiN2 (300 - 900 K, RB34, see Fig. 9-1 and Table 9-4) [15], AlN (0.4 - 1800 K, (almost) pure single crystal along the c-axis) [41] and
β-Si3N4 (300 - 1700 K) [51].
Table 9-8: The estimates for the maximum achievable thermal diffusivity aThe usig the data
of Table 9-7 for MgSiN2, AlN and β-Si3N4 (at 300, 600 and 900 K) and resulting
thermal conductivity κ The (obtained from the molar density ρm and heat capacity
Cp), compared with corresponding highest experimentally observed thermal
conductivity κ Exp.
Material a ρm Cp = CV κThe κExp [m2 s-1][mol m-3][J mol-1 K-1][W m-1 K-1][W m-1 K-1] 300 K -5 4 MgSiN2 1.10 - 1.18 10 3.90 10 61.7 26 - 28 23 AlN 7.31 - 8.22 10-5 7.94 104 30.6 178 - 200 246 - 285 -5 4 β-Si3N4 3.83 - 4.52 10 2.29 10 90.6 79 - 94 106 - 122 600 K -5 4 MgSiN2 0.46 - 0.49 10 3.88 10 88.1 16 - 17 15 AlN 2.46 - 2.77 10-5 7.91 104 44.0 86 - 96 96 -5 4 β-Si3N4 1.54 - 1.81 10 2.28 10 144.5 51 - 60 63 900 K -5 4 MgSiN2 0.29 - 0.31 10 3.86 10 95.6 11 - 11 11 AlN 1.48 - 1.66 10-5 7.87 104 47.7 56 - 62 55 -5 4 β-Si3N4 0.96 - 1.13 10 2.27 10 157.0 34 - 40 41
223 Chapter 9.
For MgSiN2 ceramics the highest experimentally obtained thermal conductivity at 300 K does not exceed 25 W m-1 K-1 despite the fact that already considerable effort has been made to improve the thermal conductivity [9, 34, 36, 37, 63, 64]. As the predicted value of 26 - 28 W m-1 K-1 is close to this value it can be concluded that the highest experimentally observed value is close to the intrinsic one. From Table 9-8 and Fig. 9-2 it is obvious that a further reduction of the defect concentration in the MgSiN2 lattice will not result in a significant increase of the thermal diffusivity/conductivity, because for the best samples the intercept with the T-axis (Fig. 9-2) is already very close to the theoretical value of 89 K (Table 9-7).
The estimates for β-Si3N4 are in reasonable agreement with the highest experimentally observed values indicating that the thermal conductivity cannot be significantly increased. The measured value of 122 W m-1 K-1 at 300 K [51] is somewhat higher than the expected 106 W m-1 K-1 estimated from the measured thermal conductivity along the c-axis (180 W m-1 K-1) and a-axis (69 W m-1 K-1) for a single grain [57]. However, the estimate at 300 K (178 - 200 W m-1 K-1) for the thermal conductivity of AlN is significantly lower than the observed experimental values for isotropic materials (246 - 266 W m-1 K-1). Evidently the intercept with the T-axis is underestimated. This is caused by underestimation of the reduced Debye temperature as also optic phonons contribute to the heat conduction [10] whereas only acoustic phonons are considered. This underestimation of the reduced Debye ~ temperature θ also results in a too low value for the maximum achievable thermal conductivity as estimated with the Slack equation [10]. When using the a -1 versus ~ T method, both the slope A' (= bA/θ ) and the (theoretical) intercept with the T-axis ~ ~ (= θ /2b) are related to θ . However, especially at high temperatures the influence ~ of θ on the estimate is limited as a -1 = A'T + B' ≈ A'T and A' is determined experimentally (see Table 9-8: 600 and 900 K estimates). In general the theoretical estimates are in good agreement (within 20 %) with the best experimentally observed values (Table 9-8), unless also optic phonons contribute substantially to the heat conduction (like in the case of AlN), resulting in
224 A new method for estimation of the intrinsic thermal conductivity
~ an underestimation of the intercept with the T-axis (= θ /2b). However, at higher ~ 1 θ temperatures (T / 3 × 4 0 ) the exact value of the intercept with the T-axis becomes less important, and therefore also the influence of optic phonons contributing to the heat conduction, as a -1 = A'T + B' ≈ A'T (high T ) resulting in a better agreement between the estimated and the experimentally observed thermal conductivity. At high temperatures the accuracy of the estimate is consequently determined by the error in the slope A'. Furthermore, these calculations directly indicate that the thermal conductivity of AlN and β-Si3N4 is relatively high whereas that of MgSiN2 is limited. So, for applications where a high thermal conductivity is required β-Si3N4 (and AlN) is a more interesting compound than
MgSiN2.
5.3. Comparison with other estimates
If the here presented estimated value of the maximum achievable thermal -1 -1 conductivity of MgSiN2 ceramics at 300 K of 26 - 28 W m K is compared with previous estimates which have been made for MgSiN2 ceramics at 300 K (Table 9-9), it is noticed that the presented value is the lowest of all. Especially the difference with the first estimated values using the theory of Slack is considerable. During time these values were adjusted downwards, as more accurate input parameters became available [38, 39, 58]. The estimate of the thermal conductivity for AlN (178 - 200 W m-1 K-1) is much lower than that of 319 W m-1 K-1 at 300 K [41]. In the meantime this value of 319 W m-1 K-1 has been widely interpreted as being the true intrinsic value for AlN. However, this value was obtained by correcting the measured thermal conductivity for defect scattering by oxygen impurities using the experimental value of 285 W m-1 K-1 measured on a single crystal providing the thermal conductivity along the c-axis [41]. This axis has the lowest thermal expansion coefficient [65] and therefore it is expected to show the highest thermal conductivity, as it is empirically known that in general the direction with the lowest thermal expansion
225 Chapter 9.
Table 9-9: Theoretical estimates of the thermal conductivity for MgSiN2, AlN and β-Si3N4 ceramics
at 300 K. For comparison the highest measured values (κ Exp) are also given (* probably based on n-2/3 dependence of the Slack equation [11] and the intrinsic estimate of about 300 W m-1 K-1 for AlN).
