2. Lattice vibrations I a - Atoms vibrate about their rest position. - The vibration of a single atom depends on the neighbor atoms. Within a crystal all vibrations are correlated = lattice vibrations or lattice modes. - The shortest wavelength in a certain direction is equal to the shortest crystallographic translation in that direction. rest position bond = spring Thermal properties Most solids expand during heating. Thermal expansion is due to the change in the amplitude of lattice vibrations Longitudinal vibration along a 1-D atom chain

3. Thermal properties The average value for the distance d is displaced to the right with increasing temperature => expansion of the interatomic distance. Reason for the expansion: Anharmonic vibration of atoms about their rest position Atom vibrates assymetrically from 0°K position reference atom average d Lattice vibrations II d min d max

4. interatomic distance r d min = o°K =d max = o°K T1T1 T2T2 T3T3 T4T4 d min,T3 average r T 4 > T 3 > T 2 > T 1 Temp Position enveloppe of second atom Thermal properties Lattice vibrations III d max, T3 x x

5. Thermal properties Thermal expansion I The thermal expansion of a single dimension in a solid is characterized by the linear thermal expansion coefficient. The subscript T indicates that  is temperature dependent in most cases! The dependence over certain temperature intervalls is often linear, allowing the use of the right expression with an average coefficient:   const.    (T) T 0 2004006008001000   °C -1 x 10 -6  0 2 4 6 8 10 12  200°C  800°C x x x Example of a constant and three temperature dependent thermal expansion coefficient.

6. Thermal properties Thermal expansion II Analogue to the linear expansion coefficient, a volume expansion coefficient can be defined: For simple structure, the thermal expansion coefficient is related to the bond strength: where z is the charge of the cation and n its coordination number. Material Steel 13.0 Aluminium 22.0 Al 2 O 3 7.2 - 8.9 MgO(s) 13.5 Zr O 2 (tet) 12.0 Zr O 2 (mcl) 7.0 Cordierite 2.1 ZrWO 4 -2.0  (°C -1 )x 10 6 Average linear thermal expansion coefficients. Some ceramic material have particularly low expansion coefficient.

7. Thermal properties Linear thermal expansion T (C°) 0 2004006008001000  l/l(%) 0 0.2 0.4 0.6 0.8 Al Cu Zr O 2 Al 2 O 3 Cordierite Si 3 N 4 1.0 1.2 Linear thermal expansion parallel to the c-axis of a variety of materials. The expansion coefficient at each temperature is given by the slope of the curve

8.  V/V(%) 0 2004006008001000 1.0 2.0 3.0 4. 0 Cristobalite  -Si O 2  -Si O 2 mcl Zr O 2 tet Zr O 2 Pyrex Volume thermal expansion for materials with phase transformations and Pyrex glass. Thermal properties Volume thermal expansion

9. Al 2 O 3 30.0 - 35.0 AlN 200.0-280.0 Cordierite 4.0 SiC 84.0-93.0 Cordierite 4.0 Steel 45 Aluminium 239 Diamond2000 Thermal properties Thermal conduction Thermal conduction in a solid is described by equations which are analogous to Fick‘s equation for diffusion: Q: heat T: temperature A: surface The heat transferred per unit time across a surface A is proportional to the temperature gradient across the surface. The proportionality factor is the thermal conductivity k th. In metals, heat is mostly transported by conduction electrons, in ceramics lattice vibration (phonons) are mainly responsible for heat transport. For high theramic conductivity, the following properties are usually required: (1) low atomic mass, (2) strong bonding, (3) simple crystal structure, and (4) low anharmonicity. There are only few ionic materials that fulfill these requirements, most of them have diamond like structures: BeO, BN, SiC, Si, GaP etc. Material Average thermal conductivities. Among the ceramic materials, there are low as well as high conductivity examples. k th (W m -1 K -1

10. Thermal shock resistance I Thermomechanical properties Thermal inhomogeneities in a ceramic body induces stresses due to differential thermal expansion. The inhomogeneities will be large for ceramics with large thermal expansion coefficients and/or small thermal conductivities surface TsTs TaTa TiTi Temperature distribution in a plate which is cooled from both surfaces area under tension area under compression The thermal stresses can be large enough to induce cracking of the material. The thermal shock resistance of a material is characterized by the largest instant temperature difference a material can withstand without failure:  : Poisson‘s ratio  : failure strength E: Youngs modulus  : thermal expansion coefficient

