⇦Chinese Physics Letters, 2021, Vol. 38, No. 2, Article code 026501 Scaling Behavior between Heat Capacity and Thermal Expansion in Solids Meibo Tang (汤美波)*, Xiuhong Pan (潘秀红), Minghui Zhang (张明辉), and Haiqin Wen (温海琴) Affiliations Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, China Received 2 October 2020; accepted 14 December 2020; published online 27 January 2021 Supported by the National Natural Science Foundation of China (Grant No. 51001113).
*Corresponding author. Email: mbtang@mail.sic.ac.cn Citation Text: Tang M B, Pan X H, Zhang M H, and Wen H Q 2021 Chin. Phys. Lett. 38 026501    Abstract We experimentally analyze the heat capacity and thermal expansion of reference solids in a wide temperature range from several Kelvin to melting temperature, and establish a universal double-linear relation between the experimental heat capacity $C_{\rm p}$ and thermal expansion $\beta$, which is different from the previous models. The universal behavior between heat capacity and thermal expansion is important to predict the thermodynamic parameters at constant pressure, and is helpful for understanding the nature of thermal properties in solids. DOI:10.1088/0256-307X/38/2/026501 © 2021 Chinese Physics Society Article Text Heat capacity (HC) and coefficient of thermal expansion (CTE) can be used to characterize changes in macroscopic variables, for example, temperature-dependent enthalpy and volume. They are the most fundamental and important thermodynamic parameters for solids, and have been widely studied by many theoretical and experimental methods.[1,2] In general, heat capacity depends upon temperature $T$ and conditions of heating: at constant-volume $C_{\rm v}$ or constant-pressure $C_{\rm p}$. Conventionally, $C_{\rm v}$ is smaller than $C_{\rm p}$. HC data are usually obtained under constant, generally atmospheric, pressure, i.e. experimental HC usually refers to $C_{\rm p}$. At low temperatures, HC is very sensitive to the atomic structure of the solids and to the quantum mechanical nature of the atomic motions, which was developed by Einstein and Debye.[3,4] HC of solids near room temperature is usually close to a constant $3R$ ($R$: gas constant), which is the Dulong–Petit law. HC of solids at high temperatures is larger than the theoretical value $3R$, which is due to thermal expansion and is usually explained by anharmonic effect.[5] However, direct relationship between experimental HC and CTE is rarely reported. There is a well-known relationship between $C_{\rm v}$ and $C_{\rm p}$: $$ C_{\rm p} = C_{\rm v}+ KVT \beta^{ 2},~~ \tag {1} $$ where $V$ is the molar volume, $\beta$ the coefficient of volumetric thermal expansion at constant pressure and is defined as $\beta =\frac{1}{V}\cdot \frac{dV}{dT}$, $T$ is the temperature, and $K$ is the isothermal bulk modulus.[6] Values of $C_{\rm p}$, $\beta$, $K$, $T$ and $V$ are measured by experiments, $C_{\rm v}$ can only be determined according to Eq. (1) or theoretical models. Another relationship in terms of the $C_{\rm v}$ and $\beta$, suggested by Grüneisen, is $$ C_{\rm v} = KV \beta /\gamma,~~ \tag {2} $$ where $\gamma$ is the Grüneisen constant.[7] From Eqs. (1) and (2), the relation of $C_{\rm p}$ and $\beta$ should be quasi-parabolic. Actually, the Grüneisen constant is calculated by Eqs. (1) and (2). Because values of $C_{\rm v}$ and $\gamma$ cannot be directly measured by experiments, the correlative behavior between $C_{\rm p}$ and $\beta$ cannot be determined by the relationships or the models. By studying the experimental results, Bodryakov reported the linear behavior between the HC and CTE for some materials at low temperatures or low HC (below 3$R$).[8,9] He also reported a trilinear behavior between the HC and CTE for solid KCl from zero to melting temperature.