14 Thermal Properties of Materials 14.1·Fundamentals The thermal properties of materials are important whenever heat- ing and cooling devices are designed.Thermally induced expan- sion of materials has to be taken into account in the construc- tion industry as well as in the design of precision instruments. Heat conduction plays a large role in thermal insulation,for ex- ample,in homes,industry,and spacecraft.Some materials such as copper and silver conduct heat very well;other materials,like wood or rubber,are poor heat conductors.Good electrical con- ductors are generally also good heat conductors.This was dis- covered in 1853 by Wiedemann and Franz,who found that the ratio between heat conductivity and electrical conductivity (di- vided by temperature)is essentially constant for all metals. The thermal conductivity of materials varies only within five orders of magnitude (Figure 14.1).This is in sharp contrast to the variation in electrical conductivity,which spans about twenty-five orders of magnitude (Figure 11.1).The thermal con- ductivity of metals and alloys can be readily interpreted by mak- ing use of the electron theory,elements of which were explained in previous chapters of this book.The electron theory postulates that free electrons perform random motions with high velocity over a large number of atomic distances.In the hot part of a metal bar they pick up energy by interactions with the vibrating lattice atoms.This thermal energy is eventually transmitted to the cold end of the bar. In electric insulators,in which no free electrons exist,the con- duction of thermal energy must occur by a different mechanism. This new mechanism was found by Einstein at the beginning of
14 The thermal properties of materials are important whenever heating and cooling devices are designed. Thermally induced expansion of materials has to be taken into account in the construction industry as well as in the design of precision instruments. Heat conduction plays a large role in thermal insulation, for example, in homes, industry, and spacecraft. Some materials such as copper and silver conduct heat very well; other materials, like wood or rubber, are poor heat conductors. Good electrical conductors are generally also good heat conductors. This was discovered in 1853 by Wiedemann and Franz, who found that the ratio between heat conductivity and electrical conductivity (divided by temperature) is essentially constant for all metals. The thermal conductivity of materials varies only within five orders of magnitude (Figure 14.1). This is in sharp contrast to the variation in electrical conductivity, which spans about twenty-five orders of magnitude (Figure 11.1). The thermal conductivity of metals and alloys can be readily interpreted by making use of the electron theory, elements of which were explained in previous chapters of this book. The electron theory postulates that free electrons perform random motions with high velocity over a large number of atomic distances. In the hot part of a metal bar they pick up energy by interactions with the vibrating lattice atoms. This thermal energy is eventually transmitted to the cold end of the bar. In electric insulators, in which no free electrons exist, the conduction of thermal energy must occur by a different mechanism. This new mechanism was found by Einstein at the beginning of Thermal Properties of Materials 14.1 • Fundamentals
272 14.Thermal Properties of Materials Rubber, Cork H20 Sulfur Wood, Glass, Fe Cu Asbestos Nylon Concrete NaCl Ge Si Al Ag Diamond 10-1 101 102 103 Lm·K Phonon conductors Electron conductors FIGURE 14.1.Room- the 20th century.He postulated the existence of phonons or lat- temperature thermal tice vibration quanta,which are thought to be created in large conductivities for numbers in the hot part of a solid and partially eliminated in the some materials.See cold part.The transfer of heat in dielectric solids is thus linked also Table 14.3. to a flow of phonons from hot to cold.Figure 14.1 indicates that in a transition region both electrons as well as phonons may con- tribute,in various degrees,to thermal conduction.Actually, phonon-induced thermal conduction occurs even in metals,but its contribution is negligible compared to that of electrons. Other thermal properties are the specific heat capacity,and a related property,the molar heat capacity.Their importance can best be appreciated by the following experimental observations: Two substances with the same mass but different values for the specific heat capacity require different amounts of thermal en- ergy to reach the same temperature.Water,for example,which has a relatively high specific heat capacity,needs more thermal energy to reach a given temperature than,say,copper or lead of the same mass.Specifically,it takes 4.18 J!to raise 1 g of water by 1 K.But the same heat raises the temperature of 1 g of cop- per by about 11 K.In short,water has a larger heat capacity than copper.