Estimated Value Reference Estimation Method Based On [W m-1 K-1]
-1 -1 MgSiN2 κExp = 23 W m K [15] 26 - 28 this work a -1 versus T 27 - 35 [10] modified Slack equation 34 [10] standard Slack equation 26 ± 4 [15] thermal diffusivity measurements → 37 ± 13 [15] modified Slack equation
time time 35 - 50 [63] defect scattering 40 - 70 [63] Slack equation 75 [66] Slack equation 120 [8] not specified* -1 -1 AlN κExp = 266 W m K [62] 178 - 200 this work a -1 versus T 124 - 209 [10] modified Slack equation → 128 [10] standard Slack equation
time time 319 [41] defect scattering 320 [2] scaling factor Mδθ 3 -1 -1 β-Si3N4 κExp = 106 W m K [57] 79 - 94 this work a -1 versus T 84 - 116 [10] modified Slack formula → 124 [10] standard Slack formula
time time 177 [68] two-phase composite model 200 - 320 [5] Slack equation coefficient has the highest thermal conductivity [67]. Also the observed slope value in the a -1 versus T plot for this sample (Fig. 9-3) differed from the typically observed value confirming the anisotropic behaviour of the thermal conductivity. Therefore, the present author has the feeling that the estimate of 319 W m-1 K-1 [41]
226 A new method for estimation of the intrinsic thermal conductivity
related to one direction is too high for isotropic AlN, and the intrinsic thermal conductivity for isotropic AlN equals about the highest experimentally observed thermal conductivity of 246 - 266 W m-1 K-1 [4, 62] for isotropic samples. So, the here presented estimated value of 178 - 200 W m-1 K-1 is lower as compared to the true intrinsic value of about 246 - 266 W m-1 K-1 for isotropic material [4, 62]. However, this estimate of 178 - 200 W m-1 K-1 (at 300 K) is in much better agreement with the highest experimentally observed value than the estimates obtained using the Slack formula or the modified Slack formula (Table 9-9). The present estimate of 79 - 94 W m-1 K-1 for the intrinsic thermal conductivity of β-Si3N4 is significantly lower than the first reported estimate of 200 - 320 W m-1 K-1 [5], whereas it is in good agreement with previous estimates (Table 9-9) based on the Slack equation, the modified Slack equation, and the experimentally measured value on a single grain (106 W m-1 K-1 [57]), indicating that the present estimate is only slightly lower than the intrinsic value of β-Si3N4. The measured thermal conductivity along the c-axis of 180 W m-1 K-1 is in good agreement with the value estimated using a two-phase composite model resulting in a value of 177 W m-1 K-1 [68] along the c-axis. This provides some further confidence that the intrinsic value equals about 106 W m-1 K-1 [57] indicating that the value of 122 W m-1 K-1 [51] was measured on a somewhat anisotropic sample.
5.4. Limitations, accuracy and reliability
It should be noted that the new estimation method based on equation (5) was obtained by approximating an already simple description of the (temperature dependence of the) thermal diffusivity (equation (4)) of a pure phonon conductor. Furthermore the value of b = 2 in equation (4), which was used to calculate the ~ intercept with the T-axis (= θ /2b), may differ somewhat from 2 and vary from substance to substance [17]. Leibfreid et al. [20] suggest that for a FCC lattice b = √5/3. However, some scarce experimental results confirm the value of b ≈ 2 (2.3, 2.7 and 2.1 for solid helium, diamond and sapphire, respectively [18]). Also the choice of θ = θ 0 is somewhat arbitrary. In general the Debye temperature θ can
227 Chapter 9.
be obtained from elastic constants or heat capacity data [31, 58] resulting in θ E and θ C, respectively. Assuming that the acoustic phonons are the major heat carriers the high temperature limit of the Debye temperature based on the acoustic phonons
A θ∞ is needed [11] to evaluate the reduced Debye temperature. In a previous paper
A [10] a more appropriate estimate for θ∞ based on heat capacity data was presented resulting in somewhat lower values for θ . However, if also optic phonons contribute to the heat conduction, which is to some extend always the case for n > 1, this estimate for θ is too low. So, for practical use the choice of θ = θ 0 seems to be a good compromise as it can be easily obtained from elastic constants, and in combination with the assumption that b = 2 seems to results in reasonable estimates for the intercept with the T-axis. The presented estimation method seems to be more reliable than the theoretical Slack equation [11] or the modification of this equation [10] (Table 9-9). This can be explained in view of the influence of the accuracy of the reduced Debye temperature on the resulting estimate. The (more complicated) Slack equation is very sensitive for relatively small deviations in the input parameters [10], whereas the here presented method is relatively easy and less sensitive for small deviations in slope and intercept. In general, the Slack equation is especially useful when no samples for thermal diffusivity/conductivity measurements are available whereas the a -1 versus T plots give a more accurate indication of the maximum achievable thermal conductivity, and moreover can be used to guide the optimisation of the processing in order to obtain the desired thermal conductivity.
6. Conclusions
A new simple method for estimating the maximum achievable (intrinsic) thermal conductivity of non-metallic compounds was presented based on temperature dependent thermal diffusivity measurements. Its strength is that non-optimised samples can be used to provide a good impression of the intrinsic thermal
228 A new method for estimation of the intrinsic thermal conductivity
conductivity. It was successfully applied to MgSiN2 and β-Si3N4 providing some evidence for its general applicability. For AlN too low estimates were obtained due to the fact that optic phonons, which are not considered when using this method, contribute substantially to the heat conduction. However, in general the estimates are accurate within 20 % and become more accurate with increasing temperature, independent of the fact whether or not optic phonons contribute substantially to the heat conduction. Furthermore, the method is a useful tool for optimising the processing as it enables discrimination between the lattice characteristics, defects and microstructure.
References
1. A. Roosen, Modern Substrate Concepts for the Microelectronic Industry, Electroceramics IV 2, Aachen (Germany), September 5 - 7 1994, edited by R. Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann (Augustinus Buchhandlung, 1994) 1089. 2. M.P. Borom, G.A. Slack and J.W. Szymaszek, Thermal Conductivity of Commercial Aluminum Nitride, Bull. Am. Ceram. Soc. 51 (1972) 852. 3. W. Werdecker and F. Aldinger, Aluminum Nitride - An Alternative Ceramic Substrate for High Power Applications in Microcircuits, IEEE Trans. Compon., Hybrids, Manuf. Technol., CHMT-7 (1984) 399 - 404. 4. T.B. Jackson, A.V. Virkar, K.L. More, R.B. Dinwiddie, Jr. and R.A. Cutler, High-Thermal-Conductivity Aluminum Nitride Ceramics: The Effect of Thermodynamic, Kinetic, and Microstructural Factors, J. Am. Ceram. Soc. 80 (1997) 1421. 5. J.S. Haggerty and A. Lightfoot, Oppertunities for enhancing the thermal
conductivities of SiC and Si3N4 ceramics through improved processing, Ceram. Eng. Sci. Proc. 16, 19th Annual Conference on Composites, Advanced Ceramics, Materials, and Structures-A, Cocoa Beach, Florida, USA, January 8 - 12 1995, edited by J.B. Wachtman (The American Ceramic Society, Westerville, OH, 1995) 475.
