11MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesThermal properties¾ Heat capacity • atomic vibrations, phonons • temperature dependence• contribution of electrons¾ Thermal expansion• connection to anharmonicity of interatomic potential• linear and volume coefficients of thermal expansion¾ Thermal conductivity• heat transport by phonons and electrons¾ Thermal stresses2MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesHeat capacityThe heat capacity, C, of a system is the ratio of the heat added to the system, or withdrawn from the system, to the resultant change in the temperature:C = ΔQ/ΔT = dQ/dT [J/deg]¾ This definition is only valid in the absence of phase transitions¾ Usually C is given as specific heat capacity, c, per gram or per mol¾ Heat capacity can be measured under conditions of constant temperature or constant volume. Thus, two distinct heat capacities can be defined:VVdTqC⎟⎠⎞⎜⎝⎛δ=PPdTqC⎟⎠⎞⎜⎝⎛δ=- heat capacity at constant volume- heat capacity at constant pressureCPis always greater than CV- Why?Hint: The difference between CPand Cvis very small for solids and liquids, but large for gases.3MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesHeat capacityHeat capacity is a measure of the ability of the material to absorb thermal energy. Thermal energy = kinetic energy of atomic motions + potential energy of distortion of interatomic bonds.The higher is T, the large is the mean atomic velocity and the amplitude of atomic vibrations Æ larger thermal energyVibrations of individual atoms in solids are not independent from each other. The coupling of atomic vibrations of adjacent atoms results in waves of atomic displacements. Each wave is characterized by its wavelength and frequency. For a wave of a given frequency ν, there is the smallest “quantum” of vibrational energy, hν, called phonon.Thus, the thermal energy is the energy of all phonons (or all vibrational waves) present in the crystal at a given temperature.Scattering of electrons on phonons is one of the mechanisms responsible for electrical resistivity (Chapter 18)4MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesTemperature dependence of heat capacityHeat capacity has a weak temperature dependence at high temperatures (above Debye temperature θD) but decreases down to zero as T approaches 0K.The constant value of the heat capacity of many simple solids issometimes called Dulong – Petit lawIn 1819 Dulong and Petit found experimentally that for many solids at room temperature, cv≈ 3R = 25 JK-1mol-1This is consistent with equipartition theorem of classical mechanics: energy added to solids takes the form of atomic vibrations and both kinetic and potential energy is associated with the three degrees of freedom of each atom. T/θDCv, J K-1mol-125MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesTemperature dependence of heat capacityThe low-T behavior can be explained by quantum theory. The first explanation was proposed by Einstein in 1906. He considered a solid as an ensemble of independent quantum harmonic oscillators vibrating at a frequency ν. Debye advanced the theory by treating the quantum oscillators as collective modes in the solid (phonons) and showed thatcv~ AT3 at T → 0KQuantized energy levelsΔE/kTne~P−ΔE << kT - classical behaviorΔE ≥ kT - quantum behaviorEnergyΔE = hν6MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesHeat capacity of metals – electronic contributionIn addition to atomic vibrations (phonons), thermal excitation of electrons can also make contribution to heat capacity.To contribute to bulk specific heat, the valence electrons would have to receive energy from the thermal energy, ~kT. Thus, only a small fraction of electrons which are within kT of the Fermi level makes a contribution to the heat capacity. This contribution is very small and insignificant at room temperature. The electron contribution to cvis proportional to temperature, cvel= γTand becomes significant (for metals only) at very low temperatures (remember that contribution of phonons cv~ AT3 at T → 0K).7MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesHeat capacity of various materials (at RT)8MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesThermal expansionMaterials expand when heated and contract when cooled()TTTllllllflfΔα=−α=Δ=−0000where l0is the initial length at T0, lfis the final length at Tfαlis the linear coefficient of thermal expansionSimilarly, the volume change with T can be described as ()TTTVVVVVVfVfΔα=−α=Δ=−0000where αVis the volume coefficient of thermal expansionFor isotropic materials and small expansions, αV≈ 3αl()000203032020303033333llVVllllllllllllVffΔ+=Δ+≈Δ+Δ+Δ+=Δ+==0003llVVVfΔ+≈00003llVVVVVfΔ≈Δ=−TTlVΔα≈Δα 339MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesPhysical origin of thermal expansiontypical interatomic interaction potentials are asymmetric (anharmonic) Potential Energy0Interatomic distance rincrease of the average value of interatomic separationRising temperature results in the increase of the average amplitude of atomic vibrations. For an anharmonic potential, this corresponds to the increase in the average value of interatomic separation, i.e. thermal expansion.10MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesPhysical origin of thermal expansionsymmetric (harmonic) potentialPotential Energy0Interatomic distance rthe average value of interatomic separation does not changeThermal expansion is related to the asymmetric (anharmonic) shape of interatomic potential. If the interatomic potential is symmetric (harmonic), the average value of interatomic separation does not change, i.e. no thermal expansion.11MSE 2090: Introduction to Materials Science Chapter 19, Thermal PropertiesThermal expansion of various materialstendency to expand upon heating is counteracted by contraction related to ferromagnetic properties of this alloy (magnetostriction)The stronger the interatomic bonding (deeper the potential energy curve), the smaller is the thermal expansion.The values of αland αVare increasing with rising TNegative thermal expansion:liquid water contracts when heated from 0 to 4°C.ZrVPO7, ZrW2O8, quartz at very low