(The large heat capacity of water is,incidentally,the rea- son for the balanced climate in coastal regions and the heating of North European countries by the warm water of the Gulf Stream.)We need to define the various versions of heat capaci- ties for clarification. The heat capacity,C',is the amount of heat,do,that needs to be transferred to a substance in order to raise its temperature by a certain temperature interval.The unit for the heat capacity is J/K. IThe unit of energy is the joule;see Appendix II.Obsolete units are the calorie (1 cal =4.18 J),or the British thermal unit(BTU)which is the heat required to raise the temperature of one pound of water by one de- gree fahrenheit (1 BTU 1055 J)
the 20th century. He postulated the existence of phonons or lattice vibration quanta, which are thought to be created in large numbers in the hot part of a solid and partially eliminated in the cold part. The transfer of heat in dielectric solids is thus linked to a flow of phonons from hot to cold. Figure 14.1 indicates that in a transition region both electrons as well as phonons may contribute, in various degrees, to thermal conduction. Actually, phonon-induced thermal conduction occurs even in metals, but its contribution is negligible compared to that of electrons. Other thermal properties are the specific heat capacity, and a related property, the molar heat capacity. Their importance can best be appreciated by the following experimental observations: Two substances with the same mass but different values for the specific heat capacity require different amounts of thermal energy to reach the same temperature. Water, for example, which has a relatively high specific heat capacity, needs more thermal energy to reach a given temperature than, say, copper or lead of the same mass. Specifically, it takes 4.18 J1 to raise 1 g of water by 1 K. But the same heat raises the temperature of 1 g of copper by about 11 K. In short, water has a larger heat capacity than copper. (The large heat capacity of water is, incidentally, the reason for the balanced climate in coastal regions and the heating of North European countries by the warm water of the Gulf Stream.) We need to define the various versions of heat capacities for clarification. The heat capacity, C, is the amount of heat, dQ, that needs to be transferred to a substance in order to raise its temperature by a certain temperature interval. The unit for the heat capacity is J/K. FIGURE 14.1. Roomtemperature thermal conductivities for some materials. See also Table 14.3. 272 14 • Thermal Properties of Materials Sulfur Wood, Asbestos Rubber, Cork Nylon H2O Glass, Concrete NaCl SiO2 Fe Ge Si Al Cu Ag Diamond 103 102 101 10–1 1 K W m · K Phonon conductors Electron conductors 1The unit of energy is the joule; see Appendix II. Obsolete units are the calorie (1 cal 4.18 J), or the British thermal unit (BTU) which is the heat required to raise the temperature of one pound of water by one degree fahrenheit (1 BTU 1055 J)
14.1·Fundamentals 273 The heat capacity is not defined uniquely,that is,one needs to specify the conditions under which the heat is added to the sys- tem.Even though several choices for the heat capacities are pos- sible,one is generally interested in only two:the heat capacity at constant volume C'and the heat capacity at constant pressure Cp.The former is the most useful quantity because Cv is ob- tained immediately from the energy,E,of the system.The heat capacity at constant volume is defined as: (14.1) On the other hand,it is much easier to measure the heat capac- ity of a solid at constant pressure than at constant volume.For- tunately,the difference between Cp and Cv for solids vanishes at low temperatures and is only about 5%at room temperature. The specific heat capacity is the heat capacity per unit mass: c=C (14.2) 17n where m is the mass of the system.Again,one can define it for constant volume or constant pressure.It is a material constant and is temperature-dependent.Characteristic values for cp are given in Table 14.1.The unit of the specific heat capacity is J/g.K.We note from Table 14.1 that the cp values for solids are considerably smaller than the specific heat capacity of water. Combining Egs.(14.1)and (14.2)yields: △E=△Tncv, (14.3) which expresses that the thermal energy (or heat)which is trans- TABLE 14.1.Experimental thermal parameters of various substances at room temperature and ambient pressure Specific heat Molar Molar heat Molar heat capacity,(cp)(atomic)mass capacity (Cp)capacity (Cv) Substance J g J 8·K mol mol·K ol·K Al 0.897 27.0 24.25 23.01 Fe 0.449 55.8 25.15 24.68 Ni 0.456 58.7 26.8 24.68 Cu 0.385 63.5 24.48 23.43 Pb 0.129 207.2 26.85 24.68 Ag 0.235 107.9 25.36 24.27 C(graphite) 0.904 12.0 10.9 9.20 Water 4.184 18.0 75.3