[10] Andritsos et al. analyzed the anharmonic effect on the $C_{\rm v}$ by molecular dynamics simulation, and shows that the $C_{\rm v}$ is dependent on the temperature and CTE.[5] Askerov and Cankurtaran theoretically studied the temperature-dependent $C_{\rm p}$ and $\beta$ by Debye approximation, and the relationship between the $C_{\rm p}$ and $\beta$ is complicated and temperature-dependent.[11] Drebushchak et al. studied the relationship between the $C_{\rm p}$ and $\beta$ by the Lennard–Jones potential, and the pressure-dependent $C_{\rm p}$.[12,13] Garai theoretically predicted a linear correlation between the $C_{\rm p}$ and $\beta$ below the Debye temperature.[14] The studies of Raju et al. show that the ratio $\beta /C_{\rm p}$ is approximated by a temperature-independent constant, the temperature-dependent bulk modulus, enthalpy and volume are estimated, and are close to the experimental data.[15] Although there are some experimental results or models about the correlation between the $C_{\rm p}$ and $\beta$, the systemic studies about the universal relationship between the experimental $C_{\rm p}$ and $\beta$ for general solids in wide temperature ranges are lacking. Our previous study shows a new heat capacity model by the correlation between the experimental heat capacity $C_{\rm p}$ and coefficient of thermal expansion ($\beta$) in Al$_{2}$O$_{3}$ and Cu.[16] In this work, we quantitatively analyze the experimental HC $C_{\rm p}$ and the coefficient of volumetric thermal expansion ($\beta$) in a series of reference solid materials with the melting temperature $T_{\rm m}$ from 3695 K to 84 K, and find that there is a critical HC $C_{\rm p 0}$ for each of the reference solids: the experimental $C_{\rm p}$ is linearly dependent on the CTE $\beta$, below or above the critical value, respectively. The linear behavior is different from the quasi-parabolic result by Eqs. (1) and (2). The average value of the critical HC $C_{\rm p 0}$ in the reference solid materials is $25.17 \pm 0.91$ J$\cdot$mol$^{-1}\cdot$K$^{-1}$, which is close to the Dulong–Petit value 3$R$. The universal thermodynamic behaviors are helpful to predict the macroscopic physical properties in general solids.
cpl-38-2-026501-fig1.png
Fig. 1. (a) Temperature-dependent heat capacity $C_{\rm p}$ (or $C_{\rm v}$) and the coefficient of volumetric thermal expansion $\beta$ for tungsten (data from White and Minges); (b) correlation between the $C_{\rm p}$ (or $C_{\rm v}$) and $\beta$ for tungsten (square and triangle: data from White and Minges;[17] circle: data from Bodryakov[8]), solid lines: linear-fitting results (blue line: $C_{\rm p}$ is below about 3$R$; green line: $C_{\rm p}$ is above about 3$R$); dashed lines: the coordinates of the intersection of the blue and green solid lines.
Collating and Analyzing the Experimental Data. The experimental heat capacity $C_{\rm p}$, the calculated constant-volume heat capacity $C_{\rm v}$, and the coefficient of volumetric thermal expansion ($\beta$) of pure metal: tungsten (W) up to melting temperature are shown in Fig. 1(a).[17] When the temperature is close to zero, both HC and CTE are close to zero. Above zero Kelvin, both HC and CTE quickly increase with increasing temperature. After the HC is close to the Dulong–Petit value 3$R$, both HC and CTE still gradually increase with temperature, but the changes in both HC and CTE with temperature obviously decrease. Thus, the CTE varies with temperature in the same manner as the experimental HC. The constant-volume heat capacity $C_{\rm v}$ is close to the experimental heat capacity $C_{\rm p}$ below 3$R$. Above 3$R$, the difference between the $C_{\rm p}$ and $C_{\rm v}$ gradually increases with the increasing temperature. By the temperature-dependent heat capacity $C_{\rm p}$ (or $C_{\rm v}$) and thermal expansion $\beta$, we can obtain the $C_{\rm