# Thermodynamic Relationships for Perfectly Elastic Solids Undergoing Steady-State Heat Flow

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

_{P})/∂P = −1/B; c

_{P}= nΞ times thermal expansivity divided by density; c

_{P}= c

_{V}nΞ/B. Implications of our validated formulae are briefly covered.

## 1. Introduction

^{3}in an isotropic medium. The unit vector denotes the specific direction of heat flow. Because any matrix representation can be diagonalized, the one-dimensional Cartesian form on the right-hand side (RHS) embodies the physics of heat transport.

_{SB}= 5.670 × 10

^{−8}Wm

^{−2}K

^{−4}describes a blackbody (see Section 2.1). Temperature is thus defined by heat loss to the surroundings. In classical thermostatics, T is related to heat content Q, but not in a simple way, e.g., [6].

_{P}is specific heat (on a per mass basis). Thermal conductivity governs the thermal evolution of a system, embodying how much heat is flowing and how fast. When changes in T are small, Equation (3) simplifies to:

^{−1}∂V/∂T. This historical identity thus portrays the response of a static property to compression (P on the left-hand side, LHS) as arising from changes caused by heating (T and α on the RHS). Yet, diverse observations show that solids respond to heating and to compression in different ways, as embodied in the quasi-harmonic model of solids [8]. In particular, examining accurate experimental measurements of the P dependence of κ for 20 different solids that also had accurate material properties suggests that ∂(lnc

_{P})/∂P depends simply on the inverse of the isothermal bulk modulus, B

_{T}= −V(∂V/∂P)

^{−1}[7].

#### 1.1. Different Behaviors of Solids and Gases May Affect Thermostatic Equations

#### 1.2. Purpose and Limitations of the Paper

#### 1.3. Organization of the Paper and Key Results

## 2. Theoretical Description of Solids Conducting Heat in Steady State

#### 2.1. Link of Temperature to Heat Flux

_{B}is Boltzmann’s constant.

_{BB}and emissivity (ξ) is independent of both ν and T. Metals and graphite were used in classic experiments (Figure 2) because these strongly absorb and have optical functions that vary slowly with ν and T. Transparent material (e.g., silicate glasses) also have emissions, but these are related to I

_{BB}in a complicated manner that depends on the size of the object, absorption characteristics, surface reflections, and thermal gradients [11]. Gases are extremely transparent and were historically considered not to emit.

**Figure 2.**Emission curves of cavity radiation at 1370 K from Coblentz [12] compared to a near-IR absorption spectrum of natural fluorite (green curve, with an arbitrary y-scale). Dashed line = raw data, labeled “prismatic”. Solid curve with small dots = corrected data. Solid line with circles = the ideal Planck curve. Arrows indicate points Coblentz [12] used to fit the blackbody curve and determine the maximum. He omitted regions connected with atmospheric absorptions, in which features are partly due to use of natural fluorite as a prism, and in which material contains impurity bands.

#### 2.1.1. Wien’s Law

^{−1}was experimentally determined. The irrational number w

_{3}(~2.821439) on the RHS was derived from I

_{BB}(Equation (8)) by numerically solving a transcendental equation [13,14]. Thus, ascertaining T from Equation (9) implicitly assumes a broad and skewed spectrum of a greybody (Figure 2), whereby ν

_{max}differs from c/λ

_{max}.

_{peak}= c/λ

_{peak}with an intensity that is symmetric, or nearly so, about the characteristic frequency. Energy with a certain narrow frequency range is used to stimulate specific processes, e.g., laser light causes electronic transitions whereas sound waves cause low-frequency motions. However, heating a material requires redistributing the energy that is applied in some specified frequency range, which may be quite narrow, to the wide range of frequencies that comprise the thermal emissions of the material (Figure 2).

#### 2.1.2. Repercussions of Temperature Depending on Emitted Flux and Spectral Properties

- The hallmark of a hot dense body is that it emits heat over a wide spectral range (Figure 2). This unavoidable loss signifies that its state is dynamic, not static.
- Temperature governs the total flux emitted, with the following caveat:
- Because thermal emissions depend on the spectral properties of the material, Q may also depend on characteristics beyond the static physical properties considered in the historical model.

#### 2.2. Connection of Steady-State Behavior with Coincident Adiabatic and Isothermal Conditions

#### 2.2.1. Spherical Coordinates

_{surroundings}= ℑ

_{in}: Figure 3a). Steady state requires:

_{in}= ℑ

_{out}= constant

_{in}and κ are low, thus approaching large regions of constant T.