The heat capacity is not defined uniquely, that is, one needs to specify the conditions under which the heat is added to the system. Even though several choices for the heat capacities are possible, one is generally interested in only two: the heat capacity at constant volume Cv and the heat capacity at constant pressure Cp. The former is the most useful quantity because Cv is obtained immediately from the energy, E, of the system. The heat capacity at constant volume is defined as: Cv E T v . (14.1) On the other hand, it is much easier to measure the heat capacity of a solid at constant pressure than at constant volume. Fortunately, the difference between Cp and Cv for solids vanishes at low temperatures and is only about 5% at room temperature. The specific heat capacity is the heat capacity per unit mass: c C m (14.2) where m is the mass of the system. Again, one can define it for constant volume or constant pressure. It is a material constant and is temperature-dependent. Characteristic values for cp are given in Table 14.1. The unit of the specific heat capacity is J/g K. We note from Table 14.1 that the cp values for solids are considerably smaller than the specific heat capacity of water. Combining Eqs. (14.1) and (14.2) yields: E Tm cv, (14.3) which expresses that the thermal energy (or heat) which is trans- 14.1 • Fundamentals 273 TABLE 14.1. Experimental thermal parameters of various substances at room temperature and ambient pressure Specific heat Molar Molar heat Molar heat capacity, (cp) (atomic) mass capacity (Cp) capacity (Cv) Substance g J K m g ol mo J l K mo J l K Al 0.897 27.0 24.25 23.01 Fe 0.449 55.8 25.15 24.68 Ni 0.456 58.7 26.80 24.68 Cu 0.385 63.5 24.48 23.43 Pb 0.129 207.2 26.85 24.68 Ag 0.235 107.9 25.36 24.27 C (graphite) 0.904 12.0 10.90 9.20 Water 4.184 18.0 75.30
274 14.Thermal Properties of Materials ferred to a system equals the product of mass,increase in tem- perature,and specific heat capacity. A further useful material constant is the heat capacity per mole. It compares materials that contain the same number of mole- cules or atoms.The molar heat capacity is obtained by multi- plying the specific heat capacity cv(or cp)by the molar mass,M (see Table 14.1): C,=G=cw·M, (14.4) where n is the amount of substance in mol. We see from Table 14.1 that the room-temperature molar heat capacity at constant volume is approximately 25 J/mol.K for most solids.This was experimentally discovered in 1819 by Du- long and Petit.The experimental molar heat capacities for some materials are depicted in Figure 14.2 as a function of tempera- ture.We notice that some materials,such as carbon,reach the Dulong-Petit value only at high temperatures.Some other ma- terials such as lead reach 25 J/mol.K at relatively low tempera- tures. All heat capacities are zero at T=0K.The Cy values near T= O K climb in proportion to T3 and reach 96%of their final value at a temperature Op,which is defined to be the Debye tempera- ture.Op is an approximate dividing point between a high- temperature region,where classical models can be used for the interpretation of Cv,and a low-temperature region,where quan- tum theory needs to be applied.Selected Debye temperatures are listed in Table 14.2. 25 Pb Cu Cv mol.K Carbon FIGURE 14.2.Temperature depen- dence of the molar heat capacity 100 200 300 400 500 Cy for some materials. T [K]
ferred to a system equals the product of mass, increase in temperature, and specific heat capacity. A further useful material constant is the heat capacity per mole. It compares materials that contain the same number of molecules or atoms. The molar heat capacity is obtained by multiplying the specific heat capacity cv (or cp) by the molar mass, M (see Table 14.1): Cv C n v cv M, (14.4) where n is the amount of substance in mol. We see from Table 14.1 that the room-temperature molar heat capacity at constant volume is approximately 25 J/mol K for most solids. This was experimentally discovered in 1819 by Dulong and Petit. The experimental molar heat capacities for some materials are depicted in Figure 14.2 as a function of temperature. We notice that some materials, such as carbon, reach the Dulong–Petit value only at high temperatures. Some other materials such as lead reach 25 J/mol K at relatively low temperatures. All heat capacities are zero at T 0 K. The Cv values near T 0 K climb in proportion to T3 and reach 96% of their final value at a temperature )D, which is defined to be the Debye temperature. )D is an approximate dividing point between a hightemperature region, where classical models can be used for the interpretation of Cv, and a low-temperature region, where quantum theory needs to be applied. Selected Debye temperatures are listed in Table 14.2. 274 14 • Thermal Properties of Materials 25 Cv J mol. K Pb Cu Al Carbon 100 200 300 400 500 T [K] FIGURE 14.2. Temperature dependence of the molar heat capacity Cv for some materials.
