8 Thermal Properties This chapter considers composites for thermal applications,including the basic principles related to the thermal behavior of materials.The applications include thermal conduction,heat dissipation,thermal insulation,heat retention,heat stor- age,rewritable optical discs,and shape-memory actuation. 8.1 Thermal Expansion Upon heating,the length of a solid typically increases.Upon cooling,its length typically decreases.This phenomenon is known as thermal expansion. The coefficient of thermal expansion(CTE,also abbreviated to a)is defined as a=(1/Lo)(△L/△T), (8.1) where Lo is the original length and L is the length at temperature T.In other words, a is the fractional change in length per unit change in temperature(i.e.,strain per unit change in temperature).Equation 8.1 can be rewritten as △L/L。=AT. (8.2) Thermal expansion is reversible,so thermal contraction occurs upon cooling,and the extent of thermal contraction is also governed by a. The change in dimensions that occurs upon changing the temperature may be undesirable when the dimensional change results in thermal stress;i.e.,internal stress associated with the fact that the dimensions are not the same as those the component should have when it is not constrained.An example concerns a joint between two components that exhibit different thermal expansion coefficients. The joint is created at a given temperature.Subsequent to the bonding,the joint is heated or cooled.Upon heating or cooling,the bonding constrains the dimensions of both components,thus causing both components to be unable to attain the dimensions that they would do if they were not bonded.When the thermal stress is too high,deformation such as warpage may occur.Debonding can even occur. If thermal cycling occurs,the thermal stress is cycled,thus resulting in thermal fatigue when the number of thermal cycles is sufficiently high. Thermal expansion can be advantageously used to obtain a tight fit between two components.For example,a pipe(preferably one with a rather high coefficient of 277
8 Thermal Properties This chapter considers composites for thermal applications, including the basic principles related to the thermal behavior of materials. The applications include thermal conduction, heat dissipation, thermal insulation, heat retention, heat storage, rewritable optical discs, and shape-memory actuation. 8.1 Thermal Expansion Upon heating, the length of a solid typically increases. Upon cooling, its length typically decreases. This phenomenon is known as thermal expansion. The coefficient of thermal expansion (CTE, also abbreviated to α) is defined as α = (1/Lo)(ΔL/ΔT) , (8.1) where Lo is the original length and L is the length at temperature T. In other words, α is the fractional change in length per unit change in temperature (i.e., strain per unit change in temperature). Equation 8.1 can be rewritten as ΔL/Lo = αΔT . (8.2) Thermal expansion is reversible, so thermal contraction occurs upon cooling, and the extent of thermal contraction is also governed by α. The change in dimensions that occurs upon changing the temperature may be undesirable when the dimensional change results in thermal stress; i.e., internal stress associated with the fact that the dimensions are not the same as those the component should have when it is not constrained. An example concerns a joint between two components that exhibit different thermal expansion coefficients. The joint is created at a given temperature. Subsequent to the bonding, the joint is heated or cooled. Upon heating or cooling, the bonding constrains the dimensions of both components, thus causing both components to be unable to attain the dimensions that they would do if they were not bonded. When the thermal stress is too high, deformation such as warpage may occur. Debonding can even occur. If thermal cycling occurs, the thermal stress is cycled, thus resulting in thermal fatigue when the number of thermal cycles is sufficiently high. Thermal expansion can be advantageously used to obtain a tight fit between two components. For example, a pipe (preferably one with a rather high coefficient of 277
