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Theoretical analyses of thermal shock and thermal expansion coefficients of metals and ceramics
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CERAMICS INTERNATIONAL
Ceramics International 41 (2015) 1145–1153 www.elsevier.com/locate/ceramint
Theoretical analyses of thermal shock and thermal expansion coefficients of metals and ceramics Yoshihiro Hiratan Department of Chemistry, Biotechnology, and Chemical Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima 890-0065, Japan Received 25 July 2014; accepted 9 September 2014 Available online 28 September 2014
Abstract The theoretical expression for the critical flux (If) of energy input to fracture a solid material was derived using available material parameters. The If value becomes larger in the material with larger κ (thermal conductivity) and KIC (fracture toughness) values and smaller β (thermal expansion coefficient), E (Young's modulus), a (size of fracture origin) and x (thickness of material). The following useful relationship was also derived for β, E and Cp (heat capacity at a constant pressure): β¼ Cp/3E. This relation agreed with the experimentally measured β value in the material with a low Debye temperature and a small temperature dependence of Young's modulus at a high temperature. The theoretical equation representing the temperature dependence of β value was derived. This equation gives the interpretation that the gradual increase of the measured β value is due to the decrease of Young's modulus at a high temperature. Further discussion provided a more accurate equation which does not contain a Young's modulus and is composed of Cv (heat capacity at a constant volume), Cp, ν (Poisson's ratio) and temperature. A very good agreement was observed between the measured and calculated β values of some metals. It was also found that the measured β of alumina ceramics was very close to the average value calculated using Young's modulus in each direction of single crystal Al2O3. & 2014 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Keywords: C. Thermal expansion; C. Thermal properties; C. Thermal shock resistance
1. Introduction A thermal expansion coefficient (TEC) of metal or ceramics is measured as a function of heating temperature and the average value in a wide temperature range is used in the design of materials application. The metal or ceramics with a remarkably low TEC is attractive as a high temperature structural material because of its high thermal shock resistance [1–5]. Fortunately many measured data of TECs of metals and ceramics are available on a metal handbook [6] or chemical handbook [7]. These data are used in the calculation of thermal shock resistance. However, few theoretical analyses have been proposed to predict the magnitude of TEC [8]. This paper derived a useful relationship among a flux of energy input to a solid material (I), an induced stress (σ), a TEC (β), a Young's modulus (E) and a thermal conductivity (κ), and succeeded finally in relating a TEC to the heat capacity at a n
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http://dx.doi.org/10.1016/j.ceramint.2014.09.042 0272-8842/& 2014 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
constant pressure (Cp) and a Young's modulus. The calculated TEC was compared to the measured TEC and the difference between two TEC values was discussed. The above discussion is useful to understand the essence of thermal expansion coefficient. 2. Thermal shock resistance A linear TEC (β) of a material with length L, which is heated uniformly at a temperature T, is defined by Eq. (1) 1 ∂L ∂ε ∂L β¼ ¼ ∂ε ð1Þ L ∂T P ∂T P L where ε is the strain [9]. On the other hand, a Young's modulus represents the slope of σ (applied stress) – ε relation by Eq. (2). ∂σ ð2Þ E¼ ∂ε P Furthermore, a flux of energy input to a material with thickness x (Fig. 1) is related to the product of a thermal conductivity (κ)
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The dimension, J/m3 K, expresses the unit of heat capacity per unit volume at a constant pressure (Cp1). Since the βE value corresponds to the characteristic of one direction of a material, βE is related to Cp1 by Eq. (9) βΕ ¼ 13C p1
ð9Þ
Eq. (9) indicates that the β value becomes smaller in a material with a smaller Cp1 and a larger E. On the other hand, κ (J/s mK) is related to α (thermal diffusibility, m2/s) and Cv1 (heat capacity per unit volume at a constant volume, J/m3 K) by Eq. (10) [10]. κ ¼ 13C v1 α Fig. 1. Temperature gradient in a solid material with thickness x at a given flux of energy (I).
and a temperature gradient (∂T/∂x)P by Eq. (3). ∂T I ¼ κ ∂x P
ð3Þ
The product of Eqs. (1)–(3) provides an interesting relationship of Eq. (4). ∂σ βE ¼ I ð4Þ ∂x P κ P Eq. (4) expresses the stress gradient induced in the material at a given temperature gradient. When β, E and κ are treated as constant values, the integration of Eq. (4) at a constant pressure leads to the relationship of Eq. (5). βE σ¼ Ix ð5Þ κ The induced σ becomes smaller at smaller β, E, I and x values and at a larger κ value. When the material is fractured at a given temperature gradient, the fracture mechanics is applied to Eq. (5), resulting in Eq. (6), βE K IC σ f ¼ − I f x ¼ pffiffiffi ð6Þ κ Y a where KIC is the fracture toughness, a the size of fracture origin and Y the shape factor of fracture origin. That is, the critical flux of energy causing the fracture is given by Eq. (7). κ K IC pffiffiffi ð7Þ If ¼ βE Y a x The If value becomes larger at larger κ and KIC values and at smaller β, Ε, a and x values. Eq. (7) is useful to estimate the thermal shock resistance of metals or ceramics. This equation will be compared with the experimentally measured If value in future. 3. Thermal expansion coefficient In Eqs. (6) or (7), the βE/κ value at a constant pressure may be treated as a material constant. The dimension of the unit of βE value is as follows: m N J ð8Þ ðβΕ Þ ¼ ¼ 3 mK m2 m K
ð10Þ
The combination of Eqs. (9) and (10) leads to Eq. (11) βΕ C p1 1 ¼ ð11Þ κ C v1 α The substitution of Eq. (11) for Eq. (5) results in Eq. (12) Cp1 Ix σ¼ ð12Þ Cv1 α When an energy is input along one direction to a solid material, the induced stress becomes smaller in a thinner material (x) with a larger α value. The Cp1/Cv1 ratio is close to unity at a low temperature and gradually increases above unity at a higher temperature as discussed in Section 4.2. Furthermore, the β value in Eq. (9) is expressed by the heat capacity at a constant pressure (Cp2) of the dimension, J/mol K, density (ρ, g/m3) and molecular weight (M, g/mol) of a material by Eq. (13). C p1 Cp2 ρ ð13Þ ¼ β¼ 3EM 3E That is, the β value is related to the measurable parameters of a material. The β value calculated by Eq. (13) is compared with the measured β value in the next section to discuss the usefulness of Eq. (13). 4. Comparison of calculated and measured TECs at a low temperature 4.1. Calculated results The Cp2, ρ and E of metals and ceramics at 293–298 K are listed on a chemical handbook [7,11] and a metal handbook [6]. Those data at room temperature were used in Eq. (13) to calculate the β value. On the other hand, the β values measured in a temperature range of 273–373 K are also presented in the handbooks [6,7]. Fig. 2 shows the relationship between the calculated and measured β values for 31 metals and 10 ceramics. In Fig. 2, the straight lines with the slopes of β(measured)/β(calculated) ¼ 1, 1.5, 2.0 and 3.0 are presented to evaluate the correspondence of both the β values. At first, the magnitude of the calculated β values was the same order as the measured β values. As seen in Fig. 2, the calculated β values in a wide range of C (diamond), TiN, mullite, Zr, Nb, V, BaTiO3, Bi, Pb and In were very close to the measured β values. The other data are distributed in the range of β(mea.)/β(cal.) ¼ 1.5– 3.0. That is, the calculated β values showed a tendency to become smaller than the measured β values. This result is
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discussed based on the temperature difference in the measurement and calculation of β values.
EðTÞ o1 EðT 0 Þ
4.2. Discussion
Since Young's modulus of metal or ceramics decreases at a higher temperature [14], the relation of Eq. (17) is derived. As a result, the β(cal.T0)/β(mea.T) ratio of Eq. (14) becomes smaller than unity.
Since the β value depends on the heating temperature [8,12,13], the β(cal.T0)/β(mea.T) ratio is presented by Eq. (14), βðcal:T 0 Þ Cp2 ðT 0 ÞρðT 0 Þ 3EðTÞ C p2 ðT 0 Þ EðTÞ ¼ ffi βðmea:TÞ 3EðT 0 Þ Cp2 ðTÞρðT Þ C p2 ðTÞ EðT 0 Þ
ð14Þ
The T0 and T in Eq. (14) are the temperatures where the β values are calculated and measured, respectively. The ρ(T0)/ρ(T) ratio is close to unity because of the small values of β(cal.) and β(mea.) (o50 10 6 K 1). According to Debye's model [8,14], Cv2 in the dimension of J/mol K at a constant volume of solid approaches a constant value 3R (=24.94 J/mol K, R: gas constant) at T»θ (Debye temperature). The Cp2 (J/mol K)–Cv2 (J/mol K) relation is expressed by Eq. (15) for an isotropic body, ð3βÞ VT κT 2
C p2 ¼ C v2 þ
ð15Þ
where V is the molar volume of solid and κT is the isothermal compressibility (κT ¼ ( 1/V)(∂V/∂σ)T, where σ is the stress applied to the material). Since the Cv2 value is larger than the (3β)2VT/κT value, the Cp2 value increases gradually along the Cv2 value with increasing heating temperature. That is, the following relations are derived for Eq. (14): Case 1. T0 (measurement) C p2 ðT 0 Þ o1 C p2 ðTÞ
(¼ 298 K) o θ
(Debye
temperature) o T ð16Þ
Fig. 2. Relationship between the measured TEC and the calculated TEC of metals and ceramics at 293–298 K.