14.2.Interpretation of the Heat Capacity by Various Models 275 TABLE 14.2.Debye temperatures of some materials Substance ep(K) Pb 95 Au 170 Ag 230 270 Cu 340 Fe 360 Al 375 Si 650 1850 GaAs 204 InP 162 14.2.Interpretation of the Heat Capacity by Various Models The classical (atomistic)theory for the interpretation of the heat capacity postulates that each atom in a crystal is bound to its site by a harmonic force similar to a spring.A given atom is thought to be capable of absorbing thermal energy,and in doing so it starts to vibrate about its point of rest.The amplitude of the oscillation is restricted by electrostatic repulsion forces of the nearest neighbors.The extent of this thermal vibration is there- fore not more than 5 or 10%of the interatomic spacing,de- pending on the temperature.In short,we compare an atom with a sphere which is held at its site by two springs [Figure 14.3(a)]. The thermal energy that a harmonic oscillator of this kind can absorb is proportional to the absolute temperature of the envi- ronment.The proportionality factor has been found to be the Boltzmann constant kB(see Appendix II).The average energy of the oscillator is then: E=kBT. (14.5) Now,solids are three-dimensional.Thus,a given atom in a cu- bic crystal also responds to the harmonic forces of lattice atoms in the other two directions.In other words,it is postulated that each atom in a cubic crystal represents three oscillators [Figure 14.3(b)],each of which absorbs the thermal energy kBT.There- fore,the average energy per atom is: E=3kBT. (14.6)
The classical (atomistic) theory for the interpretation of the heat capacity postulates that each atom in a crystal is bound to its site by a harmonic force similar to a spring. A given atom is thought to be capable of absorbing thermal energy, and in doing so it starts to vibrate about its point of rest. The amplitude of the oscillation is restricted by electrostatic repulsion forces of the nearest neighbors. The extent of this thermal vibration is therefore not more than 5 or 10% of the interatomic spacing, depending on the temperature. In short, we compare an atom with a sphere which is held at its site by two springs [Figure 14.3(a)]. The thermal energy that a harmonic oscillator of this kind can absorb is proportional to the absolute temperature of the environment. The proportionality factor has been found to be the Boltzmann constant kB (see Appendix II). The average energy of the oscillator is then: E kBT. (14.5) Now, solids are three-dimensional. Thus, a given atom in a cubic crystal also responds to the harmonic forces of lattice atoms in the other two directions. In other words, it is postulated that each atom in a cubic crystal represents three oscillators [Figure 14.3(b)], each of which absorbs the thermal energy kBT. Therefore, the average energy per atom is: E 3kBT. (14.6) 14.2 • Interpretation of the Heat Capacity by Various Models 275 TABLE 14.2. Debye temperatures of some materials Substance )D (K) Pb 95 Au 170 Ag 230 W 270 Cu 340 Fe 360 Al 375 Si 650 C 1850 GaAs 204 InP 162 14.2 • Interpretation of the Heat Capacity by Various Models
276 14.Thermal Properties of Materials ○w○w○ r o (a) (b) FIGURE 14.3.(a)A one- We consider now nNo atoms,where No is the Avogadro number; dimensional harmonic see Appendix II.Then the total internal energy of these atoms is: oscillator and (b)a three-dimensional E 3nNokBT. (14.7) harmonic oscillator. Finally,the molar heat capacity is given by combining Eqs.(14.1), (14.4),and (14.7),which yields: Cy 3NokB. (14.8) Inserting the numerical values for No and kg(see Appendix II) into Eq.(14.8)yields Cv=24.95J/mol·K, i.e.,about 25 J/mol.K which is quite in agreement with the ex- perimental findings at high temperatures(Figure 14.2). It is satisfying to see that a simple model involving three har- monic oscillators per atom can readily explain the experimen- tally observed heat capacity.However,one shortcoming is im- mediately evident:the calculated molar heat capacity turned out to be temperature-independent,according to Eq.(14.8),and also independent of the material.This discrepancy with the observed behavior(Figure 14.2)was puzzling to scientists at the turn of the 20th century and had to await quantum theory to be explained properly.Einstein postulated in 1907 that the energies of the above-mentioned classical oscillators should be quantized,i.e., he postulated that only certain vibrational modes should be al- lowed,quite in analogy to the allowed energy states of electrons. These lattice vibration quanta were called phonons. The term phonon stresses an analogy with electrons or pho- tons.As we know from Chapter 13,photons are quanta of elec- tromagnetic radiation,i.e.,photons describe (in the appropriate