11. Failure pattern in Al 2 O 3 disk exposed to a temperature difference of 700 (°K)) Material SiAlON 900 Si 3 N 4 500 Al 2 O 3 200 Ti 3 SiC 2 >1400 Fused SiO 2 1600 Li-Al-silicate glass ceramics 670  T (°K) Thermal shock resistance II Thermomechanical properties The critical temperature difference is determined by measuring the retained strength of samples after they were heated to various temperature T and quenched in water. Above a certain critical temperature T, the retained strength decreases dramatically relative to the initial strength. The temperature difference relative to the quench medium (room temperature) is indicated as measure of the thermal shock resistance. T 

12. Thermomechanical properties Engine catalysator substrates: cordierite honeycombs Catalysator substrate materials require both a small thermal expansion coefficient and a high thermal shock resistance. Cordierite honeycombs have a very low thermal expansion coefficient which is very low and with a critical temperature of 1100°C an exceptionally good thermal shock resistance.

13. Chip substrate and packaging Thermomechanical properties Chip substrates and packages must have good mechanical resistance, high thermal conductivity to evacuate the heat, good sealing capacity and thermal expansion compatible with the chip materials. Intel Pentium (IV) microprocessor Ceramic Cap Bonding WiresIC Chip Substrate Pins Chip Bond Metallization Section through a micro chip Courtesy, Intel

14. 5KW 18KW 1.5KW 500W 4004 8008 8080 8085 8086 286 386 486 Pentium® proc 0.1 1 10 100 1000 10000 100000 19711974197819851992200020042008 Power (Watts) Courtesy, Intel Power dissipation Thermomechanical properties Evolution of power delivery and dissipation in microprocessors

15. Thermal properties Aluminium nitride as chip packaging material I The most remarkable property exhibited by AlN is its high thermal conductivity - in ceramic materials second only to beryllia. At moderate temperatures (~200 ｡ C) its thermal conductivity exceeds that of copper. This high conductivity coupled with high volume resistivity and dielectric strength leads to its application as substrates and packaging for high power or high- density assemblies of microelectronic components. One of the controlling factors which limits the density of packing of electronic components is the need to dissipate heat arising from ohmic losses and maintain the components within their operating temperature range. Substrates made from AlN provide more efficient cooling than conventional and other ceramic substrates, hence their use as chip carriers and heat sinks. AlNBeO Al 2 O 3 Bending strength (MPa) 350 200 320 Dielectric constant 8.86.8 9.8 TCE (×10-6/°C) 4.6 7.5 7.6 Thermal cond. (W/mK) 250250 18 AlN has wurtzite type structure

16. Thermal properties Aluminium nitride: thermal properties The value for the thermal conductivity of polycristalline AlN depends strongly on the starting material and manufacturing procedures.

17. Thermal properties Chemical stability of AlN Under ambient conditions, AlN is not stable. It reacts with both oxygen and water vapour, the latter reaction is more prominent: 4AlN + 8H 2 O = 2Al 2 O 3 + 4NH 4 + O 2 Commercially available AlN powder has often an oxidized layer that diminshes the thermal properties of the material, because during sintering a AlON phase develops along the grain boundaries. This problem can be solved by adding yttria as sintering aid. Yttria forms a liquid phase at sintering temperature, that dissolves Al 2 O 3. Because the liquid does not wet AlN it is concentrated at the triple junction. The presence of the liquid phase enhances also the sinterability of AlN, which is difficult to densify >95%.

18. Thermomechanical properties High temperature mechanical behavior t t t t t t t t Stress/ strain curves and mechanical anologons for „endmember“ materials Elastic material Viscous material Visco-elastic material F stress vs. time strain vs. time strain rate vs. stress Newtonian Non-Newtonian plastic material mechanical model

19. Thermomechanical properties Creep deformation I At high temperature, ceramics under load will undergo a (viscous type) deformation called creep. Creep strain vs. time for constant load  time primary secondary tertiary Rupture instantaneous deformation Three creep regimes are typically observed: - primary region: instantaneous increase followed by decreasing strain rate - secondary region: steady state creep - tertiary region: increasing strain rates just before rupture

20. Creep strain vs. time for constant load and different temperature or constant temperatures and different loads (T m melting temperature) time T 3 or  3 T 2 or  2 T 1 or  1 T < 0.4T m Thermomechanical properties Creep deformation II  Creep mechanisms in the steady state regime: - Diffusional creep: volume diffusion for high T and/or large grains: Nabbaro-Herring creep grain boundary diffusion for low T and or small grains: Cobble creep - Dislocation creep:for high T - Grain sliding, cavitation: for high T and small grain size T 1 <T 2 <T 3  = const;  1 <  2 <  3 T = const