#### 2.2.2. Longitudinal Flow in Cylindrical Geometry and in Cartesian Systems

_{source}at x = 0 to T

_{sink}at x = L. It is immaterial whether the flux is radiatively applied (as in laser-flash analysis, LFA) used to measure D) or is supplied by electrical heating, or by contact with a hot plate. This equivalence has been amply demonstrated by benchmarking LFA against conventional heat transport measurements of metals, e.g., [15].

#### 2.3. Equations of State, Elastic Behavior, and Work

#### 2.3.1. Classical Definitions and Their Link to Mathematical Constraints

_{T}is the bulk modulus. Its P dependence is likewise specific to the material of interest. Their second-order cross-derivatives are interdependent:

_{P}the relevant parameter, whereas setting dT = 0 in Equation (15) makes B

_{T}the defining property. Thus, Equations (11) and (12) describe behavior along an isobar and isotherm, respectively. The above equations constitute the EOS of a material.

#### 2.3.2. Rigidity and Its Relationship to EOS Formulations for Solids

_{T}or B

_{aco}, is not tied to heat. For this reason, the shear velocities are unrelated to the thermal Grüneisen parameter [17] which connects B

_{T}with B

_{aco}in the historical model (Section 2.3.6).

_{11}, c

_{44}, and the off-diagonal element c

_{12}. Because only three parameters are needed for isotropic solids, the elastic moduli are related:

#### 2.3.3. Irrelevance of Friction to a Static Model and Implications for Work-Heat Relations

#### 2.3.4. Connection of the EOS with Perfectly Frictionless Elastic Behavior

_{0}or ρ

_{0}), plus knowledge of α(T), B(P), and either cross-derivative (Section 2.3.1). Features of perfectly frictionless elastic solids (PFES) are summarized as follows:

- The perfectly frictionless elastic approximation is static: time is not involved and systems are fully restorable. That is, the ideal system is reversible (Figure 4b), although in a real system changes are made via manipulating and changing the surroundings.
- Because reversibility of the system and an instantaneous response to changing conditions are central to the PFES approximation, adding heat to the system has no effect other than raising temperature, after which P and/or V respond, in accord with imposed experimental constraints and the EOS. The time-dependent nature of heat uptake (Section 2.3.5) explains why this is the driver of change.
- Independence of mass and heat (Figure 4) and conservative behavior require separate treatment of variables related to mass occupying space (i.e., the EOS and shear modulus, G, which governs shape) and to heat occupying space (i.e., the heat content Q, storativity C, or a specific heat). Yet, the latter three parameters may depend on the size of the box (V), and thus on P (or T) conditions, as well as on B (or α) which describe volumetric changes.

#### 2.3.5. Uptake of Heat during Frictionless Elastic Behavior

#### 2.3.6. Why Rigid Solids under Steady State Have One Bulk Modulus

_{S}in the historic model, where S is entropy. This equality is not true for gases, due to their lack of rigidity combined with heat being carried by the molecules during their translational motions.

_{S}. Since S is defined as Q/T in reversible experiments, Bs is referred to as the adiabatic bulk modulus. Because heat is irrelevant to elasticity experiments, we instead use the notation B

_{aco}, for acoustic bulk modulus, when referring to such data in Section 3.

_{aco}= B

_{T}, contrary to much literature, which posits that:

_{aco}= B

_{T}(1 + αγ

_{th}T), historic.

#### 2.3.7. Young’s Modulus and Work in a PFES

#### 2.4. Behavior of Heat in Perfectly Frictionless Elastic Solids during Steady-State Conduction

#### 2.4.1. Specific Heat Definitions

_{V}data for solids are lacking, we focus on c

_{P}.

#### 2.4.2. Incremental Responses for a PFES

#### 2.4.3. Pressure Derivatives of Specific Heat during Steady State

_{T}and the high T case where c

_{P}is nearly constant, which reduces Equation (28) to:

#### 2.4.4. Pressure Derivatives of Storativity during Steady State

_{P}leads to a strong dependence of C on B

_{T}:

#### 2.4.5. Temperature Derivative of Specific Heat during Steady State from Stefan’s Law

_{P}and V. Hence, to a high degree of accuracy, the solution to Equation (42) and thus to Equation (39) provides a new equation:

_{P}and found equality at low T but a linear dependence at high T [27,28,29,30]. Bodryakov and colleagues [27,28,29,30] explained the discontinuous behavior on the basis of vibrations being the main energy reservoir in a solid, and did not consider elastic energy. Our derivation of Equation (43) suggests continuous behavior, but we have not yet incorporated the rigidity of solids.