We consider now nN0 atoms, where N0 is the Avogadro number; see Appendix II. Then the total internal energy of these atoms is: E 3nN0kBT. (14.7) Finally, the molar heat capacity is given by combining Eqs. (14.1), (14.4), and (14.7), which yields: Cv 3N0kB. (14.8) Inserting the numerical values for N0 and kB (see Appendix II) into Eq. (14.8) yields Cv 24.95 J/mol K, i.e., about 25 J/mol K which is quite in agreement with the experimental findings at high temperatures (Figure 14.2). It is satisfying to see that a simple model involving three harmonic oscillators per atom can readily explain the experimentally observed heat capacity. However, one shortcoming is immediately evident: the calculated molar heat capacity turned out to be temperature-independent, according to Eq. (14.8), and also independent of the material. This discrepancy with the observed behavior (Figure 14.2) was puzzling to scientists at the turn of the 20th century and had to await quantum theory to be explained properly. Einstein postulated in 1907 that the energies of the above-mentioned classical oscillators should be quantized, i.e., he postulated that only certain vibrational modes should be allowed, quite in analogy to the allowed energy states of electrons. These lattice vibration quanta were called phonons. The term phonon stresses an analogy with electrons or photons. As we know from Chapter 13, photons are quanta of electromagnetic radiation, i.e., photons describe (in the appropriate FIGURE 14.3. (a) A onedimensional harmonic oscillator and (b) a three-dimensional harmonic oscillator. 276 14 • Thermal Properties of Materials z x y (a) (b)
14.2.Interpretation of the Heat Capacity by Various Models 277 frequency range)classical light.Phonons,on the other hand,are quanta of the ionic displacement field which (in the appropriate frequency range)represent classical sound. The word phonon conveys the particle nature of an oscillator. Moreover,Einstein also postulated a particle-wave duality.This suggests phonon waves which propagate through the crystal with the speed of sound.Phonon waves are not electromagnetic waves; they are elastic waves,vibrating in a longitudinal and/or in a transversal mode. The quantum theoretical treatment of the heat capacity,as de- veloped by Einstein and improved by Debye is too involved for the present book.The result of the Einstein theory may be given here,nevertheless: exp Cy=3NokB (14.9) exp We discuss Cy for two special temperature regions.For large temperatures the approximation e*=1 +x can be applied,which yields Cv=3NokB(see Problem 14.6)in agreement with (14.8), i.e.,we obtain the classical Dulong-Petit value.For T-0,C ap- proaches zero,again in agreement with experimental observa- tions.Thus,the temperature dependence of Cy is now in quali- tative accord with the experimental findings.One minor discrepancy,however,has to be noted:At very small tempera- tures the experimental C.decreases by T3,as stated above.The Einstein theory predicts,instead,an exponential decrease.The Debye theory which we shall not discuss here alleviates this dis- crepancy by postulating that the individual oscillators interact with each other. At very high and very low temperatures,the phonon theory does not yield a complete description of the observed behavior. The reason for this is that at these temperatures the free elec- trons (if present)provide a noticeable contribution to Cv.Again, quantum mechanical considerations as developed in Section 11.1 need to be applied.The electron contribution yields (for mono- valent metals)the following expression: Cg=ok话r 2 EF (14.10) where Er is the Fermi energy (see Section 11.1).We notice a lin- ear relationship between heat capacity and temperature.Figure 14.4 summarizes the experimentally observed C-values as well
