Thermal properties. Heat capacity atomic vibrations, phonons temperature dependence contribution of electrons

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1 Therma properties Heat capacity atomic vibrations, phonons temperature dependence contribution of eectrons Therma expansion connection to anharmonicity of interatomic potentia inear and voume coefficients of therma expansion Therma conductivity heat transport by phonons and eectrons Therma stresses MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 1

2 Heat capacity The heat capacity, C, of a system is the ratio of the heat added to the system, or withdrawn from the system, to the resutant change in the temperature: C = ΔQ/ΔT = dq/dt [J/deg] This definition is ony vaid in the absence of phase transitions Usuay C is given as specific heat capacity, c, per gram or per mo Heat capacity can be measured under conditions of constant temperature or constant voume. Thus, two distinct heat capacities can be defined: C C P δq = dt δq = dt P - heat capacity at constant voume - heat capacity at constant pressure C P is aways greater than C - Why? Hint: The difference between C P and C v is very sma for soids and iquids, but arge for gases. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 2

3 Heat capacity Heat capacity is a measure of the abiity of the materia to absorb therma energy. Therma energy = kinetic energy of atomic motions + potentia energy of distortion of interatomic bonds. The higher is T, the arge is the mean atomic veocity and the ampitude of atomic vibrations arger therma energy ibrations of individua atoms in soids are not independent from each other. The couping of atomic vibrations of adjacent atoms resuts in waves of atomic dispacements. Each wave is characterized by its waveength and frequency. For a wave of a given frequency ν, there is the smaest quantum of vibrationa energy, hν, caed phonon. Thus, the therma energy is the energy of a phonons (or a vibrationa waves) present in the crysta at a given temperature. Scattering of eectrons on phonons is one of the mechanisms responsibe for eectrica resistivity (Chapter 18) MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 3

4 Temperature dependence of heat capacity Heat capacity has a weak temperature dependence at high temperatures (above Debye temperature θ D ) but decreases down to zero as T approaches K. C v, J K -1 mo -1 T/θ D The constant vaue of the heat capacity of many simpe soids is sometimes caed Duong Petit aw In 1819 Duong and Petit found experimentay that for many soids at room temperature, c v 3R = 25 JK -1 mo -1 This is consistent with equipartition theorem of cassica mechanics: energy added to soids takes the form of atomic vibrations and both kinetic and potentia energy is associated with the three degrees of freedom of each atom. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 4

5 Temperature dependence of heat capacity The ow-t behavior can be expained by quantum theory. The first expanation was proposed by Einstein in 196. He considered a soid as an ensembe of independent quantum harmonic osciators vibrating at a frequency ν. Debye advanced the theory by treating the quantum osciators as coective modes in the soid (phonons) and showed that c v ~ AT 3 at T K Quantized energy eves ΔE << kt - cassica behavior ΔE kt - quantum behavior Energy P ΔE = hν n ~ e ΔE/kT MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 5

6 Heat capacity of metas eectronic contribution In addition to atomic vibrations (phonons), therma excitation of eectrons can aso make contribution to heat capacity. To contribute to buk specific heat, the vaence eectrons woud have to receive energy from the therma energy, ~kt. Thus, ony a sma fraction of eectrons which are within kt of the Fermi eve makes a contribution to the heat capacity. This contribution is very sma and insignificant at room temperature. The eectron contribution to c v is proportiona to temperature, c v e = γt and becomes significant (for metas ony) at very ow temperatures (remember that contribution of phonons c v ~ AT 3 at T K). MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 6

7 Heat capacity of various materias (at RT) MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 7

8 8 MSE 29: Introduction to Materias Science Chapter 19, Therma Properties Therma expansion Materias expand when heated and contract when cooed ( ) T T T f f = α Δ = α Δ = where is the initia ength at T, f is the fina ength at T f α is the inear coefficient of therma expansion Simiary, the voume change with T can be described as ( ) T T T f f Δ = α = α Δ = where α is the voume coefficient of therma expansion For isotropic materias and sma expansions, α 3α ( ) f f Δ + = Δ + + Δ Δ + Δ + = + Δ = = 3 f Δ + 3 f Δ Δ = T T Δ α Δ α 3