#### 2.4.6. Heat Uptake Provides Non-Dissipative Work

_{1}(T) describes the process of thermal expansion. When T is low, the solid is stiff because the bond lengths are small and bonding is strong. As T rises, the bonds lengthen and weaken. At high T, with weaker bonding, the same increment of Q added as at low T should cause greater expansion. Clearly, the structure of the solid should affect the function c

_{1}.

_{P}, and ρ describe the bulk solid, so the structure is immaterial to these measurable quantities. The desired quantity, F, is related to Ξ, the number of atoms, and the number of bonds around each atom (i.e., atomic coordination of the structure). For example, diatomics have 2 atoms which share 1 bond, so F is proportional to Ξ/2. The same holds for the monatomic diamond structure, for which each atom is bonded to 4 others, mutually. Monatomics with the bcc structure have 2 atoms in the unit cell, which are bonded to 8 others, which double counts the bonds: thus F is proportional to 2Ξ/4. The 4 metal atoms in an fcc unit cell have 12 nearest neighbors, again double counting, so F is proportional to 4Ξ/6. Corundum has Al cations which are 6-coordinated, so Ξ/3 describes the force per cation. For the polyatomics with multiple sites, and given the above assumption of spherical atoms, F is estimated as being proportional to Ξ times the number of cations (N) divided by the number of atoms in the formula unit (Z):

#### 2.4.7. Ratio of Specific Heats

_{V}is not measured for solids.

## 3. Evaluation of New and Old Formulations via Comparison with Experimental Data

#### 3.1. Comparison of Bulk Moduli from Acoustic and Volumetric Studies

#### 3.1.1. Techniques

_{T}/∂P to 0 at P = 0 [32]. Rather, values for instantaneous derivatives depend on the accuracy with which V and P are measured, the spacing in P between data acquisition points, and the absence of deformation.

_{T}

_{,0}) and 1st order derivative (B′ = ∂B

_{T}/∂P), suffice to delineate V(P). Large ranges in pressure are needed to establish the latter parameter, because it is the 2nd order pressure derivative of V. Additionally, uncertainties increase with P. Hence, B′ = 4 is commonly assumed. Although applying a certain form for the EOS is useful for comparisons, this approach introduces uncertainties by restricting parameter space. Convolution of B

_{T}

_{,0}with B′ in EOS fits is a mathematical consequence of using only these two coefficients.

#### 3.1.2. Bulk Moduli for Solids at NTP

_{aco}. The calculated difference depends strongly on α-values near NTP, which are well-constrained for metals [37] and fairly large. Although the historical correction term of 1.6% is close to the experimental uncertainty in bulk moduli for individual metals, it is larger than the uncertainty of 0.5% of the fit for these 36 metals (see insets in Figure 5a). On average, the historical correction is unnecessary.

_{aco}= B

_{T}. Applying historic Equation (23) to B

_{aco}predicts that bulk moduli should be only 0.6% lower than the trend in the data: this correction term is small because silicates and oxides have low α. Incompressible diamond (elemental C) and stishovite (SiO

_{2}with the rutile structure) greatly influence the fit. Because α is low for insulators, little difference exists between data and the historic prediction, Equation (23).

**Figure 5.**Comparison of data on bulk modulus from compilations of data from different experimental techniques. The x-axes depict XRD results from [32]: (

**a**) metallic elements. Elasticity data (color points and line) mostly from [36]; supplemented by data on Pb and In [38] and Zn [35]. Calculations use γ

_{th}from [36]; recommended values of α from [37]; and c

_{P}from [39]; (

**b**) electrical insulators and the non-metallic elements Si and C. Elasticity data on Si from [40]: otherwise from [41]. Additional XRD data, e.g., on BaF

_{2}, from [42,43,44]. Calculations use γ

_{th}and α from [45].