ð17Þ
Case 2. θ (Debye temperature)oT0 (¼ 298 K)oT (measurement) C p2 ðT 0 Þ ~1 C p2 ðTÞ
ð18Þ
EðTÞ o1 EðT 0 Þ
ð19Þ
In this case, the Cp2(T0)/Cp2(T) ratio approaches unity but the E(T)/E(T0) ratio is smaller than unity. The β0(cal.T0)/β(mea.T) ratio reflects mainly the ratio of Young's moduli at two temperatures (T0 and T). Fig. 3 shows the β(cal.T0)/β(mea.T) ratio against the T0 (¼ 298 K)/θ ratio. The Debye temperature (θ) is also listed on a metal handbook [6]. The data in Fig. 3 are grouped into 5 classes according to the T0 (¼ 298 K)/Tm (melting point) ratio of metals and ceramics. As seen in Fig. 3, β(cal.T0)/β(mea.T) ratio of each group increases at a higher T0/θ ratio. Although it is difficult to specify the heating temperature for the measured TEC values, the tendency in Fig. 3 is explained by the relations of Cases 1 and 2. The regions of (T0/θ)o1 and (T0/θ)41 correspond to Cases 1 and 2, respectively. The calculated TEC value approaches the measured TEC value in the metal or ceramics with a lower Debye temperature as compared with T0 (=298 K). This tendency in each class of a similar T0/Tm ratio reflects mainly the change of Cp2(T0)/Cp2(T) ratio, because the E (T)/ E(T0) ratios as a function of T0/Tm ratio may be a similar value in each group. The above tendency also depends on the
Fig. 3. Ratio between the measured and the calculated TECs as a function of T0 (¼298 K)/θ (Debye temperature) ratio. The Tm in Fig. 3 means the melting point of metal or ceramics.
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T0/Tm ratio at a similar T0/θ ratio. In the material with a high melting point, the calculated TEC approaches the measured TEC. This observation reflects the influence of E(T)/E(T0) ratio in Case 1 or 2. The decrease of Young's modulus at a higher temperature becomes smaller in the metal or ceramics with a high melting point. Therefore, the E(T)/E(T0) ratio approaches unity in the material with a high melting point. As discussed above, the key factors affecting the β(cal.T0)/β(mea.T) ratio are the Debye temperature and the melting point of metal or ceramics. In the material with a low Debye temperature and a high melting point, the calculated CTE approaches the measured CTE.
small 1000 K/Tm ratio. That is, the correspondence between β (cal.)/β(mea.) values becomes higher in the material with a small temperature dependence of Young's modulus at a high temperature. The some data of β(cal.)/β(mea.) ratio 4 1 may be due to the relation of θ o T(mea.) o 1000 K (Cp2(1000 K)/ Cp2(T)4 1 in Eq. (20)).
5. Comparison of calculated and measured TECs at high temperatures In Case 2 in Section 4, the β(cal.T0)/β(mea.T) ratio reflects mainly the E(T)/E(T0) ratio. However, it is difficult to measure the accurate Young's modulus at a high temperature. Most of the reported Young's moduli of metals and ceramics are measured at room temperature. The following relation was compared because of the difficulty of the measurement of the parameters in Eq. (13) at high temperatures. In Eq. (20), E (298 K) was used instead of E (1000 K). βðcal:1000 KÞ Cp2 ð1000 KÞ ρð1000 KÞ 3E ðTÞ ¼ βðmea:TÞ 3Eð298 KÞ C p2 ðTÞ ρðTÞ Cp2 ð1000 KÞ ΕðTÞ ffi C p2 ðTÞ Εð298 KÞ
ð20Þ
In the condition of θ o 1000 K o T(mea.) or θ o T(mea.) o 1000 K, the Cp2(1000 K)/Cp2(T) ratio is close to unity (see Section 6). That is, the β(cal.1000 K)/β(mea.T) ratio mainly reflects the ratio of Young's moduli (E(T)/E(298 K) o 1) in Eq. (20). Fig. 4 shows the β(mea.T K) – β(cal.1000 K) relation of metals and ceramics. Most of the data are seen in the slope range of β(mea.T K)/β(cal.1000 K) ¼ 1–3. This tendency was similar to the plots in Fig. 2. The data in Fig. 4 are grouped into 4 classes by the nature of chemical bonding as defined in our previous paper [15]. AD-1: weak chemical bond (mixing of metal bond and covalent bond, Au, Pb, Cu, In, Ga, Be), AD-2: intermediate chemical bond (mixing of metal bond and ionic bond, Ag, Bi, Cd, Zn, Sb, Zr), AD-3: strong chemical bond (mixing of metal bond, covalent bond and ionic bond, Pt, W, Ta, Hf, Pd, Mo, Ni, Co, Nb, Fe, Cr, V, Ti), and ceramics (mixing of covalent bond and ionic bond, Al2O3, TiO2, MgO, C(diamond), SiC). Fig. 5 shows the β(cal.1000 K)/β(mea.T) ratio of each group as a function of the 1000 K/Tm ratio. The calculated TEC approaches the measured TEC in the materials with a high melting point. In other words, the plots in Fig. 5 suggest that the E(T)/E(298 K) ratio in Eq. (20) decreases drastically with the decrease in melting point. The β(cal.1000 K)/β(mea.T) ratio shifted to a larger ratio of 1000 K/Tm in the order of ceramics o AD-3o AD-1, AD-2. When the material has a stronger chemical bond, the melting point becomes higher. As a result, the calculated TEC approaches the measured TEC at a
Fig. 4. Relationship between the measured TEC and the TEC calculated using a heat capacity in atmospheric pressure at 1000 K.
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fig. 5. Ratio between the measured TEC and the TEC calculated using a heat capacity at 1000 K as a function of 1000 K/Tm (melting point) ratio of metals and ceramics. The data in Fig. 5 are grouped into 4 classes (AD-1, 2, 3 and ceramics) by the nature of chemical bonding. See text and Ref. [15] for the detailed explanation.
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6. Temperature dependence of thermal expansion coefficient
The approximation of 1/κT C E and the relationship between Cp1 and Cp2 in Eq. (13) are coupled with Eq. (15) to give Eq. (24) because of ρV/M ¼ 1.
6.1. Derivation of basic equation of TEC
Cp1 ¼ C v1 þ ð3βÞ2 ET
In Figs. 2–5, we could see the influence of θ, E, Tm and the strength of chemical bond on the TECs of metals and ceramics at 298 K and 1000 K. In this section, we discuss the more general expression of TEC as a function of heating temperature. The Cp2–Cv2 relation for an isotropic body is given by Eq. (15). The isothermal compressibility of the material in a state of hydrostatic compression or tension (κT, in Eq. (15)) is related to the bulk modulus K by Eq. (21), 1 ∂V ∂εV 1 ð21Þ κT ¼ ¼ ¼ V ∂σ T Κ ∂σ T
Since 3β is equal to Cp1/E in Eq. (13), Eq. (25) is derived.
where εv is the strain of bulk. The bulk modulus (Κ) is related to Young's modulus (E) and Poisson's ratio (ν) by Eq. (22) [16]. E Κ¼ 3ð1 2νÞ
ð22Þ
Therefore, the combination of Eqs. (21) and (22) leads to the relationship of Eq. (23). 1 E ¼ K ¼ kT 31 2ν
ð23Þ
Since Poisson's ratio of metals at room temperature is around 1/3 [16], the bulk modulus is almost equal to Young's modulus. Fig. 6 compares the measured E values and the K values at σ ¼ 0.98 GPa and at room temperature for 27 materials. The straight line corresponds to the slope 1 of the E/K ratio. It is found that the data are scattered around the straight line, and this result basically supports the theoretical relationship of 1/κT C E in Eq. (23). 500
Cp1 ¼
ð24Þ
Cv1 13 β T
ð25Þ
The Cp1 value depends on the β value and heating temperature, and increases at a higher temperature. The condition of β¼ 0 K 1 leads to the relation Cp1 ¼ Cv1. The temperature dependence of Cv1( ¼ Cv2ρ/M) at a constant volume is analyzed theoretically in the Einstein and Debye equations [8,14], as discussed in Section 4.2. The relation by Eq. (25) is useful to understand more the meaning of Eq. (12). The substitution of Eq. (25) for Eq. (12) results in Eq. (26). σ¼
Ix ð1 3 β TÞα
ð26Þ
When a flux of energy is input to the surface of the material, the (1–3βT) value is calculated to be 0.885–0.996 under the conditions of β¼ 1 10 6 30 10 6 K 1 at T (surface temperature)¼ 1237 K. That is, the (1–3βT) value is around unity and the induced stress depends mainly on the α, x and I values. Fig. 7 shows the following normalized relation of Eq. (25): Cp2/Cv2 ratio – 1/(1–3β(mea.)T) plot at 1000 K. As a Cv2 value, 3R value (24.94 J/mol K) is used because the Debye temperatures of many metals are lower than 1000 K. That is, the Cv2 values of many metals approach 3R at a high temperature. In Fig. 7, the Cp2 per 1 mol-atom of ceramics is also presented. The some straight lines with different slopes are presented to evaluate the plotted data. The data were also grouped into 4 classes as shown in Fig. 5, depending on the nature of 1.60
400
Young's modulus, E (GPa)
1.40
300 1.20
200 1.00
100
0.80
0
0
100
200
300
400
500
1.00
1.02
1.04
1.06
1.08
1.10
Bulk modulus, K (GPa) Fig. 6. Comparison of the measured Young's modulus and the bulk modulus determined from the isothermal compressibility (κT) of metals and ceramics.
Fig. 7. Heat capacity (Cp2) normalized by 3R (24.94 J/mol K) as a function of 1/(1–3β(mea.)T) at 1000 K in Eq. (25).
