Applied Physics A

, 126:102 | Cite as

Caloric and isothermal equations of state of solids: empirical modeling with multiply broken power-law densities

  • Roman TomaschitzEmail author
  1. 1.ViennaAustria


Empirical equations of state (EoSs) are developed for solids, applicable over extended temperature and pressure ranges. The EoSs are modeled as multiply broken power laws, in closed form without the use of ascending series expansions; their general analytic structure is explained and specific examples are studied. The caloric EoS is put to test with two carbon allotropes, diamond and graphite, as well as vitreous silica. To this end, least-squares fits of broken power-law densities are performed to heat capacity data covering several logarithmic decades in temperature, the high- and low-temperature regimes and especially the intermediate temperature range where the Debye theory is of limited accuracy. The analytic fits of the heat capacities are then temperature integrated to obtain the entropy and caloric EoS, i.e. the internal energy. Multiply broken power laws are also employed to model the isothermal EoSs of metals (Al, Cu, Mo, Ta, Au, W, Pt) at ambient temperature, over a pressure range up to several hundred GPa. In the case of copper, the empirical pressure range is extended into the TPa interval with data points from DFT calculations. For each metal, the parameters defining the isothermal EoS (i.e. the density–pressure relation) are inferred by nonlinear regression. The analytic pressure dependence of the compression modulus of each metal is obtained as well, over the full data range.


Multi-parameter equation of state (EoS) Caloric EoS of carbon allotropes Specific heat of vitreous silica Thermal EoS and compression modulus of metals High-pressure regime Multiply broken power laws 

1 Introduction

The aim of this paper is to develop analytic equations of state (EoSs) for solids which can reproduce empirical data sets covering several orders in temperature and pressure, including the extended crossovers between the asymptotic low and high pressure and temperature regimes. The proposed EoSs are multiply broken power laws, which do not involve truncated series expansions in density, pressure or temperature (frequently used in empirical EoSs, cf. e.g. the reviews [, , , , ]) and are, therefore, equally suitable for the mentioned asymptotic regions.

At first, we study caloric EoSs, i.e. the temperature dependence of the internal energy, obtained by integration of the isochoric heat capacity. We work out three specific examples, obtaining closed analytic expressions for the heat capacities of diamond, graphite and vitreous silica by least-squares regression. The isochoric heat capacities are structured as multiply broken power laws [, , ],
with positive amplitudes b0 and bk, k=1,,n, and bk<<bk+1. The exponents βk, k=1,,n are positive, β0 is real, and the exponents ηk can be positive or negative. CV(T) is analytic on the positive real axis and consists of n+1 approximate power-law segments, Tβ0, Tβ0+β1sign(η1),,Tβ0+k=1nβksign(ηk) in the intervals T<<b1, b1<<T<<b2,,bn<<T, respectively. The amplitudes bk define the break points between the power-law segments and the exponents ηk determine the extent of the transitional regions. Power laws in log–log plots appear as approximately straight segments, and the exponents ηk determine the extent of the transitional regions between the power-law segments.

The isochoric heat capacities of diamond, graphite and vitreous SiO2, discussed in Sect. 2, are special cases of the broken power-law density (1). In the case of diamond, CV(T) interpolates between the constant Dulong–Petit high-temperature limit and the Debye T3 low-temperature slope. In the intermediate temperature range, the broken power law (1) is sufficiently adaptable to model deviations from the Debye theory such as the boson peaks of glasses. In the case of graphite, CV(T) interpolates between the linear low-temperature slope of the degenerate electron gas and the classical Dulong–Petit regime, and in the case of the SiO2 glass, the linear low-temperature slope is replaced by a power law with non-integer index close to one, generated by a fermionic two-level system. In the latter two examples, a phononic cubic temperature slope does not emerge in the empirical data sets, as the low-temperature regime is dominated by Fermi systems.