229 Chapter 9.
6. K. Watari, K. Hirao, M.E. Brito, M. Toriyama and S. Kanzaki, Hot Isostatic
Pressing to Increase Thermal Conductivity of Si3N4 Ceramics, J. Mater. Res. 14 (1999) 1538. 7. K. Hirao, K. Watari, M.E. Brito, M. Toriyama and S. Kanzaki, High Thermal Conductivity of Silicon Nitride with Anisotropic Microstructure, J. Am. Ceram. Soc. 79 (1996) 2485. 8. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent
Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non- Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh, 1994) 239. 9. I.J. Davies, T. Shimazaki, M. Aizawa, H. Suemasu, A. Nozue and K. Itatani, Physical Properties of Hot-Pressed Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 276. 10. Chapter 8; R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, Room
Temperature Estimates of the Thermal Conductivity of MgSiN2, AlN and
β-Si3N4 using a Modified Slack Equation, to be published. 11. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1979) 1. 12. F.R. Chavat and W.D. Kingery, Thermal Conductivity: XIII, Effect of Microstructure on Conductivity of Single-Phase Ceramics, J. Am. Ceram. Soc. 40 (1957) 306. 13. D.-M. Liu and B.-W. Lin, Thermal Conductivity in Hot-Pressed Silicon Carbide, Ceram. Int., 22 (1996) 407. 14. W.D. Kingery, Thermal Conductivity: XII, Temperature Dependence of Conductivity for Single-Phase Ceramics J. Am. Ceram. Soc. 38 (1955) 251. 15. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modeling of the Thermal th Diffusivity/Conductivity of MgSiN2 Ceramics, ITCC 24 and ITES 12, 24 International Thermal Conductivity Conference and 12th International Thermal Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 26 - 29,
230 A new method for estimation of the intrinsic thermal conductivity
1997, edited by P.S. Gaal and D.E. Apostolescu (Technomic Publishing Co., Inc., Lancaster, 1999) 3. 16. R. Berman, Thermal Conduction in Solids, Clarendon Press, Oxford (1976). 17. P. Debye, Zustandsgleichung und Quantenhypotheses mit einem Anhang über Wärmeleitung in: Vorträge über die Kinetische Theorie der Materie und der Elektrizität (Teubner, Berlin, 1914), pp. 19 - 60. 18. R. Berman, The Thermal Conductivity of Dielectric Solids at Low Temperatures, Advances in Phys. 2 (1953) 103. 19. P.G. Klemens, Theory of the Thermal Conductivity of Solids, Thermal Conductivity 1, edited by R.P. Tye (Academic Press, London, 1969). 20. G. Leibfried and E. Schlömann, Wärmeleitung in elektrisch isolierenden Kristallen, Nachr. Akad. Wiss. Göttingen, Math. Physik. Kl. IIa (1954) 71. 21. P.G. Klemens, The thermal conductivity of dielectric solids at low temperatures, Proc. Roy. Soc. (London), A208 (1951) 108. 22. P.G. Klemens, Thermal Conductivity and Lattice Vibrational Modes, Solid State Phys. 7, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1958) 1. 23. K. Watari, K. Ishazaki and F. Tsuchiya, Phonon Scattering and Thermal Conduction Mechanisms of Sintered Aluminium Nitride Ceramics, J. Mater. Sci. 28 (1993) 3709. 24. R. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen, Ann. Phys. 3 (1929) 1055. 25. M. Roufosse and P.G. Klemens, Thermal Conductivity of Complex Dielectric Crystals, Phys. Rev. B. 7 (1973) 5379. 26. J.R. Drabble and H.J. Goldsmid, in International Series of Monographs on Semiconductors 4, Thermal Conduction in Semiconductors, edited by H.K. Henisch (Pergamon Press, Oxford, 1961), p. 141. 27. A.K. Collins, M.A. Pickering and R.L. Taylor, Grain size dependence of the thermal conductivity of polycrystalline chemical vapor deposited β-SiC at low temperatures, J. Appl. Phys. 68 (1990) 6510.
231 Chapter 9.
28. P.G. Klemens, Thermal Resistance due to Point Defects at High Temperatures, Phys.Rev. 119 (1960) 507. 29. V. Ambegaokar, Thermal Resistance due to Isotopes at High Temperatures, Phys. Rev. 114 (1959) 488. 30. P.G. Klemens, Thermal Conductivity of Lattice Vibrational Modes, Solid State Phys. 7, edited by F. Seitz and D. Turnbull (Academic Press Inc., Publishers, New York, 1958) 1. 31. P. Debije, Zur Theorie der spezifischen Wärmen, Ann. Physik 39 (1912) 789. 32. R.C. Enck and R.D. Harris, Temperature Dependence of Thermal Conductivity of Electronic Ceramics by an Improved Flash Diffusivity Technique, Mat. Res. Soc. Proc. 167, Advanced Electronic Packaging Materials, Boston, Massachusetts, USA, November 27 - 29 1989, edited by A.T. Barfknecht, J.P. Partridge, C.J. Chen and C.-Y. Li (Materials Research Society, Pittsburgh, 1990) 235. 33. M.C. Roufosse and P.G. Klemens, Lattice Thermal Conductivity of Minerals at High Temperatures, J. Geophys. Res. 79 (1974) 703. 34. Chapter 3; R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar,
Preparation, Characterisation and Properties of MgSiN2 Ceramics, to be published. 35. 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. 36. H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Densification of MgSiN2, Ceram. Trans. 51 Int. Conf. Cer. Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995) 585. 37. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan,
Hot-pressing of MgSiN2 ceramics, J. Mat. Sci. Lett. 13 (1994) 1314.
232 A new method for estimation of the intrinsic thermal conductivity
38. Chapter 7; R.J. Bruls, H.T. Hintzen, G. de With, R. Metselaar and J.C. van Miltenburg, The temperature dependence of the Grüneisen parameter of
MgSiN2, AlN and β-Si3N4, submitted to J. Phys. Chem. Solids. 39. Chapter 5; R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,
Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998) 7871. 40. R.A. Swalin, Thermodynamics of Solids, second edition (John Wiley & Sons, Inc., New York, 1972) 82. 41. G.A. Slack, R.A. Tanzilli, R.O. Pohl and J.W. Vandersande, The Intrinsic Thermal Conductivity of AlN, J. Phys. Chem. Solids 48 (1987) 641. 42. M. Rohde and B. Schulz, Thermal Conductivity of Irradiated and Non-irradiated Fusion Ceramics, 21st International Thermal Conductivity Conference and 9th International Thermal Expansion Symposium (ITCC 21 and ITES 9), 1990, edited by C.J. Cremers and H.A. Fine (Plenum Publishing, New York, 1990) 509. 43. T.B. Jackson, K.Y. Donaldson and D.P.H. Hasselman, Temperature Dependence of the Thermal Diffusivity/Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 73 (1990) 2511. 44. P.S. de Baranda, Thermal Conductivity of Aluminum Nitride, Ph. D. Thesis, Rutgers University, Piscataway, NJ, USA (1991) pp. 1 - 294. 45. P.S. de Baranda, A.K. Knudsen and E. Ruh, Effect of CaO on the Thermal Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 76 (1993) 1751. 46. P.S. de Baranda, A.K. Knudsen and E. Ruh, Effect of Silica on the Thermal Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 76 (1993) 1761. 47. A. Witek, M. Bockowski, A. Presz, M. Wróblewski, S. Krukowski, W. Wlosinski and K. Jablonski, Synthesis of oxygen-free aluminum nitride ceramics, J. Mater. Sci. 33 (1998) 3321. 48. K. Tsukuma, M. Shimada and M. Koizumi, Thermal Conductivity and
Microhardness of Si3N4 with and Without Additives, Am. Ceram. Soc. Bull. 60 (1981) 910.