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chemical bonding. In each group, the Cp2/Cv2 ratio increases as β(mea.) value increases. More discussion is described in the next section. At a similar β value, the Cp2/Cv2 ratio becomes larger in the order of ceramics o AD-1, AD-2 o AD-3. The Cp2/Cv2 ratio for ceramics is near the line with a slope of unity. However, the Cp2/Cv2 ratio for metals is distributed around the lines with the slopes of 1.1–1.4. This result may be explained by the influence of ν in Eq. (22). In Eq. (25), the ν value is approximated to be 1/3. 6.2. Reliability of derived TEC equation The coupling of Eqs. (13), (15) and (23) gives the following accurate expression of β value (Eq. (27)): β¼
β0 1 hβ T
ð27Þ
where β0 is equal to Cv1/3E and h is equal to 1/(1 2ν). The solution of β for Eq. (27) is given by Eq. (28), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 4hβ0 T ð28Þ β¼ 2hT Eq. (28) is useful to predict the temperature dependence of TEC under the condition of constant values of β0 and ν. Since the Cv2 value (J/mol K) of a material is close to 3R (24.94 J/mol K) at a higher temperature, the Cv1 (¼ Cv2ρ/M, J/m3 K) is allowed to be treated as a constant value. That is, the β0 value is calculated to be 3Rρ/3ME for a constant E value with no temperature dependence. Fig. 8(a) shows the β value calculated by Eq. (28) for Ce0.8Sm0.2O1.9 as a function of heating temperature. The following values were used to determine the β0 value: Cv2 ¼ 24.94 (J/mol-atom K) 2.9(mol-atom/Ce0.8Sm0.2O1.9 composition)¼ 72.33 (J/mol – Ce0.8Sm0.2O1.9 K), ρ¼ 7.077 (g/cm3) [17], E¼ 168 (GPa) [18], M¼ 172.56 (g/mol – Ce0.8Sm0.2O1.9). The calculated β0 value was 5.886 10 6 K 1. In the calculation, the h value of 3 (ν¼ 1/3) was substituted for Eq. (28). When the ν value was changed in the range of ν¼ 0–0.4 at 298–1000 K, the influence of ν on the β value was negligible (β¼ 5.94 10 6, 5.99 10 6 and 6.07 10 6 K 1 for ν¼ 0.15, 0.33 and 0.40, respectively, at 1000 K.). As seen in Fig. 8, the β(cal.) value was almost independent of the heating temperature. Fortunately, the temperature dependence of the experimental TEC value of Ce0.8Sm0.2O1.9 has been measured in our previous paper [13]. Fig. 8(c) shows the experimentally determined β value. The measured β value was larger than the calculated β value and increased at a higher temperature. The degree of increase in β value was about 3 10 6 K 1 at 300 1500 K. One reason for the discrepancy of the measured and calculated TECs is the treatment of E value included in β0 in Eq. (28). Since the Ε value decreases at a high temperature [14], the β0 value becomes larger with increasing temperature. This effect of β0 was confirmed in the calculation by decreasing the E value to 60% of the value measured at room temperature as an example. Fig. 8(b) shows the TEC for Ε¼ 101 GPa. The decrease of Young's modulus leads to the increase of β value as compared with the calculated β value of Fig. 8(a). The gradual increase of β(mea.) with heating temperature reflects the decrease of Ε value at a high temperature.
Fig. 8. Calculated TECs at (a) E ¼ 168 GPa and (b) E ¼101 GPa and (c) the measured TEC for Ce0.8Sm0.2O1.9 as a function of heating temperature.
That is, the interpretation for the observed β value is well explained by the shift of β value from the calculated line (a) for E¼ 168 GPa to the calculated line (b) for E¼ 101 GPa with increasing temperature through the many lines with different E values. That is, the temperature dependence of TEC is deeply related to the characteristics of E value at a high temperature, and the measurement of accurate E value at a high temperature is needed to predict the TEC by Eq. (28). The temperature dependence of TEC of Al2O3 ceramics is also interesting. Although there is a difference of TECs of Al2O3 between the measurement and the calculation by Eqs. (13) or (28) (see Fig. 3 or Table 1), the temperature dependence of TEC is quite similar for the measured and calculated β values. Fig. 9 shows the β(mea.T K)/β(mea. 473 K) ratio and the β(cal.T)/β(cal. 450 K) ratio as a function of heating temperature based on the available Cp2 values [7,12]. The experimental β ratio is theoretically related to Eq. (29) and the calculated β ratio has a meaning of Eq. (30) for a constant E value (380 GPa at room temperature). βðmea:T KÞ Cp2 ðT KÞ Eð473 KÞ ffi βðmea:473 KÞ Cp2 ð473 KÞ EðT KÞ
ð29Þ
βðcal:T KÞ Cp2 ðT KÞ ffi βðcal:450 KÞ Cp2 ð450 KÞ
ð30Þ
As seen in Fig. 9, both the β ratios increase gradually with increasing temperature along a similar curve by 700 K, suggesting that the E(473 K)/E(T) is close to unity. However, a deviation of the two curves occurs at a higher temperature. This result is explained by E(473 K)/E(T K) 4 1 in Eq. (29). The above comparison indicates that Eq. (30) is quite useful to estimate a β value at a high temperature when the Cp2 values and the β value at a low temperature are available.
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Table 1 Comparison of thermal expansion coefficients between the experiment at 273–373 K and the calculation by Eqs. (13), (28) and (31). Material
TEC ( 10 6 K 1) Measured
Al Ag Au Be Bi C (diamond) Cu Cd Cr Co Fe Ga Hf In Mo Mn Mg Ni Nb Pt Pd Pb Ru Sb Sn Ta Ti V W Zn Zr Al2O3 3Al2O3 2SiO2 B4C BaTiO3 MgO MgAl2O4 SiO2 SiC TiO2 TiN
23.7 19.3 14.24 15 //c 16.2, ? c 12.0 1.32 (323 K) 16.2 //c 52.6, ? c 21.4 8.4 12.6 13.75 (273–1170 K) 53 (200–291 K) 6 56 5.1 21.63 //c 27.1, ? c 24.3 12.5 7.2 8.99 10.6 29.0 6.75 //c 17.17, ? c 8.0 23.8 6.5 8.8 8.3 4.5 31.0 5.83 (253–273 K) 8.6 4.5 4.5 9.4 13.5 9.0 //c 9.0, ? c 14 4.0 8.7 3.41
Eq. (13)
Eq. (28) Eq. (31)
298 K
1000 K 400 K
400 K
7.56 11.33 10.44 10.17 12.64 0.579 9.28 13.49 4.25 5.98 4.87 13.11 4.63 54.57 2.58 6.00 13.35 6.55 7.31 5.84 8.15 29.86 2.27 6.01 7.89 4.17 6.88 7.54 2.38 9.00 6.13 2.71 3.66 1.99 9.08 3.61 3.47 5.35 1.83 4.15 2.70
9.51 13.25 11.87 19.25 14.89 1.87 10.83 14.70 6.18 8.67 11.43
7.82 11.24 10.39 17.96 12.54 2.26 9.57 13.13 4.57 6.02 4.84
20.82 22.16 12.83
5.22 58.97 3.04 8.62 16.52 8.21 8.13 6.46 9.35 32.70 2.64 6.83 9.47 4.60 8.92 9.20 2.69 10.03 6.24 4.28 5.87
4.52 54.46 2.72 5.73 13.64 6.32 7.39 5.48 7.83 28.76 2.34 5.94 7.54 4.13 6.90 7.59 2.47 8.90 6.12 4.30 5.95 1.89 11.20 4.79 5.26 9.12 3.43 5.53 3.64
11.35 4.89 5.02 8.24 3.31 5.46 3.79
15.31
Fig. 9. TEC ratios at two temperatures for the measured and calculated β values of Al2O3 ceramics.
6.55
As compared with the measurement of E value, more accurate Cp2 value is available at a high temperature. The coupling of Eqs. (13), (15) and (22) leads to Eq. (31). 1 Cv1 1 β¼ ð31Þ hT Cp1 Eq. (25) corresponds to h ¼ 3 (ν ¼ 1/3). At a higher temperature, the Cv1/Cp1 ratio is approximated by Eq. (32). Cv1 C v2 3Rð ¼ 24:94 J=mol KÞ ¼ ¼ Cp2 C p1 C p2
ð32Þ
Eq. (31) is useful to calculate the β value at a high temperature
using the available Cp2 and ν values. Fig. 10 shows the β values calculated by Eq. (31) for some metals and ceramics in the conditions of Cv2 ¼ 3R (24.94 J/mol K) and 3R/Cp2 o 1. In Fig. 10, the ν values [16], the measured β values and the measured temperature range are also presented. A relatively good agreement is seen for the measured and calculated β values of some metals at 373–400 K, indicating the usefulness of Eq. (31). However, the large temperature dependence of the calculated β values for MgO, Al2O3 and mullite is not seen in the measured values. A possible explanation for the discrepancy between the measured and calculated β values at a high temperature is the relatively high Debye temperature in ceramics [8], which gives a relatively smaller Cv2 than 3R and increases β value. Another possibility is the change of ν with heating temperature. The decrease of h (decrease of ν) in Eq. (31) increases the β value. That is, Eq. (31) is more effective when the temperature dependence of Cv1 and h is clarified. Table 1 summarizes the β values included in the figures of this paper. The measured value represents the average β value in a certain temperature range and the calculated β value indicates the TEC at a specific temperature. In the calculation of β values at 400–1000 K by Eqs. (13) and (28), the E values at room temperature and Poisson's ratio of 1/3 were used. Although care is taken to compare the β values in Table 1, the β values by Eq. (13) reflect the features of the measured values. As discussed in Section 4, the calculated β values are smaller than the measured β values. The β values by Eq. (28) were very close to those by Eq. (13). When the Cp2 data are not available as a function of temperature, Eq. (28) is useful to predict the TEC. The calculation by Eq. (31) provides a good agreement between the measured and calculated β values in some metals. This equation does not contain the Ε value and is functions of the accurate Cp2 and ν values. At this moment the simple and reliable Eq. (31) is recommended to calculate theoretically the β values of metals when the data of Cp2 and ν values are available.