In recent years, new phenomenological EoSs have been proposed to model density–pressure data sets obtained from Hugoniot shock compression of, for instance, alkali metals [, ], alkaline earths [], aluminum [, , ], iron [], copper [], tin [] and tungsten [], to mention but a few. In the second part of this paper, multiply broken power laws are employed as isothermal EoSs of metals, with emphasis on the high-pressure (GPa and TPa) regime. In this case, the pressure dependence of the mass density ρ has the analytic shape
with positive amplitudes ρ0, bk, bk<<bk+1, and real exponents ηk, k=1,,n. These parameters are temperature dependent, but will be treated as constants in the examples studied here, at ambient temperature of 300 K. Density (2) consists of n+1 power-law segments, 1, Pη1,,Pk=1nηk, in the intervals P<<b1, b1<<P<<b2,,bn<<P. In contrast to the broken power law (1), we have put β0=0 and βk=|ηk| in (2), so that ρ(P) is analytic at P=0, which is essential to model the compression modulus in the low-pressure range consistent with ultrasonic measurements. When discussing the compression modulus in Sect. 3, we will also need the logarithmic derivative of density (2), ρ(P)/ρ(P)=k=1nηk/(P+bk), to show how the low-pressure limit of EoS (2) is related to the Murnaghan EoS [, , , , , ].

In Sect. 3, the EoS (2) will be put to test by performing least-squares fits to high-pressure data sets of Al, Cu, Mo, Ta, Au, W, and Pt, which cover an extended pressure range, from ambient (zero) pressure to several hundred GPa [, , ]. In the case of copper, the available experimental pressure range (up to 450 GPa []) will be extended with data sets obtained from DFT calculations []. The least-squares fit of the EoS of copper then covers pressures reaching 60 TPa. In Sect. 4, we present our conclusions.

2 Heat capacity, caloric EoS, and entropy of diamond, graphite and vitreous silica

In this section, the modeling of empirical heat capacity data by multiply broken power-law densities is discussed. Two carbon allotropes, diamond and graphite, and a glass, v - SiO2, are studied as specific examples. Explicit formulas for the molar isochoric heat capacities are obtained by nonlinear least-squares regression, cf. Figs. 1, 2 and 3. Comparisons with the Debye theory are depicted in Fig. 4. The temperature variation of the internal energy and entropy variables obtained from the regressed heat capacities is shown in Fig. 5. Alternative attempts to find caloric EoSs for diamond and graphite are discussed in Refs. [, , , ] and references therein.
Fig. 1

Isochoric heat capacity of diamond, cf. Sect. 2.1. Data points from Ref. [] (circles), Ref. [] (squares) and Ref. [] (diamonds). The χ2 fit (solid red curve) is performed with CV(T) in (3), the fitting parameters are recorded in Table 1. For comparison, the black dashed curve is the Debye approximation (4), with constant Debye temperature of θdiamond=2186 K. The classical 3R Dulong–Petit limit is indicated by the black dotted line. The green dotted straight line depicts the tangent cTκ at the inflection point (T=95.53K, CV=0.2085J/(mol K)), with slope κ=3.547 and amplitude c=1.972×108J/(mol K1+κ)

Fig. 2

Isochoric heat capacity of graphite, cf. Section 2.2. Data points from Ref. [] (circles, Madagascar natural graphite), Ref. [] (squares, Canadian natural graphite), Ref. [] (rectangles, Acheson graphite), and Ref. [] (diamonds, Acheson graphite). The experimental isobaric CP data points have been converted into isochoric CV points, cf. (6). The χ2 fit (red solid curve) is performed with the analytic broken power law CV(T) in (7), the fitting parameters are listed in Table 1. The dotted blue asymptote CVb0T is the low-temperature limit of the heat capacity of the electron plasma. The CV(T) curve has an inflection point at T=3.146K, CV=7.170×104J/(molK). The dotted green line is the tangent cTκ at the inflection point, with amplitude c=3.702×105J/(molK1+κ) and slope κ=2.586, which is the maximum slope attained. See also Fig. 4 for the Debye approximation

Fig. 3

Isochoric heat capacity of v - SiO2, cf. Section 2.3. Data points from Ref. [] (rectangles), Ref. [] (circles), and Ref. [] (squares). The least-squares fit (red solid curve) is performed with the broken power law CV(T) in (8), the fitting parameters are recorded in Table 1. The classical 9R limit is indicated by the black dotted line. The dotted blue low-temperature asymptote CVb0T1.22 depicts the heat capacity of the two-level Fermi system discussed in Sect. 2.3, which dominates the phononic heat capacity at low temperature. The tangent cTκ at the inflection point (T=5.240K, CV=2.480×102J/(mol K)) has slope κ=3.770 and amplitude c=4.815×105J/(molK1+κ)