frequency range) classical light. Phonons, on the other hand, are quanta of the ionic displacement field which (in the appropriate frequency range) represent classical sound. The word phonon conveys the particle nature of an oscillator. Moreover, Einstein also postulated a particle-wave duality. This suggests phonon waves which propagate through the crystal with the speed of sound. Phonon waves are not electromagnetic waves; they are elastic waves, vibrating in a longitudinal and/or in a transversal mode. The quantum theoretical treatment of the heat capacity, as developed by Einstein and improved by Debye is too involved for the present book. The result of the Einstein theory may be given here, nevertheless: Cv 3N0kB k ' B & T 2 . (14.9) We discuss Cv for two special temperature regions. For large temperatures the approximation ex 1 x can be applied, which yields Cv 3N0kB (see Problem 14.6) in agreement with (14.8), i.e., we obtain the classical Dulong–Petit value. For T 0, Cv approaches zero, again in agreement with experimental observations. Thus, the temperature dependence of Cv is now in qualitative accord with the experimental findings. One minor discrepancy, however, has to be noted: At very small temperatures the experimental Cv decreases by T3, as stated above. The Einstein theory predicts, instead, an exponential decrease. The Debye theory which we shall not discuss here alleviates this discrepancy by postulating that the individual oscillators interact with each other. At very high and very low temperatures, the phonon theory does not yield a complete description of the observed behavior. The reason for this is that at these temperatures the free electrons (if present) provide a noticeable contribution to Cv. Again, quantum mechanical considerations as developed in Section 11.1 need to be applied. The electron contribution yields (for monovalent metals) the following expression: Cv el , (14.10) where EF is the Fermi energy (see Section 11.1). We notice a linear relationship between heat capacity and temperature. Figure 14.4 summarizes the experimentally observed Cv-values as well 2 N0kB 2 T 2 EF exp k ' B & T exp k ' B & T 1 2 14.2 • Interpretation of the Heat Capacity by Various Models 277
278 14.Thermal Properties of Materials Cy Classical theory Experiment Phonon model FIGURE 14.4.Schematic representation of the temperature dependence of the mo- Electronic heat capacity lar heat capacity,experimental and ac- cording to three models. 300KT as the contributions of electron theory,phonon theory,and clas- sical considerations as outlined above. 14.3.Thermal Conduction Heat conduction (or thermal conduction)is the transfer of ther- mal energy from a hot body to a cold body when both bodies are brought into contact.For best visualization we consider a bar of a material of length x whose ends are held at different tempera- tures (Figure 14.5).The heat that flows through a cross section of the bar divided by time and area(i.e.,the heat flux,Jo)is pro- E2 FIGURE 14.5.Schematic representation of a bar of a material whose ends are at different temperatures. eH←H X
as the contributions of electron theory, phonon theory, and classical considerations as outlined above. Heat conduction (or thermal conduction) is the transfer of thermal energy from a hot body to a cold body when both bodies are brought into contact. For best visualization we consider a bar of a material of length x whose ends are held at different temperatures (Figure 14.5). The heat that flows through a cross section of the bar divided by time and area (i.e., the heat flux, JQ) is pro- 278 14 • Thermal Properties of Materials 300 K Classical theory Experiment Phonon model Electronic heat capacity T cv FIGURE 14.4. Schematic representation of the temperature dependence of the molar heat capacity, experimental and according to three models. 14.3 • Thermal Conduction T E1 E2 X FIGURE 14.5. Schematic representation of a bar of a material whose ends are at different temperatures.