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Y. Hirata / Ceramics International 41 (2015) 1145–1153 Table 2 Stiffness (Young's modulus) and calculated TEC of single crystal Al2O3. Stiffness (Ref. [7]) (GPa, at 293 K) TEC ( 10 6 K 1)a 400 K 600 K 800 K 1000 K 1200 K C11 C12 C13 C14 C33 C44 C66
490.2 165.4 113.0 23.2 490.2 145.2 162.4 Averageb
2.54 2.98 3.18 3.30 7.54 8.83 9.42 9.79 11.03 12.92 13.79 14.32 57.72 62.93 67.18 69.77 2.54 2.98 3.18 3.30 8.57 10.04 10.72 11.13 7.67 8.99 9.60 9.97 6.65 7.79 8.31 8.64
3.39 10.06 14.72 71.71 3.39 11.44 10.24 8.88
Calculated by Eq. (13) using the following values: Cp2 ¼96.08 J/mol K at 400 K, 112.55 J/mol K at 600 K, 120.14 J/mol K at 800 K, 124.77 J/mol K at 1000 K, 128.25 J/mol K at 1200 K, ρ¼3.968 g/cm3, M¼ 101.96 g/mol. b The TEC value for C14 is not included. a
200
400
600
800
1000
1200
1400
1600
Temperature (K) Fig. 10. Comparison between the measured TEC and the TEC calculated by Eq. (31) for some metals and ceramics as a function of heating temperature.
7. TEC of single crystal Al2O3 The discussion of Section 6 succeeded in explaining the temperature dependence of metals (Fig. 10) or alumina ceramics (Fig. 9). However, there is still a difference of the magnitude between the reported TEC and the calculated TEC (Eqs. (13) and (28)) for some ceramics. The following discussion may provide a possible interpretation of the reported β value. Table 2 shows the stiffness (Cij, Young's modulus) of single crystal Al2O3 [7] at room temperature. The stiffness of single crystal depends on the direction of the crystal structure [19]. The TEC value for each Cij was calculated by Eq. (13) using the values indicated in Table 2 at 400–1200 K. The average β value for C11–C66 resulted in 6.65 10 6–8.88 10 6 K 1 at 400–1200 K. This average β value is very close to the reported β value of Al2O3 ceramics (8.6 10 6 K 1 in Table 1). The average β value has the following meaning: βðaverageÞ ¼
C p1 i ¼ 6; j ¼ 6 1 C p1 ¼ ∑ 3N i ¼ 1; j ¼ 1 C ij 3CðaverageÞ
ð33Þ
where N is the number of stiffness used in the calculation. The C (average) is 187 GPa in Table 2 and lower than the E value (380 GPa) measured mechanically. Eq. (33) indicates that the TEC is related to the parallel combination of Young's modulus in each direction. This Young's modulus relation is the same as a laminated composite of slab where a constant stress is applied perpendicular to the direction of the length of composite. That is, the thermal expansion of alumina grains in a sintered material without an external applied stress corresponds to the mechanical deformation under the application of a constant stress. In the usual measurement of Young's modulus, a mechanical stress is applied to induce a macroscopic strain. When the microstructure of alumina polycrystalline is treated as a composite of grains with different Young's moduli, the deformation mechanism is analyzed basically by a series law or a parallel law of Young's moduli. The average Young's modulus is larger for a series law at a constant strain than for a parallel law at a constant stress. That is, the mechanically measured Young's modulus reflects strongly
the deformation under a constant strain. When alumina grains are deformed at a same strain, their TEC values (∂ε/∂T) should be the same value. On the other hand, the thermal expansion under no external applied stress corresponds to the mechanical deformation at a constant applied stress. Each grain (each direction) can be deformed with heating according to the each TEC value as seen in Table 2. That is, the β value of Al2O3 ceramics reported in Table 1 is related to the average value of β values of single crystal Al2O3 in Table 2. The above discussion suggests the importance of Young's modulus substituted for Eqs. (13) and (28). 8. Summary (1) In this paper, the critical flux of energy (If) input to fracture a solid material was discussed and the If value was related to κ (thermal conductivity), β (liner thermal expansion coefficient), E (Young's modulus), KIC (fracture toughness), a (size of crack origin) and x (thickness of material). The If value becomes larger at larger κ and KIC values and at smaller β, E, a and x values. (2) During the above discussion, it was found that β is equal to Cp/3E (Cp: heat capacity at a constant pressure). This theoretical relation was compared with the experimentally measured β values for 31 metals and 10 ceramics. A good agreement was observed in the calculated and measured β values for C (diamond), TiN, mullite, Zr, Nb, V, BaTiO3, Bi, Pb and In. The difference of the two β values for other materials is greatly affected by the Debye temperature and melting point. In the material with a low Debye temperature and a high melting point, the calculated β value approaches the measured β value. Therefore, the correspondence between calculated and measured β values becomes higher in the material with a small temperature dependence of Young's modulus at a high temperature. (3) Based on the above discussion, the theoretical equation representing the temperature dependence of β value was derived. This equation is simple and the β value can be calculated by substituting only Young's modulus and heating temperature. The temperature dependence of
Y. Hirata / Ceramics International 41 (2015) 1145–1153
Young's modulus controls the accuracy of β value. The characteristic of measured β value as a function of heating temperature was well interpreted based on the theoretical equation. The gradual increase of measured β value is mainly explained by the decrease of Young's modulus at a high temperature. (4) Further discussion provided the more accurate equation of β value, which does not contain Young's modulus and is composed of the accurate Cv (heat capacity at a constant volume), Cp, ν (Poisson's ratio) and temperature. The calculated β values showed a very good agreement with the measured values of some metals. (5) It was also found that the measured β value of Al2O3 ceramics was very close to the average β value calculated using Young's modulus in each direction of single crystal Al2O3.
References [1] D.P.H. Hasselman, Strength behavior of polycrystalline alumina subjected to thermal shock, J. Am. Ceram. Soc. 53 (9) (1970) 490–495. [2] D.P.H. Hasselman, Unified theory of thermal shock fracture initiation and crack propagation in brittle ceramics, J. Am. Ceram. Soc. 52 (11) (1969) 600–604. [3] Y. Hirata, Y. Nakashima, Y. Nibo, S. Sameshima, S. Uchida, S. Hamauzu, S. Kurita, Synthesis and mechanical properties of the Al2O3/Fe system, in: G. Oprea (Ed.). Proceedings of the Symposium on Refractories for Next Millennium. The Minerals, Metals and Materials Society, 2000, pp. 95–108. [4] T. Hashiguchi, Y. Hirata, S. Sameshima, Mechanical properties and thermal shock resistance of hot-pressed high-speed steel, Trans. Mater. Res. Soc. Jpn. 29 (8) (2004) 3461–3464.
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[5] T. Nakazono, S. Sameshima, Y. Hirata, Thermal expansion of the mullite fiber/high-speed steel composite, Trans. Mater. Res. Soc. Jpn. 29 (8) (2004) 3465–3468. [6] Metal Handbook, in: S. Nagasaki (Ed.), second ed., The Japan Institute of Metal, Maruzen, Tokyo, 1984, pp. 12–13. [7] Chemical Handbook, Basic Part II, in: K. Hata (Ed.), third ed., The Chemical Society of Japan, Maruzen, Tokyo, 1984, pp. 21–22. [8] T. Nakamura, Ceramics and Heat, Ceramic Science Series, Gihodo, Tokyo, 1985, p. 25–57 (Japanese). [9] M.F. Asby, D.R.H. Jones, Engineering Materials 1, Pergamon Press, New York, 1980, p. 75. [10] Y. Hirata, Representation of thermal conductivity of solid material with particulate inclusion, Ceram. Int. 35 (2009) 2921–2926. [11] R.F. Davis, J.A. Pask, in: A.M. Alper (Ed.), Mullite in High Temperature Oxide Part IV, Academic Press, New York, 1971, pp. 37–75. [12] W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, second ed., John Wiley & Sons, New York, 1976, p. 583–624. [13] S. Sameshima, M. Kawaminami, Y. Hirata, Thermal expansion of rareearth-doped ceria ceramics, J. Ceram. Soc. Jpn. 110 (7) (2002) 597–600. [14] R.A. Swalin, Thermodynamics of Solid, second ed., John Wiley & Sons, New York, 1972, p. 53–67. [15] Y. Hirata, Thermal conduction model of metal and ceramics, Ceram. Int. 35 (2009) 3259–3268. [16] T.H. Courtney, Mechanical Behavior of Materials, McGraw-Hill, New York, 1990, p. 46–50. [17] S. Sameshima, H. Ono, K. Higashi, K. Sonoda, Y. Hirata, Microstructure of rare-earth-doped ceria prepared by the oxalate coprecipitation method, J. Ceram. Soc. Jpn. 108 (11) (2000) 985–988. [18] S. Sameshima, I. Ichikawa, M. Kawaminami, Y. Hirata, Thermal and mechanical properties of rare earth-doped ceria ceramics, Mater. Chem. Phys. 61 (1999) 31–35. [19] J.F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1985, p. 131–149.