233 Chapter 9.
49. K. Watari, Y. Seki and K. Ishizaki, Temperature Dependence of Thermal Coefficients for HIPped Sintered Silicon Nitride, J. Ceram. Soc. Jpn. Inter. Ed. 97 (1989) 170. 50. V.E. Peletskii, The Investigation of Thermal Conductivity of Silicon Nitride, High Temp. 31 (1993) 668 (translated from Teplofizika Vysokikh Temperatur 31 (1993) 727). 51. N. Hirosaki, Y. Okamoto, M. Ando, F. Munakata and Y. Akimunde, Effect of Grain Growth on the Thermal Conductivity of Silicon Nitride, J. Ceram. Soc. Jpn. Int. Ed. 104 (1996) 50. 52. K. Watari, M.E. Brito, M. Toriyama, K. Ishizaki, S. Cao and K. Mori,
Thermal conductivity of Y2O3-doped Si3N4 ceramics at 4 to 1000 K, J. Mater. Sci. Lett. 18 (1999) 865. 53. G.A. Slack and T.F. McNelly, AlN Single Crystals, J. Crystal Growth 42 (1977) 560. 54. R.H. Bogaard, personal communication, providing the raw data for ref. 32 and 42. 55. A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and Kinetic Effects of Oxygen Removal on the Thermal Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 72 (1989) 2031. 56. H. Buhr, G. Müller, H. Wiggers, F. Aldinger, P. Foley and A. Roosen, Phase
Composition, Oxygen Content, and Thermal Conductivity of AlN(Y2O3) Ceramics, J. Am. Ceram. Soc. 74 (1991) 718. 57. B. Li, L. Pottier, J.P. Roger, D. Fournier, K. Watari and K. Hirao, Measuring the Anisotropic Thermal Diffusivity of Silicon Nitride Grains by Thermoreflectance Microscopy, J. Eur. Ceram. Soc. 19 (1999) 1631. 58. Chapter 6; R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The
Temperature Dependence of the Young's Modulus of MgSiN2, AlN and Si3N4, accepted for publication in J. Eur. Ceram. Soc.
59. J. David, Y. Laurent and J. Lang, Structure de MgSiN2 et MgGeN2, Bull. Soc. Fr. Minéral. Cristallogr. 93 (1970) 153.
234 A new method for estimation of the intrinsic thermal conductivity
60. J.C. Nipko and C.-K. Loong, Phonon Excitations and Related Thermal Properties of Aluminum Nitride, Phys. Rev. B. 57 (1998) 10550.
61. R. Grün, The Crystal Structure of β-Si3N4; Structural and Stability
Considerations Between α- and β-Si3N4, Acta Cryst. B35 (1979) 800. 62. M. Okamoto, H. Arakawa, M. Oohashi and S. Ogihara, Effect of Microstructure on Thermal Conductivity of AlN Ceramics, J. Ceram. Soc. Jpn. Inter. Ed. 97 (1989) 1486.
63. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics 2, Faenza (Italy), October 1995, edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 289. 64. I.J. Davies, H. Uchida, M. Aizawa, and K. Itatani, Physical and Mechanical Properties of Sintered Magnesium Silicon Nitride Compacts with Yttrium Oxide Addition, Inorganic Materials 6 (1999) 40. 65. K. Wang and R.R. Reeber, Thermal Expansion of GaN and AlN, Mat. Res. Soc. Symp. Proc. 482, Nitride Semiconductors, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S. Nakamura and S. Strite (Materials Research Society, Warrendale, Pennsylvania, 1998) 863. 66. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)-nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of Properties and Performances by Improved Design and Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405. 67. W.D. Kingery, The Thermal Conductivity of Ceramic Dielectrics, Progress in Ceramic Science 2, edited by J.E. Burke (Pergamon Press Ltd., Oxford, 1962), pp. 182 - 235. 68. K. Hirao, K. Watari, M.E. Brito, M. Toriyama and S. Kanzaki, High Thermal Conductivity of Silicon Nitride with Anisotropic Microstructure, J. Am. Ceram. Soc. 79 (1996) 2485.
235 Chapter 9.
236 Chapter 10. Conclusions
In Chapter 1 the importance of non-metallic materials showing a high thermal conduction by phonons and the need for new materials with desirable thermal properties is pointed out. It is concluded that (1) for substrate applications, where a high thermal conductivity is needed, the traditional oxide materials are replaced by nitride materials, and (2) the relatively new material MgSiN2 might be potentially interesting (based on an estimated thermal conductivity of 120 W m-1 K-1 at room temperature reported in the literature).
In chapters 2 and 3 it is shown that the processing of MgSiN2 can be optimised by using pure starting materials and suitable reaction conditions.
MgSiN2 powders and ceramics with an oxygen content far below 1 wt. % are obtained as compared to previously synthesised materials containing about 4 wt. % oxygen. However, the thermal conductivity at room temperature of the resulting ceramics did not exceed 25 W m-1 K-1.
Therefore, also the theoretical thermal conductivity of MgSiN2, AlN and
β-Si3N4 was considered. It turned out that the theoretical thermal conductivity calculated using Slack's formula is relatively sensitive for small variations in the Debye temperature θ and Grüneisen parameter γ which are needed as input parameters. The Debye temperature was calculated using either heat capacity data (chapter 5 and 8) or elastic constants (chapter 6), whereas for the evaluation of the Grüneisen parameter (chapter 7) both are needed in combination with thermal expansion data (chapter 3, 4 and 7). The elastic constants are almost temperature independent and decrease almost linearly for T > 450 K whereas the thermal expansion coefficient and heat capacity show an S-shaped increase as a function of