_{aco}< B

_{T}from volumetric studies (Figure 6a), which is the opposite of the historic predictions, Equation (23). For metals, B

_{aco}tends to be slightly larger, whereas combining all data from Figure 5 provides an average difference very close to zero. Symmetry of the profile about a negligible difference (Figure 6a) points to a statistical origin for differences in bulk moduli measured on the same material with different techniques.

_{T}and B

_{aco}at NTP are smaller for metals than for insulators. These findings underscore that differences in bulk moduli at NTP for the ~100 samples in the compilations, many of which were measured multiple times, are caused by experimental uncertainties. Figure 5 and Figure 6 support our model.

#### 3.1.3. Uncertainty in Bulk Moduli Arising from Fitting Volume vs. Pressure

_{aco}with a slope of unity (Figure 7a). Substantial differences exist in ∂B/∂P for the two types of fits [31] (their Table 5). As shown below for lead, a 3rd order polynomial is needed, but P = 4.5 GPa is insufficient to constrain curvature for most metals. This is underscored by measurements of tungsten [47] for which V depends linearly on P. Thus, using an EOS for W is an inaccurate representation. Discrepancies in Figure 7a for B > 130 GPa are attributed to both curvature in V(P) being too small for accurate fitting at high B, and also the trend of being highly influenced by uncertain B of incompressible W.

_{T}

_{,0}= 45.5 ± 0.5 GPa. Results from Schulte and Holzapfel [48] are not included because a table of volumes was not presented and resolution of the points on their figures was insufficient for accurate digitization. They applied a two-parameter EOS to their own and previous data, yielding B = 42 ± 5 GPa with individual studies ranging from 39 to 51 GPa. All fits cluster about 40 to 42 GPa. Figure 8b omits this average because shockwave data were included by [48]. We excluded fits to both fcc and bcc phases.

_{T}

_{,0}. A key factor is the maximum pressure obtained. When the full stability field for lead is used, EOS fits with two parameters, give lower values for B

_{0}than fits to a 3rd order polynomial, which uses three parameters. The constraint of V/V

_{0}= 1 is not included in the free-parameter count, as this is fixed in all approaches.

_{T}

_{,0}requires meeting several conditions: dense spacing of points, volumetric data over a wide range of pressures, accurate (or at least consistent) determination of pressures, and using a fit with three parameters or more (in addition to V

_{0}).

#### 3.1.4. Comparison of Acoustic to XRD Determinations of ∂B/∂T for Solids

_{aco}does not always exactly equal B

_{T}at NTP (Section 3.1.1, Section 3.1.2 and Section 3.1.3). Therefore, we compare values of ∂B/∂T, which has a negative sign.

_{T}/∂T is negative, volumetric measurements should give a stronger response to T than elasticity measurements.

^{−1}for CsCl which is larger than, but similar to, ∂B

_{aco}/∂T = 5%K

^{−1}(Figure 9). For LiF and NaF, Equation (51) gives 4.8 and 4.2%K

^{−1}, respectively, which are smaller than ∂B

_{aco}/∂T = −10.6 and −6.9%K

^{−1}, respectively (Figure 9). Yagi’s [55] measurements of volumes provided similar ∂B

_{T}/∂T, rather than values about half the size of ∂B

_{aco}/∂T. The historic model is not supported.

_{T}at 298 K, with the EOS being as predicted by historic Equation (23), but not their 80 K value, and so reevaluated their data with an untested cryogenic calibration, attributed to in a personal communication, which yielded the desired historic result. As shown in Figure 8a, the EOS analysis of lead volumes at low P underestimates the bulk modulus, so their fitting approach only appears to agree with this historic adjustment. Rather, fitting lead volumes over the stability range of its bcc phase to a high-order polynomial agreement with B

_{aco}, and do not require amending via Equation (23). As demonstrated for the alkali halides, bulk moduli trends with T for lead from volumetric and acoustic techniques are parallel, and so the historic correction is refuted.

_{aco}than with the cryogenic volumetric studies, except for Na. The historic correction at 298 K exceeds or matches the difference between the various measurements, and thus agreement of absolute values involves random experimental uncertainties as is evident from compiled data (Figure 5, Figure 6 and Figure 7).

#### 3.2. Response of Heat Capacity at NTP to Compression

_{P}(P) measurements exist.

#### 3.2.1. Static Compression Techniques

_{P})/∂P [65] (their Figure 7) and [66] (their Figure 3). Uncertainty for the reported value is substantial and cannot be less than ~10% uncertainty for the change in resistivity with P, e.g., [67].