CERAMICS INTERNATIONAL
Ceramics International 41 (2015) 1145–1153 www.elsevier.com/locate/ceramint
Theoretical analyses of thermal shock and thermal expansion coefficients of metals and ceramics Yoshihiro Hiratan Department of Chemistry, Biotechnology, and Chemical Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima 890-0065, Japan Received 25 July 2014; accepted 9 September 2014 Available online 28 September 2014
Abstract The theoretical expression for the critical flux (If) of energy input to fracture a solid material was derived using available material parameters. The If value becomes larger in the material with larger κ (thermal conductivity) and KIC (fracture toughness) values and smaller β (thermal expansion coefficient), E (Young's modulus), a (size of fracture origin) and x (thickness of material). The following useful relationship was also derived for β, E and Cp (heat capacity at a constant pressure): β¼ Cp/3E. This relation agreed with the experimentally measured β value in the material with a low Debye temperature and a small temperature dependence of Young's modulus at a high temperature. The theoretical equation representing the temperature dependence of β value was derived. This equation gives the interpretation that the gradual increase of the measured β value is due to the decrease of Young's modulus at a high temperature. Further discussion provided a more accurate equation which does not contain a Young's modulus and is composed of Cv (heat capacity at a constant volume), Cp, ν (Poisson's ratio) and temperature. A very good agreement was observed between the measured and calculated β values of some metals. It was also found that the measured β of alumina ceramics was very close to the average value calculated using Young's modulus in each direction of single crystal Al2O3. & 2014 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Keywords: C. Thermal expansion; C. Thermal properties; C. Thermal shock resistance
1. Introduction A thermal expansion coefficient (TEC) of metal or ceramics is measured as a function of heating temperature and the average value in a wide temperature range is used in the design of materials application. The metal or ceramics with a remarkably low TEC is attractive as a high temperature structural material because of its high thermal shock resistance [1–5]. Fortunately many measured data of TECs of metals and ceramics are available on a metal handbook [6] or chemical handbook [7]. These data are used in the calculation of thermal shock resistance. However, few theoretical analyses have been proposed to predict the magnitude of TEC [8]. This paper derived a useful relationship among a flux of energy input to a solid material (I), an induced stress (σ), a TEC (β), a Young's modulus (E) and a thermal conductivity (κ), and succeeded finally in relating a TEC to the heat capacity at a n
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http://dx.doi.org/10.1016/j.ceramint.2014.09.042 0272-8842/& 2014 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
constant pressure (Cp) and a Young's modulus. The calculated TEC was compared to the measured TEC and the difference between two TEC values was discussed. The above discussion is useful to understand the essence of thermal expansion coefficient. 2. Thermal shock resistance A linear TEC (β) of a material with length L, which is heated uniformly at a temperature T, is defined by Eq. (1) 1 ∂L ∂ε ∂L β¼ ¼ ∂ε ð1Þ L ∂T P ∂T P L where ε is the strain [9]. On the other hand, a Young's modulus represents the slope of σ (applied stress) – ε relation by Eq. (2). ∂σ ð2Þ E¼ ∂ε P Furthermore, a flux of energy input to a material with thickness x (Fig. 1) is related to the product of a thermal conductivity (κ)
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The dimension, J/m3 K, expresses the unit of heat capacity per unit volume at a constant pressure (Cp1). Since the βE value corresponds to the characteristic of one direction of a material, βE is related to Cp1 by Eq. (9) βΕ ¼ 13C p1
ð9Þ
Eq. (9) indicates that the β value becomes smaller in a material with a smaller Cp1 and a larger E. On the other hand, κ (J/s mK) is related to α (thermal diffusibility, m2/s) and Cv1 (heat capacity per unit volume at a constant volume, J/m3 K) by Eq. (10) [10]. κ ¼ 13C v1 α Fig. 1. Temperature gradient in a solid material with thickness x at a given flux of energy (I).
and a temperature gradient (∂T/∂x)P by Eq. (3). ∂T I ¼ κ ∂x P
ð3Þ
The product of Eqs. (1)–(3) provides an interesting relationship of Eq. (4). ∂σ βE ¼ I ð4Þ ∂x P κ P Eq. (4) expresses the stress gradient induced in the material at a given temperature gradient. When β, E and κ are treated as constant values, the integration of Eq. (4) at a constant pressure leads to the relationship of Eq. (5). βE σ¼ Ix ð5Þ κ The induced σ becomes smaller at smaller β, E, I and x values and at a larger κ value. When the material is fractured at a given temperature gradient, the fracture mechanics is applied to Eq. (5), resulting in Eq. (6), βE K IC σ f ¼ − I f x ¼ pffiffiffi ð6Þ κ Y a where KIC is the fracture toughness, a the size of fracture origin and Y the shape factor of fracture origin. That is, the critical flux of energy causing the fracture is given by Eq. (7). κ K IC pffiffiffi ð7Þ If ¼ βE Y a x The If value becomes larger at larger κ and KIC values and at smaller β, Ε, a and x values. Eq. (7) is useful to estimate the thermal shock resistance of metals or ceramics. This equation will be compared with the experimentally measured If value in future. 3. Thermal expansion coefficient In Eqs. (6) or (7), the βE/κ value at a constant pressure may be treated as a material constant. The dimension of the unit of βE value is as follows: m N J ð8Þ ðβΕ Þ ¼ ¼ 3 mK m2 m K
ð10Þ
The combination of Eqs. (9) and (10) leads to Eq. (11) βΕ C p1 1 ¼ ð11Þ κ C v1 α The substitution of Eq. (11) for Eq. (5) results in Eq. (12) Cp1 Ix σ¼ ð12Þ Cv1 α When an energy is input along one direction to a solid material, the induced stress becomes smaller in a thinner material (x) with a larger α value. The Cp1/Cv1 ratio is close to unity at a low temperature and gradually increases above unity at a higher temperature as discussed in Section 4.2. Furthermore, the β value in Eq. (9) is expressed by the heat capacity at a constant pressure (Cp2) of the dimension, J/mol K, density (ρ, g/m3) and molecular weight (M, g/mol) of a material by Eq. (13). C p1 Cp2 ρ ð13Þ ¼ β¼ 3EM 3E That is, the β value is related to the measurable parameters of a material. The β value calculated by Eq. (13) is compared with the measured β value in the next section to discuss the usefulness of Eq. (13). 4. Comparison of calculated and measured TECs at a low temperature 4.1. Calculated results The Cp2, ρ and E of metals and ceramics at 293–298 K are listed on a chemical handbook [7,11] and a metal handbook [6]. Those data at room temperature were used in Eq. (13) to calculate the β value. On the other hand, the β values measured in a temperature range of 273–373 K are also presented in the handbooks [6,7]. Fig. 2 shows the relationship between the calculated and measured β values for 31 metals and 10 ceramics. In Fig. 2, the straight lines with the slopes of β(measured)/β(calculated) ¼ 1, 1.5, 2.0 and 3.0 are presented to evaluate the correspondence of both the β values. At first, the magnitude of the calculated β values was the same order as the measured β values. As seen in Fig. 2, the calculated β values in a wide range of C (diamond), TiN, mullite, Zr, Nb, V, BaTiO3, Bi, Pb and In were very close to the measured β values. The other data are distributed in the range of β(mea.)/β(cal.) ¼ 1.5– 3.0. That is, the calculated β values showed a tendency to become smaller than the measured β values. This result is
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discussed based on the temperature difference in the measurement and calculation of β values.
EðTÞ o1 EðT 0 Þ
4.2. Discussion
Since Young's modulus of metal or ceramics decreases at a higher temperature [14], the relation of Eq. (17) is derived. As a result, the β(cal.T0)/β(mea.T) ratio of Eq. (14) becomes smaller than unity.
Since the β value depends on the heating temperature [8,12,13], the β(cal.T0)/β(mea.T) ratio is presented by Eq. (14), βðcal:T 0 Þ Cp2 ðT 0 ÞρðT 0 Þ 3EðTÞ C p2 ðT 0 Þ EðTÞ ¼ ffi βðmea:TÞ 3EðT 0 Þ Cp2 ðTÞρðT Þ C p2 ðTÞ EðT 0 Þ
ð14Þ
The T0 and T in Eq. (14) are the temperatures where the β values are calculated and measured, respectively. The ρ(T0)/ρ(T) ratio is close to unity because of the small values of β(cal.) and β(mea.) (o50 10 6 K 1). According to Debye's model [8,14], Cv2 in the dimension of J/mol K at a constant volume of solid approaches a constant value 3R (=24.94 J/mol K, R: gas constant) at T»θ (Debye temperature). The Cp2 (J/mol K)–Cv2 (J/mol K) relation is expressed by Eq. (15) for an isotropic body, ð3βÞ VT κT 2
C p2 ¼ C v2 þ
ð15Þ
where V is the molar volume of solid and κT is the isothermal compressibility (κT ¼ ( 1/V)(∂V/∂σ)T, where σ is the stress applied to the material). Since the Cv2 value is larger than the (3β)2VT/κT value, the Cp2 value increases gradually along the Cv2 value with increasing heating temperature. That is, the following relations are derived for Eq. (14): Case 1. T0 (measurement) C p2 ðT 0 Þ o1 C p2 ðTÞ
(¼ 298 K) o θ
(Debye
temperature) o T ð16Þ
Fig. 2. Relationship between the measured TEC and the calculated TEC of metals and ceramics at 293–298 K.