9 Physica origin of therma expansion typica interatomic interaction potentias are asymmetric (anharmonic) Potentia Energy Interatomic distance r increase of the average vaue of interatomic separation Rising temperature resuts in the increase of the average ampitude of atomic vibrations. For an anharmonic potentia, this corresponds to the increase in the average vaue of interatomic separation, i.e. therma expansion. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 9

10 Physica origin of therma expansion symmetric (harmonic) potentia Potentia Energy Interatomic distance r the average vaue of interatomic separation does not change Therma expansion is reated to the asymmetric (anharmonic) shape of interatomic potentia. If the interatomic potentia is symmetric (harmonic), the average vaue of interatomic separation does not change, i.e. no therma expansion. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 1

11 Therma expansion of various materias tendency to expand upon heating is counteracted by contraction reated to ferromagnetic properties of this aoy (magnetostriction) Negative therma expansion: iquid water contracts when heated from to 4 C. ZrPO 7, ZrW 2 O 8, quartz at very ow T. The stronger the interatomic bonding (deeper the potentia energy curve), the smaer is the therma expansion. The vaues of α and α are increasing with rising T MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 11

12 Impications and appications of therma expansion Raiway tracks are buit from stee rais aid with a gap between the ends Thermostats based on bimeta strips made of two metas with different coefficient of therma expansion: A bimeta coi from a thermometer reacts to the heat from a ighter, by Hustvedt, Wikipedia MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 12

13 Therma conductivity Therma conductivity: heat is transferred from high to ow temperature regions of the materia. q k dt dx = - Fourier's aw where q is the heat fux (amount of therma energy fowing through a unit area per unit time) and dt/dx is the temperature gradient, and k is the coefficient of therma conductivity, often caed simpy therma conductivity. Units: q [W/m 2 ], k [W/(m K)] Note the simiarity to the Fick s first aw for atomic diffusion (Chapter 5): the diffusion fux is proportiona to the concentration gradient: J = D dc dx Non-steady state heat fow and atomic diffusion are described by the same equation: C t = D 2 C 2 x T t = k c ρ P 2 T 2 x MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 13

14 Mechanisms of heat conduction Heat is transferred by phonons (attice vibration waves) and eectrons. The therma conductivity of a materia is defined by combined contribution of these two mechanisms: k = k + k e where k and k e are the attice and eectronic therma conductivities. Lattice conductivity: Transfer of therma energy phonons Eectron conductivity: Free (conduction band) eectrons equiibrate with attice vibrations in hot regions, migrate to coder regions and transfer a part of their therma energy back to the attice by scattering on phonons. The eectron contribution is dominant in metas and absent in insuators. Since free eectrons are responsibe for both eectrica and therma conduction in metas, the two conductivities are reated to each other by the Wiedemann-Franz aw: L = k σt where σ is the eectrica conductivity and L is a constant MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 14

15 Wiedemann-Franz aw L = k σt MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 15

16 Effect of aoying on heat conduction in metas The same factors that affect the eectrica conductivity (discussed in Chapter 18) aso affect therma conductivity in metas. E.g., adding impurities introduces scattering centers for conduction band eectrons and reduce k. Cu-Zn aoy MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 16

17 Quest for good thermoeectric (TE) materias Thermoeectric conversion: conversion of therma to eectrica energy An appied temperature difference ΔT causes charge carriers in the materia (eectrons or hoes) to diffuse from the hot side to the cod side, resuting in current fow through the circuit and producing an eectrostatic potentia Δ. Figure of merit of TE materia: ZT =(α 2 σ/k)t where σ, κ and α are the eectrica conductivity, therma conductivity, and Seebeck coefficient defined as α = / T. Li et a., Nat. Asia Mater., 152, 21 Good TE materia: High σ (ow Joue heating), arge Seebeck coefficient (arge Δ), ow k (arge ΔT) are necessary. ZT 3 is needed for TE energy converters to compete with mechanica power generation and active refrigeration. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 17

18 Quest for good thermoeectric (TE) materias Nanostructured materias - a chance to disconnect the inkage between the therma and eectrica transport by controing scattering mechanisms grain boundaries, interfaces reduction of k, but aso the deterioration of carrier mobiity (μ). nano/microcomposite (f): nanopartices scatter phonon, whie micropartices can form a connected (percoating) network for eectron transport. The high performance of these materias is reated to nanostructure engineering Li et a., Nat. Asia Mater., 152, 21 MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 18