Fig. 4

Comparison of the molar isochoric heat capacities CV(T) of diamond, graphite and vitreous silica. The blue, red and black solid curves show the broken power laws CV(T) in (3), (7) and (8), respectively, with fitting parameters in Table 1. (The fits are depicted in Figs. 1, 2 and 3). The dashed curves are the Debye heat capacities, cf. (4). The Debye temperature of diamond is θdiamond=2186K (cD=b0, cf. (4) and Table 1), inferred from the low-temperature limit CVb0T3 of the broken power law (3). The Debye temperatures θgraphite=438.7K and θv - SiO2=323.4K have been chosen so that the Debye heat capacity (4) intersects the fitted CV(T) curves at their inflection point, since the empirical heat capacities of graphite and v - SiO2 do not exhibit a T3 slope in any temperature range

Fig. 5

Caloric EoS U(T)=0TCVdT and molar entropy S(T)=0T(CV/T)dT of diamond, graphite and vitreous silica, cf. (5). The caloric EoSs (thermal component of the internal energy) are depicted as solid curves, the entropies as dashed ones. At high temperature, U(T)T and the entropy diverges logarithmically. The isochoric heat capacities in the integrands are defined in (3), (7) and (8), with fitting parameters in Table 1. The low-temperature slopes read U(T)T1+β0 and S(T)Tβ0, with exponent β0 in Table 1

2.1 Diamond

We perform a χ2 fit to the available data sets [, , ] of the diamond heat capacity, depicted in the double-logarithmic plot in Fig. 1, by employing a broken power law of type (1) as isochoric heat capacity,
with amplitudes b1<<b2. The exponents βi, ηi, i=1,2 are positive, and β2=3+β1, so that the classical Dulong–Petit limit CV(T)3R is recovered, with gas constant R=8.314 J/(K mol). This limit also requires amplitudes in (3) related by b0=3Rb1β1/b2β2. The low-temperature limit is CV(T0)b0T3; the units used are CV[J/(K mol)], b0[J/(K4 mol)], and bi[K].
Broken power laws composed of multiple factors (1+(T/bi)βi/|ηi|)ηi are quite efficient for data sets stretching over several decades in temperature, as demonstrated in the subsequent examples. In (3), we have three successive power laws, T3, T3+β1, T3+β1β2, in the intervals T<<b1, b1<<T<<b2 and b2<<T, respectively, which appear as approximately straight segments in log–log plots, and the exponents η1, η2 determine the transitional regions around the break points b1 and b2. The parameters b1, b2, η1, η2 and β1 obtained from the least-squares fit are recorded in Table 1. In contrast to the Debye heat capacity, cf. (4), the heat capacity of diamond exhibits an inflection point in the crossover region between the low- and high-temperature regimes, see Fig. 1; the tangent of CV at the inflection point is shown as green dotted line depicting the power law T3.547.
Table 1

Fitting parameters of the molar isochoric heat capacity CV(T) of diamond, graphite and vitreous silica, see Figs. 1, 2 and 3




v - SiO2


























































The listed parameters (amplitudes bi, exponents βi, ηi) define the multiply broken power laws (3) (for diamond), (7) (graphite) and (8) (vitreous SiO2) representing the empirical heat capacities. Some of these parameters are interrelated, cf. Sect. 2. b0 is in units of J/(K1+β0mol). Also recorded are the minimum of the least-squares functional χ2=i=1N(CV(Ti)CVi)2/CVi2 and the degrees of freedom (dof: number N of data points (Ti,CVi) minus number of independent fitting parameters). The standard error of the fits, SE=(i=1N(CV(Ti)CVi)2/N)1/2, and the coefficient of determination, R2=1i=1N(CV(Ti)CVi)2/(Nσ2), with sample variance σ2=i=1N(CViC¯V)2/N and mean C¯V=i=1NCVi/N, are listed as well