#### 3.2.2. Dynamic Compression Techniques

_{P})/∂P in two different ways. First, from Equation (5):

_{P})/∂P by difference whereby uncertainties of the terms sum.

_{P})/∂P obtained by difference uncertain by ±20%. Figure 11 omits measurements of three samples: Gd melts very close to NTP; Zn has a hexagonal structure and the orientations differed in the D and κ experiments; whereas results on garnet gave positive ∂ln(c

_{P})/∂P, which is unexpected, and is probably due to large uncertainties in small derivatives for this hard insulator.

_{T}to calculate ∂ln(c

_{P})/∂P from the LHS of Equation (36) instead of the EOS approach as used by authors.

**Figure 11.**Graphs showing the response of storativity and c

_{P}to pressure: (

**a**) dependence on the inverse of B; (

**b**) Direct dependence on B. Grey diamonds and grey dashed line = directly determined storativity: sources = [70,71,72,73] where the error bar is from Gerlich and Andersson [70]. Black squares and solid line = specific heat from C, where circles = data where C did not discernably depend on pressure. Open cross and red dotted line = metal c

_{P}directly measured by calorimetry [65,66]. Aqua triangles = heat capacity obtained by difference (sources: [69,74,75,76,77,78]). Green short dashed line = ideal correspondence.

#### 3.2.3. Relationship of the Pressure Response of Specific Heat and Storativity to Bulk Moduli

_{P})/∂P decreases roughly linearly with B

^{−1}(Figure 11). Values for the slope vary with the technique (calorimetric or dynamic). The slope is uncertain, due to 10 to 20% uncertainties for the various approaches and the fact that an EOS is used to process storativity, which adds uncertainty—basically, this is also a difference approach. All data from all approaches combined (not shown) give a slope of about −1 or −100%. This slope is consistent with Equation (34), which shows that compression of the lattice controls the response. Within experimental uncertainty, the energy density is independent of pressure.

#### 3.2.4. Evaluation of the Historic Relationship of the Pressure Response of Specific Heat to Thermal Expansivity

_{P})/∂P, on average, responds strongly to compression whereas the correlation with historic Equation (6) is poor. The existence of a rough link is attributable to compressible solids also having large α; see, e.g., Anderson and Isaak [54].

#### 3.3. Connection of Thermal Expansion to Heat Uptake and Internal Strength

_{P}< Ξ < α. Because thermal expansion is small and measured as a response to T, values are impacted by the measurement range and fitting procedures, parallel to the limitations in determining B

_{T}(Section 3.1.1).

#### 3.3.1. Ambient Temperature

_{P}to the ratio ρ/Ξ without considering effects of structure after Equation (45). Agreement is reasonable for the monatomic elements, but with considerable scatter. This could be due to ~25 elements having N/Z = 1, but being anisotropic, as discussed below. The correlation for insulators is linear, with a slope differing from unity predicted by Equation (45). Its value of nearly ½ is as expected from our structural analysis of the interatomic forces (Section 2.4.6).

_{P}for ductile Nb and Ta but agrees with α/c

_{P}for brittle Mo and W. Shear being important means that some of the heat energy goes into deforming rather than solely expanding the lattice: consequently, α/c

_{P}is overestimated.

_{P}is increasingly overestimated by Equation (46). Positive discrepancies (overestimation of the energy supplied towards expansion) are associated with transverse strain being large compared to longitudinal strain. Thus, deformation accounts for departures of individual metals from the trends established for each of the fcc and bcc structures, but it does not account for their different trends.

#### 3.3.2. Temperature from a Few Kelvins to Nearly Melting

_{P}(T) averaged many data sets [27,28,29,30], which removes random errors. Because systematic errors also exist, we compare individual data sets in Figure 16a which should accurately represent each of α(T) and c

_{P}(T). Evaluating the temperature dependence of Equations (43), (45), or (46) further requires accurate data on Ξ(T). Fortunately, comparing rather few samples suffices because specific heat depends similarly on T for diverse materials, both simple (e.g., [24]) and complex [84]. Likewise, solids expand similarly as temperature climbs: for details, see Appendix A. Similar behavior of Ξ with T for different substances has also been observed (Figure 16b), leading to common use of the formula:

_{P}as a function of T for Al, Fe, Mo, Ta, Au, diamond, Si, MgO, Al

_{2}O

_{3}, Y

_{2}Al

_{3}O

_{12}, NaCl, and KCl.