278 8 Thermal Properties thermal expansion)is cooled and then,in the cold state,inserted into a larger pipe. Upon subsequent warming of the inner pipe,thermal expansion causes a tight fit between the two pipes. The energy associated with a bond depends on the bond distance(i.e.,the bond length),as shown in Fig.8.1a.The energy is lowest at the equilibrium bond dis- tance.The curve of energy vs.bond distance takes the shape of a trough that is asymmetric.The higher the temperature,the higher is the energy,and the aver- age bond distance(the midpoint of the horizontal line cutting across the energy trough)increases.Thermal expansion stems from the increasing amplitude of thermal vibrations with increasing temperature and the greater ease of outward vibration (bond lengthening)than inward vibration(bond shortening),reflecting the asymmetry in the energy versus bond length curve.This asymmetry causes the distance QR to be greater than the distance QP in Fig.8.1b,so that the outward vi- bration travels a greater distance than the inward vibration at a given temperature. The depth of the energy trough below the energy of zero is the bond energy.At the energy minimum,the bond length is the equilibrium value.A weaker bond has a lower bond energy and corresponds to greater asymmetry in the energy trough. Greater asymmetry means a higher value of the CTE.Hence,weaker bonding tends to give a higher CTE. Table 8.1 shows that the CTE tends to be high for polymers,medium for metals, and low for ceramics.This reflects the weak intermolecular bonding in polymers, Average bond distance Bond distance Equilibrium bond length Energy →Bond length %0 Weak bonding Bond Strong bonding P R energy ← b Inward Outward Figure 8.1.Dependence of the energy between two atoms on the distance between the atoms.a The average bond distance,which occurs at the midpoint of the horizontal line across the energy trough at a given energy,increases with increasing energy.b Comparison of a solid with strong bonding and one with weak bonding.The equilibrium bond length occurs at the minimum energy
278 8 Thermal Properties thermal expansion) is cooled and then, in the cold state, inserted into a larger pipe. Upon subsequent warming of the inner pipe, thermal expansion causes a tight fit between the two pipes. The energy associated with a bond depends on the bond distance (i.e., the bond length), as shown in Fig. 8.1a. The energy is lowest at the equilibrium bond distance. The curve of energy vs. bond distance takes the shape of a trough that is asymmetric. The higher the temperature, the higher is the energy, and the average bond distance (the midpoint of the horizontal line cutting across the energy trough) increases. Thermal expansion stems from the increasing amplitude of thermal vibrations with increasing temperature and the greater ease of outward vibration (bond lengthening) than inward vibration (bond shortening), reflecting the asymmetry in the energy versus bond length curve. This asymmetry causes the distance QR to be greater than the distance QP in Fig. 8.1b, so that the outward vibration travels a greater distance than the inward vibration at a given temperature. The depth of the energy trough below the energy of zero is the bond energy. At the energy minimum, the bond length is the equilibrium value. A weaker bond has a lower bond energy and corresponds to greater asymmetry in the energy trough. Greater asymmetry means a higher value of the CTE. Hence, weaker bonding tends to give a higher CTE. Table 8.1 shows that the CTE tends to be high for polymers, medium for metals, and low for ceramics. This reflects the weak intermolecular bonding in polymers, a b O Energy Bond energy Inward Bond length Outward Equilibrium bond length P Q R Weak bonding Strong bonding Energy Bond distance Average bond distance Figure 8.1. Dependence of the energy between two atoms on the distance between the atoms. a The average bond distance, which occurs at the midpoint of the horizontal line across the energy trough at a given energy, increases with increasing energy. b Comparison of a solid with strong bonding and one with weak bonding. The equilibrium bond length occurs at the minimum energy