237 Chapter 10.
the absolute temperature approaching a constant value at higher temperatures (T > θ ≈ 1000 K). From these data the Debye temperature near 0 K (chapter 6) and the Grüneisen parameter as a function of the absolute temperature (chapter 7) for
MgSiN2, AlN and β-Si3N4 were obtained. The Debye temperatures at 0 K (θ 0 ) of
MgSiN2, AlN and β-Si3N4 are about the same (θ 0 ≈ 900 - 950 K). The Grüneisen parameter increases as a function of the temperature approaching a constant value
(γ ≈ 1.0 for AlN and the structurally related MgSiN2, and γ ≈ 0.63 for β-Si3N4) at high temperatures (T/θ ≥ 0.8). Using Slack's equation (chapter 8), reasonable estimates for the thermal conductivity at the Debye temperature as compared to experimental values (within 20 %) were obtained. For extending the validity of the Slack equation below the Debye temperature, the equation was modified resulting in a more realistic description of the temperature dependence. This justified the theoretical calculation of the thermal conductivity of MgSiN2, AlN and β-Si3N4 down to the more interesting room temperature region. The resulting estimates for the maximum achievable thermal conductivity were in rough agreement with the highest experimental values providing a reasonable indication for the usefulness of the Slack equation. From the calculations it became clear that at room temperature AlN -1 -1 has a high thermal conductivity (>> 100 W m K ), MgSiN2 a low thermal -1 -1 conductivity (<< 100 W m K ) and β-Si3N4 a thermal conductivity in between (~ 100 W m-1 K-1). Considering the limitations of the Slack equation a new method for estimating the maximum achievable thermal conductivity is proposed, based on temperature dependent thermal diffusivity measurements on non-optimised samples using a linear extrapolation method (chapter 9). The estimates obtained -1 -1 with this method at room temperature for MgSiN2 (26 - 28 W m K ), AlN -1 -1 -1 -1 (178 - 200 W m K ) and β-Si3N4 (79 - 94 W m K ) are in favourable agreement with the best experimental values (23 W m-1 K-1, 266 W m-1 K-1 and 106 W m-1 K-1, respectively) indicating the general applicability of this relatively simple method. Furthermore this method is a strong tool for guiding the optimisation of the
238 Conclusions
material with respect to the desired thermal conductivity and can be used on non-optimised samples in order to get an impression of the maximum achievable value. Finally it can be concluded that, in contrast to the expectations at the beginning of this work, the thermal conductivity at room temperature of MgSiN2 ceramics is limited to 25 - 30 W m-1 K-1, as shown both experimentally and theoretically, reducing its potential for applications. Moreover, as a spin-off the described theoretical approach resulted in a new generally applicable method for estimation of the maximum achievable thermal conductivity of non-metallic materials. This method not only provides a good indication of the potential thermal conductivity of new non-optimised interesting materials, reducing the time and effort normally needed to obtain a reliable indication of the maximum achievable thermal conductivity, but can be also used for guiding the optimisation of it.
239 Chapter 10.
240 List of symbols
Lower-case symbols a [m2 s-1] : thermal diffusivity a, b, c [Å] : lattice parameters b [m] : width -1 -1 cp [J kg K ] : specific heat capacity at constant pressure d [Å] : d-value, interplanar spacing d [µm] : particle size f [s-1] : flexural frequency h [J s] : Planck's constant (6.626 10-34 J s) h [m] : height hkl [-] : Miller indices k [J K-1] : Boltzmann's constant (1.381 10-23 J K-1) k [eV K-1] : Boltzmann's constant (8.62 10-5 eV K-1) l [m] : length l [m] : phonon mean free path lgb [m] : phonon mean free path due to grain boundary scattering lpd [m] : phonon mean free path due to phonon-defect scattering lpp [m] : phonon mean free path due to phonon-phonon scattering ltot [m] : total phonon mean free path m [kg] : (sample) mass n [-] : number of atoms per primitive unit cell p [Pa] : pressure q [J] : energy s, st [-] : (total) number of atoms per formula unit / molecule
241 List of symbols
sa [-] : number of anions per formula unit / molecule t [s, h, years] : time -1 vl [m s ] : longitudinal sound velocity -1 vs [m s ] : sound velocity -1 vt [m s ] : transverse sound velocity wi : 1/yi, weight factor 2 2 1/2 wRp [-] : {Σ wi (yi (obs) - yi (calc)) / Σ wi (yi (obs)) } , weighted R-pattern x, y, z [-] : position along the x, y and z direction yi (calc) : calculated intensity at the ith step yi (obs) : observed (gross) intensity at the ith step
Upper-case symbols
-1 -1 Cp [J mol K ] : heat capacity at constant pressure o -1 -1 Cp [J mol K ] : heat capacity at standard pressure -1 -1 CV [J mol K ] : heat capacity at constant volume E [GPa] : Young's modulus
FD(θ /T) [-] : Debye function G [J mol-1 K-1] : Gibbs energy
o -1 -1 GT [J mol K ] : Gibbs energy function at standard pressure
o o -1 -1 GT - H0 [J mol K ] : energy function at standard pressure H [J mol-1 K-1] : enthalpy
HK [GPa] : Knoop hardness
HV [GPa] : Vickers hardness
o -1 -1 H0 [J mol K ] : standard formation enthalpy o o -1 -1 HT - H0 [J mol K ] : enthalpy function at standard pressure
I/I0 [%] : relative intensity M [kg mol-1] : mole mass
M [kg mol-1] : mean atomic mass
242 List of symbols
-1 23 -1 NA [mol ] : Avogadro's number (6.022 10 mol ) [N] [wt. %] : nitrogen content [O] [wt. %] : oxygen content Q [J] : energy
Qv [J, eV] : energy required for the formation of a vacancy R [J mol-1 K-1] : gas constant (8.314 J mol-1 K-1) R [-] : statistical R-value
Rp [-] : Σyi (obs) - yi (calc)/ Σ yi (obs), R-pattern S [J mol-1] : entropy
o o -1 ST (- S0 ) [J mol ] : entropy function at standard pressure T [K] : absolute temperature T [°C] : temperature W [Å3] : volume per anion bond V [m3]: volume 3 -1 V0 [m mol ] : molar volume at 0 K 3 -1 Vm [m mol ] : molar volume V [Å3] : volume of a unit cell Z [-] : number of formula units per unit cell
Greek symbols
α [K-1] : thermal expansion coefficient
-1 αa, αb, αc [K ] : thermal expansion coefficient along the a-, b- and c-axis -1 αlat [K ] : linear lattice thermal expansion coefficient -1 αlin [K ] : linear thermal expansion coefficient -1 βS [Pa ] : adiabatic compressibility -1 βT [Pa ] : isothermal compressibility 2 χ [-] : wRp/Rp, chi-square, goodness of fit δ [m, Å] : cube root of the average volume per atom
243 List of symbols
δ 3 [m3, Å3] : average volume per atom (volume of a unit cell divided by the number of atoms per unit cell) δ [-] : Anderson-Grüneisen parameter γ [-] : Grüneisen parameter
γθ [-] : Grüneisen parameter at the Debye temperature
γ∞ [K] : high temperature limit of the Grüneisen parameter η [-] : number of bonds (per anion) κ [W m-1 K-1] : thermal conductivity ν [-] : Poisson's ratio ρ [kg m-3] : density
-3 ρm [mol m ] : molar density θ [K] : Debye temperature, characteristic temperature
θ0 [K] : Debye temperature at 0 K
θ∞ [K] : high temperature limit of the Debye temperature 2θ [°] : diffraction angle
244 Summary
The objective of this work was to investigate, understand and optimise the thermal conductivity of MgSiN2 ceramics. In order to obtain a high thermal conductivity the impurity content and especially the oxygen content in the MgSiN2 lattice was considered to be of crucial importance. Therefore this work first concentrated on the optimisation of the synthesis of pure MgSiN2 powder and ceramics by suitable processing. Although, originally a high thermal conductivity (~ 120 W m-1 K-1) was theoretically expected for MgSiN2 this value could by far not be confirmed experimentally. Therefore, the theoretical method to predict the maximum achievable thermal conductivity (Slack's theory) was reconsidered. This resulted in an improved theory of Slack and moreover, the development of a new prediction method based on temperature dependent thermal diffusivity measurements. This was done with the intention to avoid putting a lot of time and effort in process optimisation of materials for which less interesting thermal properties can be expected. The improved prediction methods were also applied to the commercial materials AlN and β-Si3N4 in order to check the general validity of the used methods. So, an experimental as well as a theoretical approach is described in this thesis.