p}$ (or $C_{\rm v}$) and $\beta$ at same temperatures, and determine the correlation between the $C_{\rm p}$ (or $C_{\rm v}$) and $\beta$. Figure 1(b) shows the correlative behavior between the $C_{\rm p}$ (or $C_{\rm v}$) and $\beta$ of tungsten (data from Refs. [8,17]). There is an inflection in the $C_{\rm p}$ (or $C_{\rm v}$) vs $\beta$ data of tungsten. The value of HC at the inflection is a critical HC: $C_{\rm p 0}$. The constant-volume heat capacity $C_{\rm v}$ is linearly dependent on the CTE $\beta$ below the critical HC $C_{\rm p 0}$ (or at low temperatures), and deviates from the linear behavior above the critical HC $C_{\rm p 0}$, which is different from the theoretical models.[7] Therefore, from the correlative behavior between the $C_{\rm v}$ and $\beta$ for tungsten, the Grüneisen parameter in Eq. (2) cannot keep a constant in the wide temperature range, and the Grüneisen model is effective only below the critical HC (or at low temperatures). Above or below the critical HC, the experimental HC $C_{\rm p}$ exhibits linear behavior with the CTE $\beta$: $$ C_{\rm p} = C+E \cdot \beta,~~ \tag {3} $$ where $C$ and $E$ are constants. The parameters $C_{1}$ and $E_{1}$ (or $C_{2}$ and $E_{2}$) in Eq. (3) for tungsten are determined by linear fitting below (or above) the critical HC $C_{\rm p 0}$, respectively (the solid lines), and are listed in Table 1. The value of the critical HC $C_{\rm p 0}$ for tungsten, quantitatively determined by the intersection of the solid fitting lines (blue and green), is about 26.95 J$\cdot$mol$^{-1}\cdot$K$^{-1}$. Previous studies mainly focused on the correlation between the HC and CTE at low temperatures. In order to study the universal correlative behavior between the $C_{\rm p}$ and $\beta$ in solids, we survey the experimental heat capacity $C_{\rm p}$ and the coefficient of volumetric thermal expansion ($\beta$) in a series of reference solid materials.[9,17–29] The experimental data of the $C_{\rm p}$ and $\beta$ were measured at constant pressure, and were obtained in a wide temperature range from several Kelvin to melting temperature. The investigated solid materials include metals (W, Ta, Mo, Pt, Be, Cu, Al, and Hg), oxides (MgO and Al$_{2}$O$_{3}$), and inert monatomic solids (Xe, Kr, and Ar). Their melting temperatures are in the range from 3695 K to 84 K (as listed in Table 1).[30] The correlative behavior between the $C_{\rm p}$ and $\beta$ in the reference solid materials is shown in Fig. 2(a). The CTEs of inert monatomic solids are much larger than those of other solids. Figure 2(b) exhibits the correlative behavior between the $C_{\rm p}$ and $\beta$ for the reference solid materials at low CTE. Similarly to tungsten, all of the reference solids show an inflection in the $C_{\rm p}$ vs $\beta$ data. The critical HCs $C_{\rm p 0}$ at the inflection for the reference solids are close to 3$R$, and the HC exhibits linear behavior with the CTE above or below the critical HC, respectively. The linear behavior is different from the previous common belief: the quasi-parabolic relation by Eqs. (1) and (2).
Table 1. Thermodynamic parameters of some reference solid materials (mol: a mole of atoms): $T_{\rm m}$: melting temperature (see Ref. [30]); parameters $C_{1}$ and $E_{1}$ (or $C_{2}$ and $E_{2}$) obtained by the equation $C_{\rm p}= C+ E\beta$ ($C_{\rm p}$: the experimental heat capacity; $\beta$: the coefficient of volumetric thermal expansion) when $C_{\rm p}$ is below (or above) about 3$R$, respectively; $C_{\rm p 0}$: the critical heat capacity; $V$: the molar volume at room temperature (see Ref. [30]); $K$: the isothermal bulk modulus at room temperature (see Ref. [30]); $P_{2}$: the equivalent pressure (calculated by $E_{2}/V$).
Materials $T_{\rm m}$ $C_{1}$ $E_{1}$ $C_{2}$ $E_{2}$ $C_{\rm p 0}$ $V$ $K$ $P_{2}$ Refs.