19 Heat conduction in nonmetaic materias In insuators and semiconductors the heat transfer is by phonons and, generay, is ower than in metas. It is sensitive to structure: gasses and amorphous ceramics have ower k compared to the crystaine ones (phonon scattering is more effective in irreguar or disordered materias). Therma conductivity decreases with porosity (e.g. foamed poystyrene is used for drinking cups). Therma conductivity of poymers depends on the degree of crystainity highy crystaine poymer has higher k Corn kernes have a hard moisture-seaed hu and a softer core with a high moisture content. When kernes are heated, the inner moisture is vaporized and breaks the hu. The corn that pops and is used to make popcorn have hus that are made from ceuose that is more crystaine than non-popping corn kernes. As a resut, the hus are good therma conductors (2-3 times better than corns that do not pop) and are mechanicay stronger (higher pressure can buid up before popping). MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 19

20 Temperature dependence of therma conductivity Therma conductivity tend to decrease with increasing temperature (more efficient scattering of heat carriers on attice vibrations), but can exhibit compex non-monotonous behavior. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 2

21 Therma conductivity of various materias at RT Diamond: 231 Graphite: aong c-axis: 2 aong a-axis: 9.5 SiO 2 crystaine aong c-axis: 1.4 aong a-axis: 6.2 amorphous: 1.38 MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 21

22 Therma conductivity of poymer nanofibers Athough, normay, poymers are therma insuators (k ~.1 Wm -1 K -1 ), it has been demonstrated in atomistic simuations [Henry & Chen, PRL 11, 23552, 28] that therma conductivity of individua poymer chains can be very high. This finding has been supported by recent experimenta study of high-quaity utra-drawn poyethyene nanofibers with diameters of 5-5 nm and engths up to tens of miimeters [Shen et a., Nat. Nanotechno. 5, 251, 21] it has been demonstrated that the nanofibers conducts heat just as we as most metas, yet remain eectrica insuators. Making the nanofibers: Puing a thin thread of materia from a iquid soution - poymer moecues become very highy aigned. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 22

23 Therma conductivity of poymer nanofibers 14 Wm -1 K -1 3 times higher than k of buk poyethyene! Shen et a., Nat. Nanotechno. 5, 251, 21 Highy anisotropic unidirectiona therma conductivity: may be usefu for appications where it is important to draw heat away from an object, such as a computer processor chip. MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 23

24 Therma stresses can be generated due to restrained therma expansion/contraction or temperature gradients that ead to differentia dimensiona changes in different part of the soid body. can resut in pastic deformation or fracture. In a rod with restrained axia deformation: σ = E α ΔT where E is the eastic moduus, α is the inear coefficient of therma expansion and ΔT is the temperature change. Stresses from temperature gradient Rapid heating can resut in strong temperature gradients confinement of expansion by coder parts of the sampe. The same for cooing tensie stresses can be introduced in a surface region of rapidy cooed piece of materia. Therma stresses can cause pastic deformation (in ductie materias) or fracture (in britte materias). The abiity of materia to withstand therma stresses due to the rapid cooing/heating is caed therma shock resistance. Shock resistance parameter for britte materias (ceramics): σ f k TSR E α where σ f is fracture strength of the materia MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 24

25 Restrained therma expansion: Exampe probem A brass rod is restrained but stress-free at RT (2ºC). Young s moduus of brass is 1 GPa, α = / C At what temperature does the stress reach -172 MPa? T Δ T f if therma expansion is unconstrained σ Δ σ if expansion is constrained ε Δ RT compress = ε th = ε th = α = ( T ) f T Δ RT σ = Eεcompress = Eα ( Tf T ) = Eα ( T Tf ) T f = T σ Eα = Pa o Pa 2 1 C = 16 o C MSE 29: Introduction to Materias Science Chapter 19, Therma Properties 25

26 Summary Make sure you understand anguage and concepts: anharmonic potentia atomic vibrations, phonons eectron heat conductivity eectronic contribution to heat capacity heat capacity, C P vs. C attice heat conductivity inear coefficient of therma expansion specific heat capacity therma conductivity therma expansion therma stresses therma shock resistance voume coefficients of therma expansion 24.9 C v, J K -1 mo -1 High T, C v 3R Low T, C v cassica Duong Petit aw quantum Debye mode MSE 29: Introduction to Materias Science T Chapter 19, Therma Properties 26