The first chapters of the thesis deal with the preparation of MgSiN2 powders and ceramics. By suitable processing it is possible to control the oxygen content of
MgSiN2 powders and ceramics. As a consequence very pure, oxygen poor materials (<< 1 wt. %) could be obtained. However, the thermal conductivity at room temperature of the resulting ceramics did not exceed 25 W m-1 K-1.
The middle part of the thesis deals with the properties of MgSiN2, AlN and
β-Si3N4. Most data for AlN and β-Si3N4 could be obtained from the literature. For
MgSiN2 the specific heat, thermal expansion coefficient and Young's modulus
245 Summary
were experimentally determined as a function of the temperature. These data were used to evaluate the Debye temperature θ near 0 K and Grüneisen parameter γ, which are important input parameters for theoretical modelling of the thermal conductivity. The Debye temperatures are relatively temperature independent. The
Debye temperatures at 0 K (θ 0 ) of MgSiN2, AlN and β-Si3N4 are about the same
(θ 0 ≈ 900 - 950 K) and with increasing temperature the Debye temperature first decreases and subsequently increases approaching a constant value (θ ≈ 1000 K for
MgSiN2 and AlN, and θ ≈ 1200 K for β-Si3N4) at intermediate temperatures (T/θ / 0.3). The Grüneisen parameter increases as a function of the temperature approaching a constant value (γ ≈ 1.0 for MgSiN2 and AlN, and γ ≈ 0.63 for
β-Si3N4) at high temperatures (T/θ ≥ 0.8). The last chapters of the thesis deal with the theory of Slack and a new method for predicting the maximum achievable thermal conductivity of non- metallic solids. The assumptions made in Slack's theory are briefly discussed and some improvements concerning the temperature dependence are presented, resulting in a modified Slack theory. The second estimation method proposed in this thesis is based on temperature dependent thermal diffusivity measurements on non-optimised samples (a -1 versus T method). This method has the advantage that it directly provides a minimum value for the maximum thermal conductivity. The validity and limitations of these methods are discussed using MgSiN2, AlN and
β-Si3N4 as model compounds. The Slack equation provides a rough indication of the maximum achievable thermal conductivity whereas the a -1 versus T method provides more reliable estimates. From both estimation methods and also from the experimental results it can be concluded that, in contrast to the expectations at the beginning of this work, the thermal conductivity at room temperature of MgSiN2 ceramics is limited to 25 - 30 W m-1 K-1 reducing its potential for applications.
246 Samenvatting
De hoofddoelstelling van dit promotieonderzoek was het bestuderen, begrijpen en vervolgens optimaliseren van de warmtegeleidbaarheid van het relatief nieuwe keramische materiaal MgSiN2. Voor het verkrijgen van een hoge warmtegeleidbaarheid werd de reductie van de concentratie verontreinigingen, met name zuurstof, van groot belang geacht. Het eerste gedeelte van het onderzoek concentreerde zich derhalve op de synthese van zuiver MgSiN2 poeder en keramiek. Aangezien de na optimalisatie gemeten warmtegeleidbaarheid (< 25 W m-1 K-1 bij kamertemperatuur) veel lager was dan de (hoge) theoretisch voorspelde waarde (120 W m-1 K-1) werd de gebruikte methode voor de schatting van de maximaal haalbare warmtegeleidbaarheid (Slack formule) opnieuw in detail bekeken. Dit heeft geleid tot een verbeterde versie van de Slack formule, en bovendien in een geheel nieuwe schattingsmethode gebaseerd op thermische diffusiviteitsmetingen als functie van de temperatuur. Deze algemeen bruikbare methode werd mede ontwikkeld met het oog op toekomstige vraagstellingen met betrekking tot de thermische eigenschappen van keramische materialen. Hierdoor kan vroegtijdig, zonder al te veel tijd en moeite te spenderen aan procesoptimalisatie, worden onderkend welke materialen potentieel de gewenste thermische eigenschappen bezitten. Om de algemene geldigheid van beide schattingsmethoden te toetsen werd ook de warmtegeleidbaarheid van de commercieel interessante materialen AlN en β-Si3N4 berekend. Dus, zowel een experimentele als een theoretische aanpak is in dit proefschrift beschreven.
De eerste hoofdstukken beschrijven de synthese van zuiver MgSiN2 poeder (hoofstuk 2) en keramiek (hoofdstuk 3). Met name geschikte processing maakt het mogelijk om het zuurstofgehalte in MgSiN2 poeder en keramiek te beheersen. Dit resulteerde in zeer zuiver, zuurstofarm materiaal (<< 1 gewichts % zuurstof).