21. The vacancy concentration in the area under pressure is lower than in areas under tension. Material will, therefore, diffuse to the areas under tension changing the shape of the grain. vacancies s 11 -s 22 material flux vacancy flux shape of the grain after creep Thermomechanical properties Diffusion creep I

22. Thermomechanical properties Superplasticity I During superplastic deformation, strain is accumulated by the motion of individual grains or clusters of grains relative to each other by sliding and rolling. Grains are observed to change their neighbours and to emerge at the free surface from the interior. During deformation the grains remain equi- axed, or, if they were not equi-axed prior to deformation, become so during superplastic flow. www.nsf.gov/mps/dmr/highlights Strains of several hundred % are not unusual during deformation in the superplastic regime. This is an example of cubic stabilized ZrO2 (8 mol% Y2O3) with 5 wt% SiO2 deformed under tensile stress at 1425°C. Initial microstructure of the above sample (M.L. Mecartney)

23. Thermomechanical properties Superplasticity II If grain boundary sliding was to occur in a completely rigid system of grains then voids would develop in the microstructure. The holes or cavities would expand or contract as grains, moving in three dimensions, approached or receded from them. However, many superplastic materials do not cavitate. Grain boundary sliding is therefore accommodated. Even when cavities are observed, their distribution is far from homogeneous and while they would accommodate sliding, cavitation is not as likely an accommodation mechanism as either diffusion or dislocation activity.

24. Thermomechanical properties General steady state creep equation In the steady state regime the following equation describes creep for all possible mechanisms D: diffusion coefficient G: shear modulus b: Burgersvector of the active glide system T: temperature d: grain size r: grain size exponent s: stress p: stress exponent ! D and G are functions of temperature! Two regimes: - low stress, small grain size regime 1: diffusion creep p = 1, r = 2 for Nabarro - Herring creep, r = 3 for Cobble creep - high stress, (large grain size) regime 2: diffusion creep + dislocation creep: power law creep r = 0, p = 3 - 7 When the samples deforms superplastic (very small grain size, high temperature) the constitutive equations depends heavily on the mechanisms that accomodate cavitation. log  regime 1 regime 2 superplasticity

25. Thermomechanical properties The deformation mechanisms in function of temperature are represented in maps of normalized stress vs. homologous temperature e.g. the fraction of the melting temperature. The topology for metals and ceramics are similar, but the strain rate contours are shifted to higher temperature for ceramics. (Poirier, 1990) Deformation maps

26. Mechanical strength as function of temperature for different ceramic materials: SC = silicon carbide, SN = silicon nitride, PSZ = partially stabilized zirconia, HP = hot pressed, S= sintered, RB = reaction bonded. Inco = Inconel one of the best high temperature steels available. (Yanagida, 1996) Thermomechanical properties High temperature strenght of ceramics

27. Thermomechanical properties Corrosion resistance The good high temperature mechanical strength and creep behavior of ceramics is paired with a very good resistance to different types of corrosion.

28. Thermomechanical properties High temperature application of structural ceramics I High temperature structural ceramics e.g. silicon nitride and carbides are used for parts in turbine engines (right, KRIOCERA silicon nitride turbines) or are applied as coating onto metallic turbine parts (top: gas turbine in thermal power plant).

29. Thermomechanical properties High temperature application of structural ceramics II Melting and shaping of metals requires containers withstanding corrosion and high temperature. Refractory materials such as olivine and magnesia are used as brick linings in steel furnaces (grey shaded area in the left image).

30. The change in the Helmholtz free energy is given by Thermomechanical properties Diffusion creep II At constant temperature typical for creep experiments it follows that Multiplying both sides by the volume of an atom gives V/V a is nothing else than the number of atoms of the sample e.g. With Q the free energy of formation, and K contains all preexponential terms The change of the Helmholtz energy with composition (N) is equal to the change of chemical potential. Equating p with the applied stress , it follows Consider now the concentration of vacancies under a free and a stressed surface. The concentration under a free, flat surface is given by With C  the vacancy concentration under the surface subjected to the normal stress  

31. Thermomechanical properties Diffusion creep III Assuming a box with faces A and B under stress   and     negative sign: compressive stresses, positive sign tensile stresses)respectively, the concentration of vacancies under A and B are given as BB A A The concentration difference, which will be the driving force for diffusion between faces A and B is given by Assuming   = -   These vacancy concentration gradients are responsible for the diffusion during creep experiments in the Nabarro-Herring and Cobble creep regimes.