(K) (J$\cdot$mol$^{-1}\cdot$K$^{-1}$) (10$^{6}$ J/mol) (J$\cdot$mol$^{-1}\cdot$K$^{-1}$) (10$^{6}$ J/mol) (J$\cdot$mol$^{-1}\cdot$K$^{-1}$) (cm$^{3}$/mol) (GPa) (GPa)
W 3695 0.24(6) 1.827(9) 10.69(14) 1.112(7) 26.95 9.47 310 117  [17]
Ta 3293 0.007(1) 1.3060(2) 6.322(6) 0.9752(2) 24.93 10.85 200 90  [18]
MgO 3125 0.17(12) 0.5914(41) 13.26(50) 0.2745(102) 24.60 5.52 162 50  [19]
Mo 2896 $-0.008$(30) 1.583(4) 11.10(7) 0.9142(25) 26.28 9.38 230 97  [20]
Al$_{2}$O$_{3}$ 2327 $-0.020$(7) 0.94085(7) 14.27(4) 0.4060(14) 25.12 5.14 252 79  [21]
Pt 2042 0.27(33) 0.9550(218) 8.88(36) 0.6007(89) 23.49 9.09 230 66  [22]
Be 1560 0.00006(19) 0.46509(1) 6.209(12) 0.3494(2) 24.96 4.85 130 72  [23]
Cu 1357 0.19(5) 0.4997(23) 12.78(48) 0.2394(73) 24.36 7.11 140 34  [17]
Al 934 $-0.11$(3) 0.3608(9) 14.17(87) 0.1618(96) 25.79 10.00 76 16  [24–26]
Hg 264 0.011(7) 0.2239(1) 19.96(6) 0.0510(4) 25.84 14.09 25 3.6  [9]
Xe 161 0.057(24) 0.04712(9) 17.43(9) 0.01391(9) 24.70  [27]
Kr 116 0.30(5) 0.0319(1) 17.35(13) 0.01041(9) 25.60  [28]
Ar 84 0.10(3) 0.02112(6) 12.95(7) 0.01004(4) 24.59  [29]
cpl-38-2-026501-fig2.png
Fig. 2. (a) Correlative behavior between the $C_{\rm p}$ and $\beta$ for the reference materials listed in Table 1. (b) $C_{\rm p}$ and $\beta$ for the reference materials at low coefficient of volumetric thermal expansion. The dashed line is the Dulong–Petit value 3$R$.
The parameters $C_{1}$ and $E_{1}$ (or $C_{2}$ and $E_{2}$) of the reference solid materials, shown in Table 1, are the fitting results by Eq. (3) below (or above) the critical HC, respectively. The values of the critical HC $C_{\rm p 0}$ for the reference solid materials, determined by the intersection of the fitting lines, are shown in Fig. 3. The average value of $C_{\rm p 0}$ (the blue line in Fig. 3, the blue dashed lines are the errors) is $25.17 \pm 0.91$ J$\cdot$mol$^{-1}\cdot$K$^{-1}$, which is close to the Dulong–Petit value 3$R$ (the red line in Fig. 3).
cpl-38-2-026501-fig3.png
Fig. 3. Critical heat capacity $C_{\rm p 0}$ for the reference solid materials, which is determined by the intersection of the linear fitting lines. The blue line is the average value of $C_{\rm p 0}$, the dashed blue lines are the errors, and the dot-dashed red line is the Dulong–Petit value 3$R$.
The parameters $C_{1}$ for the reference solid materials are close to 0, which are in agreement with the experimental results: both HC and CTE increase from zero at 0 K with the increasing temperature. The CTE is scaled by the parameter $E_{1}$, and all of the reference solid materials present the same linear behavior between the $C_{\rm p}$ and scaled $\beta$ below the critical HC $C_{\rm p 0}$, as shown in Fig. 4(a). Similarly, the CTE is scaled by the parameters $C_{2}$ and $E_{2}$, and the linear behavior between the $C_{\rm p}$ and scaled $\beta$ above the critical HC $C_{\rm p 0}$ for the reference solid materials is shown in Fig. 4(b). Therefore the linear correlative behavior between the experimental $C_{\rm p}$ and $\beta$ below or above the critical HC $C_{\rm p 0}$ is a universal property in solids.
cpl-38-2-026501-fig4.png
Fig. 4. Linear behavior between the $C_{\rm p}$ and different scaled $\beta$: (a) $E_{1}\cdot \beta$ and (b) $C_{2}+E_{2}\cdot \beta$, for the reference materials listed in Table 1. The dashed line is the Dulong–Petit value 3$R$.