8.1 Thermal Expansion 279 Table 8.1.Coefficients of thermal expansion(CTEs)of various materials at 20C Material CTE(10-6/K Rubber 77 Polyvinyl chloride* 52 Leadb 29 Magnesiumb 26 Aluminum 23 Brassb 19 Silverb 18 Stainless steelb 17.3 Copperb 17 Goldb 14 Nickelb 13 Steelb 11.0-13.0 Ironb 11.1 Carbon steelb 10.8 Platinum 9 Tungstenb 4.5 Invar (Fe-Ni36)b 1.2 Concrete 12 Glass 8.5 Gallium arsenide 5.8 Indium phosphide 4.6 Glass,borosilicate 3.3 Quartz(fused) 0.59 Silicon 3 Diamond a Polymer;b metal;ceramic the moderately strong metallic bonding in metals,and the strong ionic/covalent bonding in ceramics.Silicon and diamond are not ceramics,but their CTE values are low because they are covalent network solids.Because stronger bonding tends to give a higher melting temperature,a lower CTE tends to correlate with a higher melting temperature.This is why tungsten,with a very high melting temperature of 3,410C,has a low CTE of 4.5 x 10-6/K.In contrast,magnesium,with a low melting temperature of 660C,has a high CTE of 23.Invar(Fe-Ni36)has an excep- tionally low CTE among metals because of its magnetic character(due to the iron part of the alloy)and the effect of the magnetic moment on the volume.Among ceramics,quartz has a particularly low CTE due to its three-dimensional network of covalent/ionic bonding.Glass has a higher CTE than quartz due to its lower degree of networking. The CTE of a composite can be calculated from those of its components.When the components are placed in series,as illustrated in Fig.8.2a,the change in length ALc of the composite is given by the sum of the changes in the lengths of the components: △Lc=△L1+△L2, (8.3)
8.1 Thermal Expansion 279 Table 8.1. Coefficients of thermal expansion (CTEs) of various materials at 20°C Material CTE (10−6/K) Rubbera 77 Polyvinyl chloridea 52 Leadb 29 Magnesiumb 26 Aluminumb 23 Brassb 19 Silverb 18 Stainless steelb 17.3 Copperb 17 Goldb 14 Nickelb 13 Steelb 11.0–13.0 Ironb 11.1 Carbon steelb 10.8 Platinumb 9 Tungstenb 4.5 Invar (Fe-Ni36)b 1.2 Concretec 12 Glassc 8.5 Gallium arsenidec 5.8 Indium phosphidec 4.6 Glass, borosilicatec 3.3 Quartz (fused)c 0.59 Silicon 3 Diamond 1 a Polymer; b metal; c ceramic the moderately strong metallic bonding in metals, and the strong ionic/covalent bonding in ceramics. Silicon and diamond are not ceramics, but their CTE values are low because they are covalent network solids. Because stronger bonding tends to give a higher melting temperature, a lower CTE tends to correlate with a higher melting temperature. This is why tungsten, with a very high melting temperature of 3,410°C, has a low CTE of 4.5 × 10−6/K. In contrast, magnesium, with a low melting temperature of 660°C, has a high CTE of 23. Invar (Fe-Ni36) has an exceptionally low CTE among metals because of its magnetic character (due to the iron part of the alloy) and the effect of the magnetic moment on the volume. Among ceramics, quartz has a particularly low CTE due to its three-dimensional network of covalent/ionic bonding. Glass has a higher CTE than quartz due to its lower degree of networking. The CTE of a composite can be calculated from those of its components. When the components are placed in series, as illustrated in Fig. 8.2a, the change in length ΔLc of the composite is given by the sum of the changes in the lengths of the components: ΔLc = ΔL1 + ΔL2 , (8.3)
280 8 Thermal Properties where ALI and AL2 are the changes in the lengths of the components.Only two components are shown in the summation in Eq.8.3 for the sake of simplicity. Dividing by the original length Lco of the composite gives △Lc/Lco=△L1/Lco+△L2/Lco=h△L1/L1o+V2△L2/L2o, (8.4) where Lio and L2o are,respectively,the original lengths of component 1(all of the strips of component I together)and component 2(all the strips of component 2 to- gether),and v and v2 are the volume fractions of components I and 2,respectively. In Eq.8.4,the relations L1o VILco (8.5) and L20 V2Lco (8.6) have been used.Using Eq.8.2,Eq.8.4 becomes cc△T=V1x1△T+y2a2△T, (8.7) where ac is the CTE of the composite and a and az are the CTEs of components 1 and 2,respectively.Division by AT gives ac via1+v2a2. (8.8) Equation 8.8 is the rule of mixtures expression for the CTE of the composite in the case of the series configuration shown in Fig.8.2a. For the parallel configuration of Fig.8.2b,when the component strips are per- fectly bonded to one another,the two components are constrained so that their lengths are the same at any temperature.This constraint causes each component 2 2 2 2 b Figure 8.2.Calculation ofthe CTE ofa composite with two components,labeled 1 and 2.a Series configuration,b parallel configuration