247 Samenvatting
Desondanks bleef de warmtegeleidbaarheid van de keramiek beperkt tot 25 W m-1 K-1. In de volgende hoofstukken worden enkele thermische en mechanische eigenschappen van MgSiN2, AlN en β-Si3N4 behandeld, die nodig zijn om de theoretische warmtegeleidbaarheid te berekenen. De meeste gegevens voor AlN en
β-Si3N4 waren reeds in de literatuur gerapporteerd. De thermische expansiecoëfficiënt (hoofstuk 3 en 4), soortelijke warmte (hoofdstuk 5) en de elasticiteitsmodulus (hoofdstuk 6) van MgSiN2 werden gemeten als functie van de temperatuur. Deze gegevens werden gebruikt om de Debye temperatuur θ bij 0 K,
θ 0 (hoofdstuk 6) en de Grüneisen parameter γ (hoofdstuk 7) van MgSiN2, AlN and
β-Si3N4 te bepalen. Deze twee grootheden zijn belangrijke parameters voor de theoretische modellering van de warmtegeleidbaarheid. Alle drie de materialen blijken een relatief hoge Debye temperatuur te hebben (θ 0 ≈ 900 - 950 K). De Grüneisen parameter vertoonde een (relatief sterke) temperatuursafhankelijkheid. Met stijgende temperatuur nam de Grüneisen parameter eerst toe om vervolgens bij hogere temperatuur (T/θ ≥ 0.8 met θ ≈ 1000 K) constant (γ ≈ 1.0 voor MgSiN2 en
AlN, en γ ≈ 0.63 voor β-Si3N4) te worden. De laatste twee hoofdstukken van dit proefschrift behandelen de aanpassing van de Slack formule (hoofdstuk 8) en een nieuwe methode om de maximale warmtegeleidbaarheid af te schatten (hoofdstuk 9). De toepasbaarheid en beperkingen van beide methoden worden besproken aan de hand van de resultaten voor MgSiN2, AlN en β-Si3N4. De Slack formule is een relatief simpele manier om de maximale warmtegeleidbaarheid van niet-metallische (keramische) materialen ruwweg te kunnen schatten. De aannames en beperkingen van deze formule worden kort besproken en enkele eenvoudige verbeteringen worden geïntroduceerd, welke resulteren in een gemodificeerde Slack formule. Hierdoor wordt het temperatuurgebied waarbinnen deze formule normaal toepasbaar is (T ≥ θ ) substantieel uitgebreid naar het praktisch interessante lagere kamertemperatuurgebied. De nieuwe schattingsmethode is gebaseerd op thermische diffusiviteitsmetingen aan niet-geoptimaliseerde preparaten als functie
248 Samenvatting
van de temperatuur. Deze eenvoudige methode heeft als voordeel dat vrij gemakkelijk een minimale waarde voor de maximale warmtegeleidbaarheid kan worden verkregen. De methode is betrouwbaarder dan de traditionele Slack formule. Bovendien kan deze methode gebruikt worden om de optimalisatie van de warmtegeleidbaarheid te sturen. Beide schattingsmethoden bevestigen de experimentele resultaten dat de warmtegeleidbaarheid bij kamertemperatuur van -1 -1 MgSiN2 gelimiteerd is tot 25 - 30 W m K . Dit reduceert de potentiële mogelijkheden van MgSiN2 keramiek voor toepassingen waarbij een hoge warmtegeleidbaarheid van belang is.
249 Samenvatting
250 Nawoord
Het nawoord van het proefschrift wordt altijd gebruikt om de mensen met wie je hebt samengewerkt tijdens je promotieonderzoek te bedanken. Het is dan ook in het algemeen zowat het allerlaatste wat er tijdens het tot stand komen van een proefschrift door de promovendus geschreven wordt. Vandaar dat het gehalte aan standaardzinnen in de meeste nawoorden zo hoog is. Je bent al lang blij dat 'het' erop zit. Hiermee doe je m.i. de mensen met wie je met zoveel plezier hebt samengewerkt toch wel een beetje te kort. Met deze korte (ongebruikelijke) inleiding op het nawoord probeer ik dan ook enigszins af te wijken van de standaard door niet direct met de deur in huis te vallen. Net zoals iedere promovendus ben ik dank verschuldigd aan heel veel mensen. Het tot stand komen van dit proefschrift was alleen mogelijk door de steun en hulp van deze mensen. Alleen had ik het nooit voor elkaar gekregen! Als allereerste zou ik die mensen willen bedanken die ik onverhoopt vergeet te noemen. Verder zal ik min of meer een chronologische volgorde van het verloop van het onderzoek aanhouden bij het bedanken van de diverse personen: Pim Groen (Philips Research), Bert de With en Bert Hintzen voor de eerste kennismaking met het boeiende onderwerp. Mijn eerste promotor Ruud Metselaar, die na mijn afstuderen zoveel vertrouwen in mijn werk had, dat hij mij een promotieplaats aanbood en met veel interesse mijn werk gevolgd heeft. Hierbij dien ik direct ook mijn directe begeleider Bert Hintzen te noemen die altijd enthousiast, kritisch en behulpzaam was. Direct betrokken bij mijn onderzoek waren voor korte of langere tijd Henk Eekhof, Agnieszka Kudyba-Jansen, Peter Gerharts en Tarek Gueddas. Bedankt voor de experimentele en wetenschappelijke bijdrage aan mijn werk. Veel technische en/of wetenschappelijke steun binnen de groep heb ik ontvangen van
251 Nawoord
Henk van der Weijden (ovens, glove-box), Gerrit Bezemer (ovens, TGA/DTA, glove-box), Hans de Jonge Baas (röntgendiffractie), Toon Rooijakkers (SEM preparaatbereiding), Hans Heijligers (SEM), Marco Hendriks (SEM, meting elasticiteitsconstanten m.b.v. puls-echo methode), Gerben Boon en Dick Klepper (TEM), Joost van Krevel (reflectie metingen), Paul van der Varst (meting Young's modulus m.b.v. impuls-excitatie methode) en Anneke Delsing (glove-box, thermische diffusiviteit). Mijn dank hiervoor. Verder alle medewerkers, promovendi en studenten van de capaciteitsgroep Vastestof- en Materiaalchemie (SVM) die ik de afgelopen jaren gekend heb. Waarbij ik met name mijn kamergenoten Henk, Robert, Stephan, Joost en Maru wil bedanken voor alle discussies en gezelligheid. Ook buiten de groep ben ik veel mensen dank verschuldigd: Joost van Eijk (TNO/TPD), Han van der Heijde (TNO/TPD), Hans-Joachim Sölter (Compotherm GmbH, Duitsland), Pim Groen (Philips Research, Aken), Theo Kappen (Philips Lighting, Eindhoven), Harrie van Hal (Philips Research, Eindhoven), Kees van Malsen (Universiteit van Amsterdam), Kees van Miltenburg (Universiteit Utrecht), Anil Virkar (University of Utah, USA), Dale Niesz (Rutgers University, USA), Chun Loong (Intense Pulse Neutron Source, Argonne, USA), Simine Short (Intense Pulse Neutron Source, Argonne, USA), Ron Bogaard (Purdue University, USA), Jos van Wolput (TU Eindhoven), Koji Watari (National Industrial Research Institute of Nagoya (NIRIN), Nagoya, Japan) en Naoto Hirosaki (National Institute for Research in Inorganic Materials (NIRIM), Japan). Voor de technische ondersteuning, het verbeteren van de diverse experimentele opstellingen en de bewerking van de keramische preparaten ben ik veel dank verschuldigd aan de mensen van de Faculteitswerkplaats en de Gemeenschappelijke Technische Dienst. Verder wil ik de leescommissie (bestaande uit: Ruud Metselaar, Kiyoshi Itatani (Sophia University, Tokyo, Japan), Bert Hintzen, Roger Marchand (Université de Rennes I, France) en Bert de With) hartelijk bedanken voor de vele nuttige aanwijzingen, suggesties, tips en discussies over het onderwerp. 特に，私の副査である板谷清司助教授（上智大学，日本）には，学位審査