ð17Þ
Case 2. θ (Debye temperature)oT0 (¼ 298 K)oT (measurement) C p2 ðT 0 Þ ~1 C p2 ðTÞ
ð18Þ
EðTÞ o1 EðT 0 Þ
ð19Þ
In this case, the Cp2(T0)/Cp2(T) ratio approaches unity but the E(T)/E(T0) ratio is smaller than unity. The β0(cal.T0)/β(mea.T) ratio reflects mainly the ratio of Young's moduli at two temperatures (T0 and T). Fig. 3 shows the β(cal.T0)/β(mea.T) ratio against the T0 (¼ 298 K)/θ ratio. The Debye temperature (θ) is also listed on a metal handbook [6]. The data in Fig. 3 are grouped into 5 classes according to the T0 (¼ 298 K)/Tm (melting point) ratio of metals and ceramics. As seen in Fig. 3, β(cal.T0)/β(mea.T) ratio of each group increases at a higher T0/θ ratio. Although it is difficult to specify the heating temperature for the measured TEC values, the tendency in Fig. 3 is explained by the relations of Cases 1 and 2. The regions of (T0/θ)o1 and (T0/θ)41 correspond to Cases 1 and 2, respectively. The calculated TEC value approaches the measured TEC value in the metal or ceramics with a lower Debye temperature as compared with T0 (=298 K). This tendency in each class of a similar T0/Tm ratio reflects mainly the change of Cp2(T0)/Cp2(T) ratio, because the E (T)/ E(T0) ratios as a function of T0/Tm ratio may be a similar value in each group. The above tendency also depends on the
Fig. 3. Ratio between the measured and the calculated TECs as a function of T0 (¼298 K)/θ (Debye temperature) ratio. The Tm in Fig. 3 means the melting point of metal or ceramics.
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T0/Tm ratio at a similar T0/θ ratio. In the material with a high melting point, the calculated TEC approaches the measured TEC. This observation reflects the influence of E(T)/E(T0) ratio in Case 1 or 2. The decrease of Young's modulus at a higher temperature becomes smaller in the metal or ceramics with a high melting point. Therefore, the E(T)/E(T0) ratio approaches unity in the material with a high melting point. As discussed above, the key factors affecting the β(cal.T0)/β(mea.T) ratio are the Debye temperature and the melting point of metal or ceramics. In the material with a low Debye temperature and a high melting point, the calculated CTE approaches the measured CTE.
small 1000 K/Tm ratio. That is, the correspondence between β (cal.)/β(mea.) values becomes higher in the material with a small temperature dependence of Young's modulus at a high temperature. The some data of β(cal.)/β(mea.) ratio 4 1 may be due to the relation of θ o T(mea.) o 1000 K (Cp2(1000 K)/ Cp2(T)4 1 in Eq. (20)).
5. Comparison of calculated and measured TECs at high temperatures In Case 2 in Section 4, the β(cal.T0)/β(mea.T) ratio reflects mainly the E(T)/E(T0) ratio. However, it is difficult to measure the accurate Young's modulus at a high temperature. Most of the reported Young's moduli of metals and ceramics are measured at room temperature. The following relation was compared because of the difficulty of the measurement of the parameters in Eq. (13) at high temperatures. In Eq. (20), E (298 K) was used instead of E (1000 K). βðcal:1000 KÞ Cp2 ð1000 KÞ ρð1000 KÞ 3E ðTÞ ¼ βðmea:TÞ 3Eð298 KÞ C p2 ðTÞ ρðTÞ Cp2 ð1000 KÞ ΕðTÞ ffi C p2 ðTÞ Εð298 KÞ
ð20Þ
In the condition of θ o 1000 K o T(mea.) or θ o T(mea.) o 1000 K, the Cp2(1000 K)/Cp2(T) ratio is close to unity (see Section 6). That is, the β(cal.1000 K)/β(mea.T) ratio mainly reflects the ratio of Young's moduli (E(T)/E(298 K) o 1) in Eq. (20). Fig. 4 shows the β(mea.T K) – β(cal.1000 K) relation of metals and ceramics. Most of the data are seen in the slope range of β(mea.T K)/β(cal.1000 K) ¼ 1–3. This tendency was similar to the plots in Fig. 2. The data in Fig. 4 are grouped into 4 classes by the nature of chemical bonding as defined in our previous paper [15]. AD-1: weak chemical bond (mixing of metal bond and covalent bond, Au, Pb, Cu, In, Ga, Be), AD-2: intermediate chemical bond (mixing of metal bond and ionic bond, Ag, Bi, Cd, Zn, Sb, Zr), AD-3: strong chemical bond (mixing of metal bond, covalent bond and ionic bond, Pt, W, Ta, Hf, Pd, Mo, Ni, Co, Nb, Fe, Cr, V, Ti), and ceramics (mixing of covalent bond and ionic bond, Al2O3, TiO2, MgO, C(diamond), SiC). Fig. 5 shows the β(cal.1000 K)/β(mea.T) ratio of each group as a function of the 1000 K/Tm ratio. The calculated TEC approaches the measured TEC in the materials with a high melting point. In other words, the plots in Fig. 5 suggest that the E(T)/E(298 K) ratio in Eq. (20) decreases drastically with the decrease in melting point. The β(cal.1000 K)/β(mea.T) ratio shifted to a larger ratio of 1000 K/Tm in the order of ceramics o AD-3o AD-1, AD-2. When the material has a stronger chemical bond, the melting point becomes higher. As a result, the calculated TEC approaches the measured TEC at a
Fig. 4. Relationship between the measured TEC and the TEC calculated using a heat capacity in atmospheric pressure at 1000 K.
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fig. 5. Ratio between the measured TEC and the TEC calculated using a heat capacity at 1000 K as a function of 1000 K/Tm (melting point) ratio of metals and ceramics. The data in Fig. 5 are grouped into 4 classes (AD-1, 2, 3 and ceramics) by the nature of chemical bonding. See text and Ref. [15] for the detailed explanation.
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6. Temperature dependence of thermal expansion coefficient
The approximation of 1/κT C E and the relationship between Cp1 and Cp2 in Eq. (13) are coupled with Eq. (15) to give Eq. (24) because of ρV/M ¼ 1.
6.1. Derivation of basic equation of TEC
Cp1 ¼ C v1 þ ð3βÞ2 ET
In Figs. 2–5, we could see the influence of θ, E, Tm and the strength of chemical bond on the TECs of metals and ceramics at 298 K and 1000 K. In this section, we discuss the more general expression of TEC as a function of heating temperature. The Cp2–Cv2 relation for an isotropic body is given by Eq. (15). The isothermal compressibility of the material in a state of hydrostatic compression or tension (κT, in Eq. (15)) is related to the bulk modulus K by Eq. (21), 1 ∂V ∂εV 1 ð21Þ κT ¼ ¼ ¼ V ∂σ T Κ ∂σ T
Since 3β is equal to Cp1/E in Eq. (13), Eq. (25) is derived.
where εv is the strain of bulk. The bulk modulus (Κ) is related to Young's modulus (E) and Poisson's ratio (ν) by Eq. (22) [16]. E Κ¼ 3ð1 2νÞ
ð22Þ
Therefore, the combination of Eqs. (21) and (22) leads to the relationship of Eq. (23). 1 E ¼ K ¼ kT 31 2ν
ð23Þ
Since Poisson's ratio of metals at room temperature is around 1/3 [16], the bulk modulus is almost equal to Young's modulus. Fig. 6 compares the measured E values and the K values at σ ¼ 0.98 GPa and at room temperature for 27 materials. The straight line corresponds to the slope 1 of the E/K ratio. It is found that the data are scattered around the straight line, and this result basically supports the theoretical relationship of 1/κT C E in Eq. (23). 500
Cp1 ¼
ð24Þ
Cv1 13 β T
ð25Þ
The Cp1 value depends on the β value and heating temperature, and increases at a higher temperature. The condition of β¼ 0 K 1 leads to the relation Cp1 ¼ Cv1. The temperature dependence of Cv1( ¼ Cv2ρ/M) at a constant volume is analyzed theoretically in the Einstein and Debye equations [8,14], as discussed in Section 4.2. The relation by Eq. (25) is useful to understand more the meaning of Eq. (12). The substitution of Eq. (25) for Eq. (12) results in Eq. (26). σ¼
Ix ð1 3 β TÞα
ð26Þ
When a flux of energy is input to the surface of the material, the (1–3βT) value is calculated to be 0.885–0.996 under the conditions of β¼ 1 10 6 30 10 6 K 1 at T (surface temperature)¼ 1237 K. That is, the (1–3βT) value is around unity and the induced stress depends mainly on the α, x and I values. Fig. 7 shows the following normalized relation of Eq. (25): Cp2/Cv2 ratio – 1/(1–3β(mea.)T) plot at 1000 K. As a Cv2 value, 3R value (24.94 J/mol K) is used because the Debye temperatures of many metals are lower than 1000 K. That is, the Cv2 values of many metals approach 3R at a high temperature. In Fig. 7, the Cp2 per 1 mol-atom of ceramics is also presented. The some straight lines with different slopes are presented to evaluate the plotted data. The data were also grouped into 4 classes as shown in Fig. 5, depending on the nature of 1.60
400
Young's modulus, E (GPa)
1.40
300 1.20
200 1.00
100
0.80
0
0
100
200
300
400
500
1.00
1.02
1.04
1.06
1.08
1.10
Bulk modulus, K (GPa) Fig. 6. Comparison of the measured Young's modulus and the bulk modulus determined from the isothermal compressibility (κT) of metals and ceramics.
Fig. 7. Heat capacity (Cp2) normalized by 3R (24.94 J/mol K) as a function of 1/(1–3β(mea.)T) at 1000 K in Eq. (25).