280 8 Thermal Properties where ΔL1 and ΔL2 are the changes in the lengths of the components. Only two components are shown in the summation in Eq. 8.3 for the sake of simplicity. Dividing by the original length Lco of the composite gives ΔLc/Lco = ΔL1/Lco + ΔL2/Lco = v1ΔL1/L1o + v2ΔL2/L2o , (8.4) where L1o and L2o are, respectively, the original lengths of component 1 (all of the strips of component 1 together) and component 2 (all the strips of component 2 together), and v1 and v2 are the volume fractions of components 1 and 2, respectively. In Eq. 8.4, the relations L1o = v1Lco , (8.5) and L2o = v2Lco (8.6) have been used. Using Eq. 8.2, Eq. 8.4 becomes αcΔT = v1α1ΔT + v2α2ΔT , (8.7) where αc is the CTE of the composite and α1 and α2 are the CTEs of components 1 and 2, respectively. Division by ΔT gives αc = v1α1 + v2α2 . (8.8) Equation 8.8 is the rule of mixtures expression for the CTE of the composite in the case of the series configuration shown in Fig. 8.2a. For the parallel configuration of Fig. 8.2b, when the component strips are perfectly bonded to one another, the two components are constrained so that their lengths are the same at any temperature. This constraint causes each component Figure 8.2. Calculation of the CTE of a composite with two components, labeled 1 and 2. a Series configuration, b parallel configuration
8.1 Thermal Expansion 281 to be unable to change its length by the amount dictated by its CTE(Eq.8.2). As a result,each component experiences a thermal stress.The component that expands more than the amount indicated by its CTE experiences thermal stress that is tensile,whereas the component that expands less than the amount indicated by its CTE experiences thermal stress that is compressive.The thermal stress is equal to the thermal force divided by the cross-sectional area.In the absence of an applied force, F1+F2=0, (8.9) where F and Fare the thermal forces in component I(all of the strips of com- ponent I together)and component 2(all of the strips of component 2 together), respectively.Hence, F1=-F2. (8.10) Since force is the product of stress and cross-sectional area,Eq.8.10 can be written U1VIA U2V2A, (8.11) where U and U2 are the stresses in components 1 and 2,respectively,and A is the area of the overall composite.Dividing by A gives U1=U22. (8.12) Using Eq.8.2,the strain in component 1 is given by (ac-)AT and the strain in component 2 is given by(ac-a2)AT.Since stress is the product of the strain and the modulus,Eq.7.12 becomes (ae-a1)△TM1h=-(ae-a2)△TM22, (8.13) where M and M2 are the elastic moduli of components 1 and 2,respectively. Dividing Eq.8.13 by AT gives (ac-a1)M1v1 =-(ac-a2)M2v2. (8.14) Rearrangement gives ac=(a1M1V1+c2M2V2)/(M11+M2V2). (8.15) Equation 8.15 is the rule of mixtures expression for the CTE of a composite in the parallel configuration. In the case that M M2,Eq.8.15 becomes xc=(1M+c22)/(v1+v2)=a1+a22, (8.16) since 1+V2=1. (8.17)
8.1 Thermal Expansion 281 to be unable to change its length by the amount dictated by its CTE (Eq. 8.2). As a result, each component experiences a thermal stress. The component that expands more than the amount indicated by its CTE experiences thermal stress that is tensile, whereas the component that expands less than the amount indicated by its CTE experiences thermal stress that is compressive. The thermal stress is equal to the thermal force divided by the cross-sectional area. In the absence of an applied force, F1 + F2 = 0 , (8.9) where F1 and F2are the thermal forces in component 1 (all of the strips of component 1 together) and component 2 (all of the strips of component 2 together), respectively. Hence, F1 = −F2 . (8.10) Since force is the product of stress and cross-sectional area, Eq. 8.10 can be written as U1v1A = U2v2A , (8.11) where U1 and U2 are the stresses in components 1 and 2, respectively, and A is the area of the overall composite. Dividing by A gives U1v1 = U2v2 . (8.12) Using Eq. 8.2, the strain in component 1 is given by (αc − α1)ΔT and the strain in component 2 is given by (αc − α2)ΔT. Since stress is the product of the strain and the modulus, Eq. 7.12 becomes (αc − α1)ΔTM1v1 = −(αc − α2)ΔTM2v2 , (8.13) where M1 and M2 are the elastic moduli of components 1 and 2, respectively. Dividing Eq. 8.13 by ΔT gives (αc − α1)M1v1 = −(αc − α2)M2v2 . (8.14) Rearrangement gives αc = (α1M1v1 + α2M2v2)/(M1v1 + M2v2) . (8.15) Equation 8.15 is the rule of mixtures expression for the CTE of a composite in the parallel configuration. In the case that M1 = M2, Eq. 8.15 becomes αc = (α1v1 + α2v2)/(v1 + v2) = α1v1 + α2v2 , (8.16) since v1 + v2 = 1 . (8.17)