14.3.Thermal Conduction 279 portional to the temperature gradient dT/dx.The proportionality constant is called the thermal conductivity K(or A).We thus write: J0=-K dx (14.11) The negative sign indicates that the heat flows from the hot to the cold end (Fourier Law,1822).Possible units for the heat con- ductivity are J/(m.s-K)or W/(m.K).The heat flux Jo is measured in J/(m2.s).Table 14.3 gives some characteristic values for K. The thermal conductivity decreases slightly with increasing tem- perature.For example,K for copper or Al2O3 decreases by about 20%within a temperature span of 1000C.In the same temper- ature region,K for iron decreases by 10%. For the interpretation of thermal conduction we postulate that the heat transfer in solids may be provided by free electrons as TABLE 14.3.Thermal conductivities at room temperature W Substance ·K ms-K Diamond Type IIa 2.3×103 SiC 4.9×102 Silver 4.29×102 Copper 4.01×102 Aluminum 2.37×102 Silicon 1.48×102 Brass (leaded) 1.2×102 Iron 8.02×101 GaAs 5×10 Ni-Silver 2.3×101 Al2O;(sintered) 3.5×101 SiO2 (fused silica) 1.4 Concrete 9.3×10-1 Soda-lime glass 9.5×10-1 Water 6.3×10-1 Polyethylene 3.8×10-1 Teflon 2.25×10-1 Snow(0°C) 1.6 ×10-1 Wood (oak) 1.6×10-1 Sulfur 2.0×10-2 Cork ×10-2 Glass wool 5 ×10-3 Air 2.3×10-4 a See also Figure 14.1.Source:Handbook of Chemistry and Physics,CRC Press,Boca Raton,FL (1994). b62%Cu,15%Ni,22%Zn
portional to the temperature gradient dT/dx. The proportionality constant is called the thermal conductivity K (or ). We thus write: JQ K d d T x . (14.11) The negative sign indicates that the heat flows from the hot to the cold end (Fourier Law, 1822). Possible units for the heat conductivity are J/(msK) or W/(m K). The heat flux JQ is measured in J/(m2 s). Table 14.3 gives some characteristic values for K. The thermal conductivity decreases slightly with increasing temperature. For example, K for copper or Al2O3 decreases by about 20% within a temperature span of 1000°C. In the same temperature region, K for iron decreases by 10%. For the interpretation of thermal conduction we postulate that the heat transfer in solids may be provided by free electrons as 14.3 • Thermal Conduction 279 TABLE 14.3. Thermal conductivities at room temperaturea Substance K m W K m J sK Diamond Type IIa 2.3 103 SiC 4.9 102 Silver 4.29 102 Copper 4.01 102 Aluminum 2.37 102 Silicon 1.48 102 Brass (leaded) 1.2 102 Iron 8.02 101 GaAs 5 101 Ni-Silverb 2.3 101 Al2O3 (sintered) 3.5 101 SiO2 (fused silica) 1.4 Concrete 9.3 101 Soda-lime glass 9.5 101 Water 6.3 101 Polyethylene 3.8 101 Teflon 2.25 101 Snow (0°C) 1.6 101 Wood (oak) 1.6 101 Sulfur 2.0 102 Cork 3 102 Glass wool 5 103 Air 2.3 104 a See also Figure 14.1. Source: Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL (1994). b 62% Cu, 15% Ni, 22% Zn.