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Y. Hirata / Ceramics International 41 (2015) 1145–1153
chemical bonding. In each group, the Cp2/Cv2 ratio increases as β(mea.) value increases. More discussion is described in the next section. At a similar β value, the Cp2/Cv2 ratio becomes larger in the order of ceramics o AD-1, AD-2 o AD-3. The Cp2/Cv2 ratio for ceramics is near the line with a slope of unity. However, the Cp2/Cv2 ratio for metals is distributed around the lines with the slopes of 1.1–1.4. This result may be explained by the influence of ν in Eq. (22). In Eq. (25), the ν value is approximated to be 1/3. 6.2. Reliability of derived TEC equation The coupling of Eqs. (13), (15) and (23) gives the following accurate expression of β value (Eq. (27)): β¼
β0 1 hβ T
ð27Þ
where β0 is equal to Cv1/3E and h is equal to 1/(1 2ν). The solution of β for Eq. (27) is given by Eq. (28), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 4hβ0 T ð28Þ β¼ 2hT Eq. (28) is useful to predict the temperature dependence of TEC under the condition of constant values of β0 and ν. Since the Cv2 value (J/mol K) of a material is close to 3R (24.94 J/mol K) at a higher temperature, the Cv1 (¼ Cv2ρ/M, J/m3 K) is allowed to be treated as a constant value. That is, the β0 value is calculated to be 3Rρ/3ME for a constant E value with no temperature dependence. Fig. 8(a) shows the β value calculated by Eq. (28) for Ce0.8Sm0.2O1.9 as a function of heating temperature. The following values were used to determine the β0 value: Cv2 ¼ 24.94 (J/mol-atom K) 2.9(mol-atom/Ce0.8Sm0.2O1.9 composition)¼ 72.33 (J/mol – Ce0.8Sm0.2O1.9 K), ρ¼ 7.077 (g/cm3) [17], E¼ 168 (GPa) [18], M¼ 172.56 (g/mol – Ce0.8Sm0.2O1.9). The calculated β0 value was 5.886 10 6 K 1. In the calculation, the h value of 3 (ν¼ 1/3) was substituted for Eq. (28). When the ν value was changed in the range of ν¼ 0–0.4 at 298–1000 K, the influence of ν on the β value was negligible (β¼ 5.94 10 6, 5.99 10 6 and 6.07 10 6 K 1 for ν¼ 0.15, 0.33 and 0.40, respectively, at 1000 K.). As seen in Fig. 8, the β(cal.) value was almost independent of the heating temperature. Fortunately, the temperature dependence of the experimental TEC value of Ce0.8Sm0.2O1.9 has been measured in our previous paper [13]. Fig. 8(c) shows the experimentally determined β value. The measured β value was larger than the calculated β value and increased at a higher temperature. The degree of increase in β value was about 3 10 6 K 1 at 300 1500 K. One reason for the discrepancy of the measured and calculated TECs is the treatment of E value included in β0 in Eq. (28). Since the Ε value decreases at a high temperature [14], the β0 value becomes larger with increasing temperature. This effect of β0 was confirmed in the calculation by decreasing the E value to 60% of the value measured at room temperature as an example. Fig. 8(b) shows the TEC for Ε¼ 101 GPa. The decrease of Young's modulus leads to the increase of β value as compared with the calculated β value of Fig. 8(a). The gradual increase of β(mea.) with heating temperature reflects the decrease of Ε value at a high temperature.
Fig. 8. Calculated TECs at (a) E ¼ 168 GPa and (b) E ¼101 GPa and (c) the measured TEC for Ce0.8Sm0.2O1.9 as a function of heating temperature.
That is, the interpretation for the observed β value is well explained by the shift of β value from the calculated line (a) for E¼ 168 GPa to the calculated line (b) for E¼ 101 GPa with increasing temperature through the many lines with different E values. That is, the temperature dependence of TEC is deeply related to the characteristics of E value at a high temperature, and the measurement of accurate E value at a high temperature is needed to predict the TEC by Eq. (28). The temperature dependence of TEC of Al2O3 ceramics is also interesting. Although there is a difference of TECs of Al2O3 between the measurement and the calculation by Eqs. (13) or (28) (see Fig. 3 or Table 1), the temperature dependence of TEC is quite similar for the measured and calculated β values. Fig. 9 shows the β(mea.T K)/β(mea. 473 K) ratio and the β(cal.T)/β(cal. 450 K) ratio as a function of heating temperature based on the available Cp2 values [7,12]. The experimental β ratio is theoretically related to Eq. (29) and the calculated β ratio has a meaning of Eq. (30) for a constant E value (380 GPa at room temperature). βðmea:T KÞ Cp2 ðT KÞ Eð473 KÞ ffi βðmea:473 KÞ Cp2 ð473 KÞ EðT KÞ
ð29Þ
βðcal:T KÞ Cp2 ðT KÞ ffi βðcal:450 KÞ Cp2 ð450 KÞ
ð30Þ
As seen in Fig. 9, both the β ratios increase gradually with increasing temperature along a similar curve by 700 K, suggesting that the E(473 K)/E(T) is close to unity. However, a deviation of the two curves occurs at a higher temperature. This result is explained by E(473 K)/E(T K) 4 1 in Eq. (29). The above comparison indicates that Eq. (30) is quite useful to estimate a β value at a high temperature when the Cp2 values and the β value at a low temperature are available.
Y. Hirata / Ceramics International 41 (2015) 1145–1153
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Table 1 Comparison of thermal expansion coefficients between the experiment at 273–373 K and the calculation by Eqs. (13), (28) and (31). Material
TEC ( 10 6 K 1) Measured
Al Ag Au Be Bi C (diamond) Cu Cd Cr Co Fe Ga Hf In Mo Mn Mg Ni Nb Pt Pd Pb Ru Sb Sn Ta Ti V W Zn Zr Al2O3 3Al2O3 2SiO2 B4C BaTiO3 MgO MgAl2O4 SiO2 SiC TiO2 TiN
23.7 19.3 14.24 15 //c 16.2, ? c 12.0 1.32 (323 K) 16.2 //c 52.6, ? c 21.4 8.4 12.6 13.75 (273–1170 K) 53 (200–291 K) 6 56 5.1 21.63 //c 27.1, ? c 24.3 12.5 7.2 8.99 10.6 29.0 6.75 //c 17.17, ? c 8.0 23.8 6.5 8.8 8.3 4.5 31.0 5.83 (253–273 K) 8.6 4.5 4.5 9.4 13.5 9.0 //c 9.0, ? c 14 4.0 8.7 3.41
Eq. (13)
Eq. (28) Eq. (31)
298 K
1000 K 400 K
400 K
7.56 11.33 10.44 10.17 12.64 0.579 9.28 13.49 4.25 5.98 4.87 13.11 4.63 54.57 2.58 6.00 13.35 6.55 7.31 5.84 8.15 29.86 2.27 6.01 7.89 4.17 6.88 7.54 2.38 9.00 6.13 2.71 3.66 1.99 9.08 3.61 3.47 5.35 1.83 4.15 2.70
9.51 13.25 11.87 19.25 14.89 1.87 10.83 14.70 6.18 8.67 11.43
7.82 11.24 10.39 17.96 12.54 2.26 9.57 13.13 4.57 6.02 4.84
20.82 22.16 12.83
5.22 58.97 3.04 8.62 16.52 8.21 8.13 6.46 9.35 32.70 2.64 6.83 9.47 4.60 8.92 9.20 2.69 10.03 6.24 4.28 5.87
4.52 54.46 2.72 5.73 13.64 6.32 7.39 5.48 7.83 28.76 2.34 5.94 7.54 4.13 6.90 7.59 2.47 8.90 6.12 4.30 5.95 1.89 11.20 4.79 5.26 9.12 3.43 5.53 3.64
11.35 4.89 5.02 8.24 3.31 5.46 3.79
15.31
Fig. 9. TEC ratios at two temperatures for the measured and calculated β values of Al2O3 ceramics.
6.55
As compared with the measurement of E value, more accurate Cp2 value is available at a high temperature. The coupling of Eqs. (13), (15) and (22) leads to Eq. (31). 1 Cv1 1 β¼ ð31Þ hT Cp1 Eq. (25) corresponds to h ¼ 3 (ν ¼ 1/3). At a higher temperature, the Cv1/Cp1 ratio is approximated by Eq. (32). Cv1 C v2 3Rð ¼ 24:94 J=mol KÞ ¼ ¼ Cp2 C p1 C p2
ð32Þ
Eq. (31) is useful to calculate the β value at a high temperature
using the available Cp2 and ν values. Fig. 10 shows the β values calculated by Eq. (31) for some metals and ceramics in the conditions of Cv2 ¼ 3R (24.94 J/mol K) and 3R/Cp2 o 1. In Fig. 10, the ν values [16], the measured β values and the measured temperature range are also presented. A relatively good agreement is seen for the measured and calculated β values of some metals at 373–400 K, indicating the usefulness of Eq. (31). However, the large temperature dependence of the calculated β values for MgO, Al2O3 and mullite is not seen in the measured values. A possible explanation for the discrepancy between the measured and calculated β values at a high temperature is the relatively high Debye temperature in ceramics [8], which gives a relatively smaller Cv2 than 3R and increases β value. Another possibility is the change of ν with heating temperature. The decrease of h (decrease of ν) in Eq. (31) increases the β value. That is, Eq. (31) is more effective when the temperature dependence of Cv1 and h is clarified. Table 1 summarizes the β values included in the figures of this paper. The measured value represents the average β value in a certain temperature range and the calculated β value indicates the TEC at a specific temperature. In the calculation of β values at 400–1000 K by Eqs. (13) and (28), the E values at room temperature and Poisson's ratio of 1/3 were used. Although care is taken to compare the β values in Table 1, the β values by Eq. (13) reflect the features of the measured values. As discussed in Section 4, the calculated β values are smaller than the measured β values. The β values by Eq. (28) were very close to those by Eq. (13). When the Cp2 data are not available as a function of temperature, Eq. (28) is useful to predict the TEC. The calculation by Eq. (31) provides a good agreement between the measured and calculated β values in some metals. This equation does not contain the Ε value and is functions of the accurate Cp2 and ν values. At this moment the simple and reliable Eq. (31) is recommended to calculate theoretically the β values of metals when the data of Cp2 and ν values are available.