_{P}above 200 K, where data on Ξ exist. As T further increases, α increases more strongly with T than does c

_{P}, such that the proportionality factor c

_{1}in Equation (43) grows non-linearly with T at very high T.

_{P}(T) was also observed over the Curie point of Fe (Appendix A), we propose that heat energy goes into expanding the lattice when no other process exists that can uptake the increment applied. In Fe, the additional process is electromagnetic. Section 3.3.1 argued that deformation likewise diverted heat-energy from thermal expansion. From both observations, we suggest that the process in Si involves electronic state changes. This hypothesis could be tested against impurity content for Si and Ge.

_{P}depends on T. Its derivative with T (the slope) depends on Ξ near 298 K, in accord with Equation (45). Trends are flat and similar for materials with very high Ξ. The slope steepens as Ξ decreases. Density and Young’s modulus together affect the low T intercept of α/c

_{P}. The behavior exhibited in Figure 16a supports the findings of Section 3.3.1.

_{P}correlate reasonable well with ∂Ξ/∂T for diverse materials (cf. Figure 16a,b). Insulators include extremely tough diamond, three incompressible oxides with varying structural complexities, and two soft alkali halides. Bonding ranges from ionic to covalent. Bass’s [41] summary table shows that the T derivatives of elastic properties vary considerably among studies of the same material. Non-linearity of the response contributes. Hence, uncertainties in ∂Ξ/∂T are substantial. On this basis of large experimental uncertainties, and because density changes with T are even smaller, ambient ρ was considered in Figure 16b and Figure 17.

## 4. Discussion and Implications

^{4}dependence of flux.

_{V}. Testing many different solids required use of compilations, which introduced uncertainties. Nonetheless, available data show that for solids:

- Only one bulk modulus exists, so the historically alleged difference between acoustic and volumetric moduli is unsupported. Likewise, the isothermal and adiabatic values for the 2nd Grüneisen parameter (Equation (14)) must be identical.
- Changes in heat content with pressure are controlled by the compressibility, which dominates changes in specific heat at moderate laboratory temperatures.
- Changes in heat content with temperature are described by specific heat by definition. Specific heat and thermal expansivity are linked, as the process of increasing V involves overcoming the elastic, tensile forces within the solid. Deformation solely occurs as shape changes arising from shear stresses uptake energy without expansion, confirmed by comparison of results from Equation (46) to Poisson’s ratio for cubic solids. If heat stimulates other processes, expansion is reduced as in Fe, or even reversed, as in Si.

#### 4.1. Heat Storage Reservoirs and Permissible Exchanges of Energy

#### 4.2. Key Variables

#### 4.3. Reservoirs vs. Historic State Functions

## 5. Conclusions

_{P}= Ξc

_{V}, it is apparent that their difference lies in whether pressure is externally controlled, or whether the resistance to heating is internal to the solid. Furthermore, we show that isothermal and isentropic (adiabatic) compressibilities are identical, which is consistent with thermal expansivity taking on one value (isobaric) and isentropic conditions not being germane.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{P}data exists at cryogenic temperatures. At high T, various studies report ~1–2% accuracy, yet comparisons of data sets (e.g., [27,28,29,30]) show a wider spread of 5%.

_{P}(T) data are commonly reported. Either a Debye model or one of various multiple term expressions is used. Many studies fit thermal expansivity to a formulation after Grüneisen, which also uses the Debye temperature. We sought studies with tabulated data.

_{P}with T is clear from Figure A1: both equal 0 in the limit of 0 K, thereafter increasing as ~T

^{3}, the increase of which then weakens with T, resulting in a “knee” at modest temperature and a ~linear increase at high T, which commonly steepens at very high T. Accuracy is required to resolve the gradual change in slope at very high T. In many substances, a “sway” exists due to the steepening at high T. The “knee” is always prominent, but when many data sets are shown together, the sway can be obscured. Plots of α above 298 K for 17 different metals [94,95], which were considered Touloukian et al.’s [37] and Gray’s [96] preferred values, show the sway, usually in both representations. Nb and Os do not show a sway, whereas for 7 additional metals, either the sway was obscured by a phase transition or temperatures accessed were too low for its detection. To fit α, Zhang et al. [95] used two Debye temperatures. The fits are reasonable, but do not match both the knee and the sway.