280 14.Thermal Properties of Materials well as by phonons.We understand immediately that in insula- tors,which do not contain any free electrons,the heat must be conducted exclusively by phonons.In metals and alloys,on the other hand,the heat conduction is dominated by electrons be- cause of the large number of free electrons which they contain. Thus,the phonon contribution is usually neglected in this case. One particular point should be clarified.Electrons in metals travel in equal numbers from hot to cold and from cold to hot in order that the charge neutrality is maintained.This is indi- cated in Figure 14.5 by two arrows marked E and E2.Now,the electrons in the hot part of a metal possess and transfer a high energy.In contrast to this,the electrons in the cold end possess and transfer a lower energy.The heat transferred from hot to cold is thus proportional to the difference in the energies of the electrons. The situation is quite different in phonon conductors.We know from Section 14.1 that the number of phonons is larger at the hot end than at the cold end.Thermal equilibrium thus involves, in this case,a net transfer of phonons from the hot into the cold part of a material. Returning to the portion of thermal conduction caused by elec- trons,we consider a volume at the center of the bar depicted in Figure 14.5 whose faces have the size of a unit area and whose length is 2l,where l is the mean free path between two consec- utive collisions between an electron and lattice atoms.A simple energy balance,taking in account the electrons of energy EI that travel from left to right and electrons having a lower energy,E2, drifting from right to left,yields for the classical equation for the heat conductivity of metals and alloys: K=Nw kpl (14.12) 2 Equation (14.12)thus reveals that the heat conductivity is larger the more electrons,Nv,per unit volume are involved,the larger their velocity,v,and the larger the mean free path,l,between two consecutive electron-atom collisions.The connection be- tween thermal conductivity and the heat capacity per volume is: K=3 Cl v l. (14.13) All three variables contained in Eq.(14.13)are temperature- dependent,but while C increases with temperature (Figure 14.4),I and,to a small degree,also v are decreasing.Thus,K should change very little with temperature,which is indeed ex- perimentally observed.As mentioned above,the thermal con- ductivity decreases about 10-5 W/(m.K)per degree.K also
well as by phonons. We understand immediately that in insulators, which do not contain any free electrons, the heat must be conducted exclusively by phonons. In metals and alloys, on the other hand, the heat conduction is dominated by electrons because of the large number of free electrons which they contain. Thus, the phonon contribution is usually neglected in this case. One particular point should be clarified. Electrons in metals travel in equal numbers from hot to cold and from cold to hot in order that the charge neutrality is maintained. This is indicated in Figure 14.5 by two arrows marked E1 and E2. Now, the electrons in the hot part of a metal possess and transfer a high energy. In contrast to this, the electrons in the cold end possess and transfer a lower energy. The heat transferred from hot to cold is thus proportional to the difference in the energies of the electrons. The situation is quite different in phonon conductors. We know from Section 14.1 that the number of phonons is larger at the hot end than at the cold end. Thermal equilibrium thus involves, in this case, a net transfer of phonons from the hot into the cold part of a material. Returning to the portion of thermal conduction caused by electrons, we consider a volume at the center of the bar depicted in Figure 14.5 whose faces have the size of a unit area and whose length is 2l, where l is the mean free path between two consecutive collisions between an electron and lattice atoms. A simple energy balance, taking in account the electrons of energy E1 that travel from left to right and electrons having a lower energy, E2, drifting from right to left, yields for the classical equation for the heat conductivity of metals and alloys: K Nvv 2 kBl . (14.12) Equation (14.12) thus reveals that the heat conductivity is larger the more electrons, Nv, per unit volume are involved, the larger their velocity, v, and the larger the mean free path, l, between two consecutive electron–atom collisions. The connection between thermal conductivity and the heat capacity per volume is: K 1 3 Cv el v l. (14.13) All three variables contained in Eq. (14.13) are temperaturedependent, but while Cv el increases with temperature (Figure 14.4), l and, to a small degree, also v are decreasing. Thus, K should change very little with temperature, which is indeed experimentally observed. As mentioned above, the thermal conductivity decreases about 105 W/(m K) per degree. K also 280 14 • Thermal Properties of Materials