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Y. Hirata / Ceramics International 41 (2015) 1145–1153 Table 2 Stiffness (Young's modulus) and calculated TEC of single crystal Al2O3. Stiffness (Ref. [7]) (GPa, at 293 K) TEC ( 10 6 K 1)a 400 K 600 K 800 K 1000 K 1200 K C11 C12 C13 C14 C33 C44 C66
490.2 165.4 113.0 23.2 490.2 145.2 162.4 Averageb
2.54 2.98 3.18 3.30 7.54 8.83 9.42 9.79 11.03 12.92 13.79 14.32 57.72 62.93 67.18 69.77 2.54 2.98 3.18 3.30 8.57 10.04 10.72 11.13 7.67 8.99 9.60 9.97 6.65 7.79 8.31 8.64
3.39 10.06 14.72 71.71 3.39 11.44 10.24 8.88
Calculated by Eq. (13) using the following values: Cp2 ¼96.08 J/mol K at 400 K, 112.55 J/mol K at 600 K, 120.14 J/mol K at 800 K, 124.77 J/mol K at 1000 K, 128.25 J/mol K at 1200 K, ρ¼3.968 g/cm3, M¼ 101.96 g/mol. b The TEC value for C14 is not included. a
200
400
600
800
1000
1200
1400
1600
Temperature (K) Fig. 10. Comparison between the measured TEC and the TEC calculated by Eq. (31) for some metals and ceramics as a function of heating temperature.
7. TEC of single crystal Al2O3 The discussion of Section 6 succeeded in explaining the temperature dependence of metals (Fig. 10) or alumina ceramics (Fig. 9). However, there is still a difference of the magnitude between the reported TEC and the calculated TEC (Eqs. (13) and (28)) for some ceramics. The following discussion may provide a possible interpretation of the reported β value. Table 2 shows the stiffness (Cij, Young's modulus) of single crystal Al2O3 [7] at room temperature. The stiffness of single crystal depends on the direction of the crystal structure [19]. The TEC value for each Cij was calculated by Eq. (13) using the values indicated in Table 2 at 400–1200 K. The average β value for C11–C66 resulted in 6.65 10 6–8.88 10 6 K 1 at 400–1200 K. This average β value is very close to the reported β value of Al2O3 ceramics (8.6 10 6 K 1 in Table 1). The average β value has the following meaning: βðaverageÞ ¼
C p1 i ¼ 6; j ¼ 6 1 C p1 ¼ ∑ 3N i ¼ 1; j ¼ 1 C ij 3CðaverageÞ
ð33Þ
where N is the number of stiffness used in the calculation. The C (average) is 187 GPa in Table 2 and lower than the E value (380 GPa) measured mechanically. Eq. (33) indicates that the TEC is related to the parallel combination of Young's modulus in each direction. This Young's modulus relation is the same as a laminated composite of slab where a constant stress is applied perpendicular to the direction of the length of composite. That is, the thermal expansion of alumina grains in a sintered material without an external applied stress corresponds to the mechanical deformation under the application of a constant stress. In the usual measurement of Young's modulus, a mechanical stress is applied to induce a macroscopic strain. When the microstructure of alumina polycrystalline is treated as a composite of grains with different Young's moduli, the deformation mechanism is analyzed basically by a series law or a parallel law of Young's moduli. The average Young's modulus is larger for a series law at a constant strain than for a parallel law at a constant stress. That is, the mechanically measured Young's modulus reflects strongly
the deformation under a constant strain. When alumina grains are deformed at a same strain, their TEC values (∂ε/∂T) should be the same value. On the other hand, the thermal expansion under no external applied stress corresponds to the mechanical deformation at a constant applied stress. Each grain (each direction) can be deformed with heating according to the each TEC value as seen in Table 2. That is, the β value of Al2O3 ceramics reported in Table 1 is related to the average value of β values of single crystal Al2O3 in Table 2. The above discussion suggests the importance of Young's modulus substituted for Eqs. (13) and (28). 8. Summary (1) In this paper, the critical flux of energy (If) input to fracture a solid material was discussed and the If value was related to κ (thermal conductivity), β (liner thermal expansion coefficient), E (Young's modulus), KIC (fracture toughness), a (size of crack origin) and x (thickness of material). The If value becomes larger at larger κ and KIC values and at smaller β, E, a and x values. (2) During the above discussion, it was found that β is equal to Cp/3E (Cp: heat capacity at a constant pressure). This theoretical relation was compared with the experimentally measured β values for 31 metals and 10 ceramics. A good agreement was observed in the calculated and measured β values for C (diamond), TiN, mullite, Zr, Nb, V, BaTiO3, Bi, Pb and In. The difference of the two β values for other materials is greatly affected by the Debye temperature and melting point. In the material with a low Debye temperature and a high melting point, the calculated β value approaches the measured β value. Therefore, the correspondence between calculated and measured β values becomes higher in the material with a small temperature dependence of Young's modulus at a high temperature. (3) Based on the above discussion, the theoretical equation representing the temperature dependence of β value was derived. This equation is simple and the β value can be calculated by substituting only Young's modulus and heating temperature. The temperature dependence of
Y. Hirata / Ceramics International 41 (2015) 1145–1153
Young's modulus controls the accuracy of β value. The characteristic of measured β value as a function of heating temperature was well interpreted based on the theoretical equation. The gradual increase of measured β value is mainly explained by the decrease of Young's modulus at a high temperature. (4) Further discussion provided the more accurate equation of β value, which does not contain Young's modulus and is composed of the accurate Cv (heat capacity at a constant volume), Cp, ν (Poisson's ratio) and temperature. The calculated β values showed a very good agreement with the measured values of some metals. (5) It was also found that the measured β value of Al2O3 ceramics was very close to the average β value calculated using Young's modulus in each direction of single crystal Al2O3.
References [1] D.P.H. Hasselman, Strength behavior of polycrystalline alumina subjected to thermal shock, J. Am. Ceram. Soc. 53 (9) (1970) 490–495. [2] D.P.H. Hasselman, Unified theory of thermal shock fracture initiation and crack propagation in brittle ceramics, J. Am. Ceram. Soc. 52 (11) (1969) 600–604. [3] Y. Hirata, Y. Nakashima, Y. Nibo, S. Sameshima, S. Uchida, S. Hamauzu, S. Kurita, Synthesis and mechanical properties of the Al2O3/Fe system, in: G. Oprea (Ed.). Proceedings of the Symposium on Refractories for Next Millennium. The Minerals, Metals and Materials Society, 2000, pp. 95–108. [4] T. Hashiguchi, Y. Hirata, S. Sameshima, Mechanical properties and thermal shock resistance of hot-pressed high-speed steel, Trans. Mater. Res. Soc. Jpn. 29 (8) (2004) 3461–3464.
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[5] T. Nakazono, S. Sameshima, Y. Hirata, Thermal expansion of the mullite fiber/high-speed steel composite, Trans. Mater. Res. Soc. Jpn. 29 (8) (2004) 3465–3468. [6] Metal Handbook, in: S. Nagasaki (Ed.), second ed., The Japan Institute of Metal, Maruzen, Tokyo, 1984, pp. 12–13. [7] Chemical Handbook, Basic Part II, in: K. Hata (Ed.), third ed., The Chemical Society of Japan, Maruzen, Tokyo, 1984, pp. 21–22. [8] T. Nakamura, Ceramics and Heat, Ceramic Science Series, Gihodo, Tokyo, 1985, p. 25–57 (Japanese). [9] M.F. Asby, D.R.H. Jones, Engineering Materials 1, Pergamon Press, New York, 1980, p. 75. [10] Y. Hirata, Representation of thermal conductivity of solid material with particulate inclusion, Ceram. Int. 35 (2009) 2921–2926. [11] R.F. Davis, J.A. Pask, in: A.M. Alper (Ed.), Mullite in High Temperature Oxide Part IV, Academic Press, New York, 1971, pp. 37–75. [12] W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, second ed., John Wiley & Sons, New York, 1976, p. 583–624. [13] S. Sameshima, M. Kawaminami, Y. Hirata, Thermal expansion of rareearth-doped ceria ceramics, J. Ceram. Soc. Jpn. 110 (7) (2002) 597–600. [14] R.A. Swalin, Thermodynamics of Solid, second ed., John Wiley & Sons, New York, 1972, p. 53–67. [15] Y. Hirata, Thermal conduction model of metal and ceramics, Ceram. Int. 35 (2009) 3259–3268. [16] T.H. Courtney, Mechanical Behavior of Materials, McGraw-Hill, New York, 1990, p. 46–50. [17] S. Sameshima, H. Ono, K. Higashi, K. Sonoda, Y. Hirata, Microstructure of rare-earth-doped ceria prepared by the oxalate coprecipitation method, J. Ceram. Soc. Jpn. 108 (11) (2000) 985–988. [18] S. Sameshima, I. Ichikawa, M. Kawaminami, Y. Hirata, Thermal and mechanical properties of rare earth-doped ceria ceramics, Mater. Chem. Phys. 61 (1999) 31–35. [19] J.F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1985, p. 131–149.