_{P}, much data are collected near ambient T, so the knee is well-established. Very high T data are less commonly explored. Yet, the sway is observed in many studies.

**Figure A1.**Comparison of properties describing the response of solids to heat additions. Left axis: black curves and points = volumetric thermal expansivity. Right axis: grey curves and points = specific heat. Scales were chosen to best match α and c

_{P}at cryogenic T. Properties are at ambient conditions, taken from various compilations. (

**a**) Aluminum. Squares = α [97]. Diamonds = recommended α [37]. Grey solid curve = c

_{P}compiled and evaluated by Desai [98]. Dashed = laser-flash calorimetry [99]. (

**b**) Iron. Thick vertical bars mark structural phase transitions. X = capacitance measurements of α [100]. Squares = dilatometry [101]. Diamonds = recommended c

_{P}[39]. Grey curve = c

_{P}compiled and evaluated by [102]. (

**c**) Molybdenum. + = recommended fit to α [37]. Circles = XRD results compiled and evaluated by Wang and Reeber [103]. Squares = transient interferometry [104]. Diamond = dilatometry data [105]. Grey curve = c

_{P}compiled and evaluated [106]. (

**d**) Tantalum. Diamonds = capacitance measurements of α [107]. + = fit to recommended values [37,96] by [94]. Squares = transient interferometry [108]. Light grey line = cryogenic calorimetry data [109]. Grey long dashes = laser flash calorimetry [110]. Grey short dashes = pulse calorimetry [111]. Dark grey line = pulse calorimetry [112]. (

**e**) Gold. Black dots = tabulated α [113]. Triangles, α obtained by differentiating tabulated volumes of Pamato et al. [114]. Thin line = α from 2nd order polynomial fit to V [114]. Open diamonds = recommended fit to α [94]. Squares = dilatometry and XRD data from Suh et al. [105]. Thick grey line = raw c

_{P}data [115]. Dotted line = mid-range of adiabatic calorimetry data [116]. Grey squares = pulse calorimetry [117]. (

**f**) Diamond. Black diamonds = α from Slack and Bartram [117], who combined 10 XRD studies of large natural crystals. Thin curve = recommended α [37]. Solid grey curve = c

_{P}[118]. X = DSC [119]. Dots = drop calorimetry of Victor [120], who stated air leakage occurred for the highest T points. Square with cross = Weber [121], who heated his samples in air. Dashed line = modulated DSC data [122], which are not absolute. (

**g**) Si. Black curve = recommended α [123]. Squares = single-crystal α [124]. Grey curve = c

_{P}compiled and evaluated [102]. (

**h**) Alkali halides. Solid lines = NaCl data: black = α [80]; grey = c

_{P}[81]. Dashed lines = compiled KCl data [54]. (

**i**) MgO. Solid curve, α as tabulated in [125] which has an inflection point (arrow) at 1000 K instead of a sway. The kink may be exaggerated, due to low and high T segments probing crystals and ceramics, respectively. Open squares = cryogenic data [126,127]. Short dashes = 2nd order polynomial fit to tabulated XRD data [93], acquired using an Ir wire heater. Grey dots = c

_{P}from [128]; triangles from [129], obtained by differentiating heat content; solid = Chase’s [130] review, where the high T trend is an extrapolation. (

**j**) Al

_{2}O

_{3}. Circle = α from powder XRD compiled and evaluated [131]. + = α compiled and evaluated [132]. Thin line = linear description of high T powder XRD [93]. Squares = single-crystal interferometry and twin telemicroscope measurements [133]. Grey curve = c

_{P}compiled and evaluated by [134]. (

**k**) Yttrium aluminum garnet. Diamonds = interferometry of a single-crystal [135]; squares = transparent polycrystal [136]. Black curve = from XRD [137]. Grey curves = DSC data: solid = [138]; dashed = [139].

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**Figure 1.**Summary and comparison of the characteristics of solids and gases most relevant to heat and its flow. The shear modulus, G, describes a special type of stored energy in solids, which is part of the elastic energy, the main reservoir. Atoms are shown as balls, with dotted arrows indicating direction of long-distance motions. Sine waves without arrowheads indicate local, back-and-forth, microscopic motions.