252 Nawoord
委員会のメンバーとしてMgSiN2粉体およびそのセラミックスのプロセスに 関連した多くの貴重なご質問およびご助言を頂きましてお礼を申し上げま す。(Especially, my second promotor Prof. K. Itatani I would like to thank for being a member of my Ph. D. committee, his comments on the manuscript as well as his many motivating critical questions and remarks concerning the processing of the MgSiN2 powder and ceramics). Je voudrais remercier le Professeur Marchand d'avoir accepté d'être membre de mon jury de thèse et d'avoir pris le temps de discuter et commenter ma thèse. Voor de natuurkundige inbreng en de daaruit voortvloeiende nuttige discussies en begripsvorming ben ik prof. G. de With grote dank verschuldigd. Shell Nederland B.V. wil ik bedanken voor de financiële steun (Shell reisdonatie) die het deelnemen aan een congres in de V.S. en het bezoeken van diverse Amerikaanse universiteiten en instituten mogelijk maakte. De nodige ontspanning en gezelligheid vond ik ondermeer in de muziek. Vooral het lidmaatschap van het Eindhovens Studenten Muziek Gezelschap Quadrivium heb ik altijd als ontzettend leuk, inspirerend en vooral gezellig ervaren. Ook de vrienden buiten de muziek wil ik bedanken voor al hun steun. Vooral als het een keer niet mee zat dan stonden ze altijd met raad en daad voor me klaar. Mijn ouders wil ik bedanken voor hun betrokkenheid, steun en vertrouwen. Mijn "broertje" Dominique wil ik bedanken voor alle steun maar vooral voor zijn gevoel voor humor en Karin voor de getoonde interesse en betrokkenheid. Mijn vriendin Marianne ben ik heel veel dank verschuldigd voor alle steun, liefde, geduld, het "er zijn", en het me op sleeptouw nemen altijd wanneer dat nodig was.
253 Curriculum Vitae
Richard Joseph Bruls werd geboren op 26 mei 1972 te Sittard. Na het behalen van zijn Atheneum B diploma in 1990 aan het Serviam Lyceum Scholengemeenschap te Sittard, studeerde hij Scheikundige Technologie aan de Technische Universiteit Eindhoven. Na het behalen van zijn ingenieurstitel door het afronden van zijn afstudeeropdracht getiteld "Investigation of the Thermal Diffusivity/Conductivity of Hot-Pressed MgSiN2 Ceramics" in februari 1996 bij de vakgroep Vastestof- en Materiaal Chemie, werd in aansluiting daarop onder begeleiding van prof.dr. R. Metselaar en dr. H.T. Hintzen een promotie onderzoek gestart op hetzelfde onderwerp. De resultaten van dit onderzoek zijn beschreven in dit proefschrift.
Richard Joseph Bruls was born in Sittard (The Netherlands) on May 26th 1972. In 1990, after completing the secondary school at the Serviam Lyceum Scholengemeenschap in Sittard, he started his study Chemical Engineering at the Eindhoven University of Technology. In February 1996 he finished his graduation work entitled "Investigation of the Thermal Diffusivity/Conductivity of
Hot-Pressed MgSiN2 Ceramics" at the Laboratory of Solid State and Materials Chemistry and obtained his Masters Degree. Subsequently, he started his Ph.D. study in the same field under supervision of Prof. R. Metselaar and Dr. H.T. Hintzen. The results of this investigation are described in this thesis.
254 List of publications
H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder
Preparation and Densification of MgSiN2, Ceramic Transactions, Vol. 51, Ceramic Processing Science and Technology (Friedrichshafen, Germany, September 11 - 14, 1994), editors H. Hausner, G.L. Messing and S. Hirano, 1995, pp. 585 - 589.
H.T. Hintzen, R.J. Bruls and R. Metselaar, The Thermal Conductivity of MgSiN2 Ceramics, Fourth Euro Ceramics (Faenza, Italy, October 2 - 6, 1995), Vol. 2, Basic Science - Developments in Processing of Advanced Ceramics - Part II, editor C. Galassi, 1995, pp. 289 - 294.
H.T. Hintzen, R. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2 Ceramics, The American Ceramic Society 98th Annual Meeting Abstracts, (Indianapolis, IN, USA, April 14 - 17, 1996), p. 249.
R.J. Bruls, H.T. Hintzen and R. Metselaar, Modeling of the Thermal
Diffusivity/Conductivity of MgSiN2 Ceramics, Thermal Conductivity 24, Thermal Expansion 12 (Pittsburgh, USA, October 26 - 29, 1997), editors P.S. Gaal and D.E. Apostolescu, 1999, pp. 3 - 14.
R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg, Heat Capacity of
MgSiN2 between 8 and 800 K, J. Phys. Chem. B, Vol. 102, 1998, pp. 7871 - 7876.
C.M. Fang, R.A. de Groot, R.J. Bruls, H.T. Hintzen and G. de With, Ab initio Band
Structure Calculations of Mg3N2 and MgSiN2, J. Phys.: Condens. Matter, Vol. 11, 1999, pp. 4833 - 4842.
255 List of publications
R.J. Bruls, H.T. Hintzen and R. Metselaar, Preparation and Characterisation of
MgSiN2 Powders, J. Mater. Sci., Vol. 34, 1999, pp. 4519 - 4531.
R. Metselaar and R. Bruls, The Thermal Conductivity of MgSiN2 in Comparison nd with AlN and Si3N4, The American Ceramic Society 102 Annual Meeting Abstracts, (St. Louis, Missouri, USA, April 30 - May 3, 2000), p. 271.
R.J. Bruls, H.T. Hintzen, R. Metselaar and C.-K. Loong, Anisotropic Thermal
Expansion of MgSiN2 from 10 to 300 K as Measured by Neutron Diffraction, J. Chem. Phys. Solids, Vol. 61, 2000, pp. 1285 - 1293.
R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar, Preparation,
Characterisation and Properties of MgSiN2 Ceramics, to be published.
R.J. Bruls, H.T. Hintzen, G. de With, R. Metselaar and J.C. van Miltenburg,
Thermodynamic Grüneisen Parameter of MgSiN2, AlN and β-Si3N4, submitted to J. Chem. Phys. Solids.
R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The Temperature
Dependence of the Young's Modulus of MgSiN2, AlN and Si3N4, accepted for publication in J. Eur. Ceram. Soc.
R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, Estimates of the Maximum
Achievable Thermal Conductivity of MgSiN2, AlN and β-Si3N4 using a Modified Slack Equation, to be published.
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, AlN and β-Si3N4 ceramics, to be published.
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