Thermal properties Lattice vibrations I Most solids expand

Thermal properties Lattice vibrations I Most solids expand during heating. Thermal expansion is due to the change in the amplitude of lattice vibrations a rest position - 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. bond = spring Longitudinal vibration along a 1 -D atom chain - The shortest wavelength in a certain direction is equal to the shortest crystallographic translation in that direction.

Thermal properties Lattice vibrations II Atom vibrates assymetrically from 0°K position dmin reference atom dmax average d 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

Thermal properties Lattice vibrations III average r Position enveloppe of second atom Temp dmin, T 3 T 4 T x 3 T 2 T 1 interatomic distance r x dmax, T 3 T 4> T 3 > T 2 > T 1 dmin = o°K=dmax = o°K

Thermal properties Thermal expansion I (T) 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: 12 x °C-1 x 10 -6 The thermal expansion of a single dimension in a solid is characterized by the linear thermal expansion coefficient. 10 8 800°C x 6 4 x 200°C 2 0 const. 0 200 400 600 800 Example of a constant and three temperature dependent thermal expansion coefficient. 1000 T

Thermal properties Thermal expansion II Analogue to the linear expansion coefficient, a volume expansion coefficient can be defined: For simple structure, thermal expansion coefficient is related to the bond strength: where z is the charge of the cation and n its coordination number. Material Steel Aluminium Al 2 O 3 Mg. O(s) Zr O 2(tet) Zr O 2(mcl) Cordierite Zr. WO 4 (°C-1 )x 10 13. 0 22. 0 7. 2 - 8. 9 13. 5 12. 0 7. 0 2. 1 -2. 0 6 Average linear thermal expansion coefficients. Some ceramic material have particularly low expansion coefficient.

Thermal properties Linear thermal expansion Cu 1. 2 Zr O 2 1. 0 l/l(%) Al Al 2 O 3 0. 8 0. 6 0. 4 Si 3 N 4 0. 2 Cordierite 0 0 200 400 600 800 1000 T (C°) 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

Thermal properties Volume thermal expansion Cristobalite -Si O 2 V/V(%) 4. 0 3. 0 -Si O 2 mcl Zr O 2 2. 0 1. 0 0 tet Zr O 2 Pyrex 200 400 600 800 1000 Volume thermal expansion for materials with phase transformations and Pyrex glass.

Thermal properties Thermal conduction in a solid is described by equations which are analogous to Fick‘s equation for diffusion: Q: heat T: temperature A: surface Material Al 2 O 3 Al. N Cordierite Si. C Cordierite Steel Aluminium Diamond kth(W m-1 K -1 30. 0 - 35. 0 200. 0 -280. 0 4. 0 84. 0 -93. 0 45 239 2000 The heat transferred per unit time across a surface A is proportional to the temperature gradient across the surface. The proportionality factor is thermal conductivity kth. 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: Be. O, BN, Si. C, Si, Ga. P etc. Average thermal conductivities. Among the ceramic materials, there are low as well as high conductivity examples.

Thermomechanical properties Thermal shock resistance I 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 Ti 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: Ta Ts surface Temperature distribution in a plate which is cooled from both surfaces area under tension area under compression : : E: : Poisson‘s ratio failure strength Youngs modulus thermal expansion coefficient

Thermomechanical properties Thermal shock resistance II 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 thermal shock resistance. T Material Si. Al. ON Si 3 N 4 Al 2 O 3 Ti 3 Si. C 2 Fused Si. O 2 Li-Al-silicate glass ceramics T (°K) 900 500 200 >1400 1600 670 Failure pattern in Al 2 O 3 disk exposed to a temperature difference of 700 (°K))

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.

Thermomechanical properties Chip substrate and packaging 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. Bonding Wires IC Chip Ceramic Cap Substrate Pins Chip Bond Metallization Section through a micro chip Intel Pentium (IV) microprocessor Courtesy, Intel

Thermomechanical properties Power dissipation 100000 18 KW 5 KW 1. 5 KW 500 W Power (Watts) 10000 1000 Pentium® proc 100 286 486 8086 10 386 8085 8080 8008 1 4004 0. 1 1974 1978 1985 1992 2000 2004 2008 Evolution of power delivery and dissipation in microprocessors Courtesy, Intel

Thermal properties Aluminium nitride as chip packaging material I The most remarkable property exhibited by Al. N 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 highdensity 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 Al. N provide more efficient cooling than conventional and other ceramic substrates, hence their use as chip carriers and heat sinks. Bending strength (MPa) Dielectric constant TCE (× 10 -6/°C) Thermal cond. (W/m. K) Al. N Be. O Al 2 O 3 350 8. 8 4. 6 250 200 320 6. 8 9. 8 7. 5 7. 6 250 18 Al. N has wurtzite type structure

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

Thermal properties Chemical stability of Al. N Under ambient conditions, Al. N is not stable. It reacts with both oxygen and water vapour, the latter reaction is more prominent: 4 Al. N + 8 H 2 O = 2 Al 2 O 3 + 4 NH 4 + O 2 Commercially available Al. N powder has often an oxidized layer that diminshes thermal properties of the material, because during sintering a Al. ON 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 Al. N it is concentrated at the triple junction. The presence of the liquid phase enhances also the sinterability of Al. N, which is difficult to densify >95%.

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

Thermomechanical properties Creep deformation I At high temperature, ceramics under load will undergo a (viscous type) deformation called creep. Rupture e secondary primary tertiary Creep strain vs. time for constant load instantaneous deformation time 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

Thermomechanical properties Creep deformation II e T 3 or 3 T 2 or 2 T 1 or 1 T < 0. 4 Tm T 1<T 2<T 3 = const; 1< 2 < 3 T = const Creep strain vs. time for constant load and different temperature or constant temperatures and different loads (Tmmelting temperature) time 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

Thermomechanical properties Diffusion creep I s 11 -s 22 vacancies material flux vacancy flux shape of the grain after creep 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.

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 equiaxed, or, if they were not equi-axed prior to deformation, become so during superplastic flow. Strains of several hundred % are not unusual during deformation in the superplastic regime. This is an example of cubic stabilized Zr. O 2 (8 mol% Y 2 O 3) with 5 wt% Si. O 2 deformed under tensile stress at 1425°C. Initial microstructure of the above sample (M. L. Mecartney) www. nsf. gov/mps/dmr/highlights

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.

Thermomechanical properties General steady state creep equation � In the steady state regime the following equation describes creep for all possible mechanisms superplasticity 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! regime 2 regime 1 log 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.

Thermomechanical properties Deformation maps 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)

Thermomechanical properties High temperature strenght of ceramics 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 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.

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).

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).

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

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 A B B The concentration difference, which will be the driving force for diffusion between faces A and B is given by A Assuming = - These vacancy concentration gradients are responsible for the diffusion during creep experiments in the Nabarro-Herring and Cobble creep regimes.