In Figs. 1 and 4, we have also indicated the Debye approximation,

The asymptotic limits of the Debye function are D(x>>1)π4/(15x3) and D(x<<1)1/3, so that CD(T0)cDT3 and CD(T)3na/mR, where na/m denotes the number of atoms per molecule. The units are cD[J/(K4 mol)] and CD[J/(K mol)] as above. The amplitude cD is taken from the least-squares fit of CV(T) in (3), cD=b0, cf. Table 1, to recover the cubic low-temperature slope. Accordingly, the Debye temperature (4) of diamond is θ=2186 K. The low-temperature amplitude cD is the only adjustable parameter of the Debye heat capacity, so that it is not surprising that the Debye approximation becomes inaccurate in the crossover region.

The temperature dependence of the caloric EoS (molar internal energy with zero-point energy subtracted) and molar entropy of diamond,
U(T)[J/ mol]=0TCVdT,S(T)[J/(K mol)]=0TCVTdT
is shown in Fig. 5, calculated by substituting CV(T) in (3) with fitting parameters recorded in Table 1.

2.2 Graphite

The heat capacity data of graphite tabulated in the experimental papers [, , , ] refer to the isobaric heat capacity CP. The conversion of isobaric (at ambient pressure) to isochoric heat capacities CV is done with the approximate formula CVCP/(1+0.526Tα(T)), where 0.526 is the dimensionless Grüneisen constant of graphite [] and α the thermal expansion coefficient perpendicular to the basal plane, cf. Refs. [, , ],

The units are α[1/K] and T[K]. The expansion α|| in the basal plane is negligible compared with α. In the case of diamond, cf. Sect. 2.1, the conversion to isochoric heat capacities has already been done in the experimental papers. In the case of vitreous silica, cf. Sect. 2.3, CVCP in the temperature range of the available data points.

The heat capacity of graphite has an electronic and a phonon component, cf. Fig. 2; at low temperature, the linear electronic heat capacity overpowers the phonon component. For the least-squares fit depicted in Fig. 2, we use a broken power law similarly structured as in (3):
where the factors are ordered by increasing magnitude of the amplitudes, b1<<b2<<b3, and the exponents βi, ηi, i=1,2,3, are positive. We put β3=1+β1β2 so that CV(T)3R, which also requires amplitudes related by b0[J/(K2 mol)]=3Rb1β1/(b2β2b3β3). The fitting parameters are bi, ηi, i=1,2,3, and β1, β2, cf. Table 1. The tangent at the inflection point of CV(T)[J/(K mol)] is depicted as dotted green line T2.586 in Fig. 2, and the electronic low-temperature asymptote CV(T)b0T is also shown in this figure.

There is no indication of a T3 slope anywhere to be seen in the data set in Fig. 2. Therefore, we choose the amplitude cD defining the Debye temperature in a way that the low-temperature T3 slope of the Debye curve CD(T) [stated in (4)] cuts through the inflection point of the empirical heat capacity CV(T), see Fig. 4. This gives a Debye temperature of θ=438.7 K [with cD=2.303×105J/(K4mol) in (4)]. A different choice of θ would just shift the T3 slope parallel to the depicted slope. It is evident from Fig. 4 that the standard Debye approximation (4) cannot give a reasonable fit to the heat capacity of graphite, irrespective of the choice of θ, except at very high temperature; see also Ref. [] and references therein for modifications of the Debye theory regarding graphite. The internal energy and entropy functions of graphite are shown in Fig. 5, obtained by integrating the empirical CV(T) in (7) (with fitting parameters in Table 1) according to Eq. (5).

2.3 Vitreous SiO2

For the heat capacity of vitreous silica [, , , , ], we use a broken power law similar to that of graphite:
where the positive amplitudes bi[K] are ordered by increasing magnitude, b1<<b2<<b3. The exponents βi and ηi, i=1,2,3, are positive, and we put β3=β0+β1β2 so that CV(T)9R, which also requires b0[J/(K1+β0 mol)]=9Rb1β1/(b2β2b3β3). The data points in Fig. 3 refer to the total heat capacity of the phonons and the fermionic two-level system which dominates the phonon heat capacity of the glass at low temperature [, ]. The low-temperature heat capacity of the two-level system is slightly different from linear, with power-law exponent of