Discussions. In metals, the heat capacity usually includes the contributions of the phonon (or lattice) and electron. The electronic heat capacity is linearly temperature-dependent at low temperatures, and is usually a small fraction of the phonon (or lattice). However, at very low temperatures, the situation reverses and the electronic specific heat becomes dominant.[31] The oxides and inert monatomic solids are insulated. The electronic heat capacity of insulated solids is zero. In this study, the reference solids include metals, oxides, and inert monatomic solids. Above several Kelvin, all of reference solids, including metals and insulated solids, show the same scaling behavior between the HC and CTE. Thus, the electronic heat capacity may be negligibly small in the temperature range, or the free electrons may induce the thermal expansion like the phonon (or lattice), and the electronic heat capacity and electronic thermal expansion also follow the scaling behavior. We define the equivalent pressure $P_{2}= E_{2}/V$, ($V$: the molar volume). Usually, the change of volume in solids from zero to melting temperature is less 5%. The values of $P_{2}$ in the reference solid materials are determined by the molar volume $V$ at room temperature, as listed in Table 1. The equivalent pressure $P_{2}$ are close to their isothermal bulk modulus $K$ at room temperature (also shown in Table 1). The equivalent pressure $P_{2}$, determined by the correlation between the $C_{\rm p}$ and $\beta$, is four orders of magnitude larger than the experimental pressure (one atmospheric pressure). Thus, compared to the equivalent pressure, the low experimental pressure weakly affects the heat capacity or thermal expansion of solids, and the scaling behavior between the $C_{\rm p}$ and $\beta$ in solids is still effective at atmospheric pressure or low pressure. Our result shows that there is a universal correlation between the experimental heat capacity and thermal expansion in solids at constant pressure. The thermodynamic variables can be predicted by the universal properties or Eq. (3). For example, the temperature-dependent thermal expansion or volume can be predicted by the heat capacity, and the temperature-dependent heat capacity or enthalpy can be predicted by the thermal expansion in solids. On the other hand, we can also predict the internal energy function at constant pressure. The HC is defined in thermodynamics: $C_{\rm p}={(\frac{dQ}{dT})}_{P}$ at constant pressure $P$ with $Q$ being the heat and $T$ the temperature. By the basic relation in thermodynamics at constant pressure $dQ=dU+ PdV$ ($dQ$: the change of the heat, $dU$: the change of the internal energy, and $dV$: the change of the volume), the temperature-dependent change of the parameters can be written as $({\frac{dQ}{dT})}_{P}=({\frac{dU}{dT})}_{P}+P({\frac{dV}{dT})}_{P}$. By $\beta =\frac{1}{V}\cdot {(\frac{dV}{dT})}_{P}$ and Eq. (3), the equation can be written as $$ \Big(\frac{dU}{dT}\Big)_{P}=C+\Big(\frac{E}{V}-P\Big)V\beta.~~ \tag {4} $$ Hence, we can obtain the internal energy function $U(T, V)$ at low constant pressure $P$ by Eq. (4) as follows: $$ U=CT+E\ln\Big(\frac{V}{V_{0}}\Big)-P(V-V_{0}),~~ \tag {5} $$ where $V_{0}$ is the molar volume at zero Kelvin, and $U=0$ at zero Kelvin. In quantum statistical physics, one of the central and most recognizable results is that the heat capacity of a solid increases from zero gradually up to near 3$R$ with increasing the temperature. However, the experimental heat capacity at high temperatures usually exceeds the theoretical value 3$R$, even is double value of 3$R$. For example, the heat capacities of W and Mo are about 48 and 53 J$\cdot$mol$^{-1}\cdot$K$^{-1}$ near their melting temperatures, respectively. The anharmonic effect of interatomic interaction is induced by the thermal expansion, and is used to explain the excess heat capacity.[1,32–34] The anharmonicity may govern many properties of solids, including the heat capacity, the thermal expansion and conductivity, phase transitions, melting, and so on. When the heat capacity is up to 3$R$, the phonon heat capacity is close to a constant. The heat capacity continues to increase with the thermal expansion. Thus, the anharmonicity can be quantified by the universal correlation between the heat capacity and thermal expansion in solids. In summary, we have, for the first time, found that the experimental $C_{\rm p}$ is linearly dependent on the $\beta$ above or below the critical HC $C_{\rm p 0}$ in the reference solid materials, and discussed the universal scaling behavior between the $C_{\rm p}$ and $\beta$. The average value of the critical HC $C_{\rm p 0}$ in solids is $25.17 \pm 0.91$ J$\cdot$mol$^{-1}\cdot$K$^{-1}$. The universal scaling behavior is effective at low pressure in both metals and insulated solids, and can be used to quantitatively determine the internal energy and anharmonicity in solids. 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