**Figure 3.**Schematics of conditions: (

**a**) Spherical symmetry, which also applies to radial flow in a very long cylinder. Matter (grey circle) emits heat in accord with its temperature (orange squiggle arrow), but emissions are actually sampled from a surface boundary layer (stippled green shell). Constant flux is maintained either by a source (star) and/or externally (blue arrow); (

**b**) Longitudinal flow in Cartesian (or cylindrical) symmetry. At steady state, flux along the special direction is a constant that is independent of position, so the axial thermal gradient is independent of time, and perpendicular slices are isothermal.

**Figure 4.**Schematics of an ideal, perfectly elastic solid: (

**a**) any given volume can contain a quantity of mass, and can independently contain some quantity of heat-energy; this independence underlies our model; (

**b**) Essence of elastic behavior. Squeezing (increasing pressure) changes V, and thus does P-V work, but does not generate heat so T is unchanged. Upon release of pressure, a perfectly elastic frictionless solid returns to its initial volume. See text for discussion of shear and shape changes; (

**c**) Receipt of small amounts of heat by a PFES. Within a short, but finite, distance, the pulse encounters vibrating ions. When energy of the applied light matches some transition energy, the affected vibrations become excited, attaining a higher energy state (e.g., an overtone). Subsequent interchanges give an overall higher vibrational energy of the collection, which imparts a higher temperature. Both steps take time.

**Figure 6.**Statistical presentation of the data from compilations. See Figure 5 for literature sources. Light grey = metals; dark grey = insulators and Si. Arrows point to various mean values: (

**a**) histogram of the difference between elasticity and volumetric measurements of bulk moduli, in percent; (

**b**) histogram of the product αγT at 298 K. Expansivity data were found for 39 of the insulators that had both types of bulk moduli measurements.

**Figure 7.**Comparison of different measures of metal bulk moduli. Length-change measurements from [31,46,47] were fit to EOS by the authors and to 2nd order polynomials here. Most acoustic determinations are from compilations listed in Figure 5. Vaidya and Kennedy [47] provide additional acoustic data: (

**a**) direct comparison. Both linear and polynomial fits are fit for tungsten, because curvature in V(P) was not resolved. Gold was not measured, but the other noble metals have relatively large B

_{T}; (

**b**) inverse comparison. The four softest metals were excluded because these not only needed 1–3 more terms for accurate fitting, but more importantly, the sigmoidal dependence of their V on P indicated deformation. We did not fit the initial slope because the lowest P data may be affected by slight deformation.

**Figure 8.**Lead volumes and bulk moduli, mostly from DAC studies: (

**a**) polynomial fit combines results from [31,49,50,51]. Double arrows denote pressure ranges. XRD experiments probed the whole stability field (to 16 GPa) but with few data points. Dotted curve = the 2nd order fit to V, where + = the corresponding B(P). Inset lists the 3rd order fit to V vs. P (solid curve), with filled squares for the resulting B(P), which is fit to the listed 3rd order polynomial. This fit gives slightly higher initial B than calculation; (

**b**) temperature dependence of bulk moduli. Diamonds = B

_{aco}(grey from [52]; black from [53]). Square in circle = result from panel a. Open squares and various triangles = several fits to neutron diffraction data [51], as labelled. Other open symbols = reported EOS values of [31,49,50,51].

**Figure 9.**Bulk modulus of alkali halide as a function of temperature. Blue curves = EOS fits of Yagi [55] to his XRD data on LiF, NaF, the low-P B1 phase of KF, and CsCl with the B2 structure. Numbers in parentheses denote previous work cited by [55]. Red squares and “VK” = length change data [57], where too few data collections were made on KF to provide a reliable B

_{T}. Broken curves = acoustic data compiled by Yagi [55], where his references 19 and 20 are incompatible with other studies. For example, Hart [58] confirmed B

_{aco}(T) from curve 22 for NaCl, i.e., the work of Jones [59]. Modified after Yagi [55] (his Figure 8) with permission.

**Figure 10.**Temperature dependence of bulk moduli for alkali metals. Filled symbols = acoustic data of [61,62,63]; grey represents previous work cited therein. Open symbols = volumetric (XRD) studies analyzed using simple forms for the EOS [64]. Open cross = length-change data [46], which are closer to acoustic results than to B from XRD. Otherwise, squares show various data on Na; circles for K; and diamonds for Rb. Arrow at 298 K shows the historic Equation (23) applied to XRD data.

**Figure 12.**Comparison of the measured P response of specific heat to the thermostatic formula (6), which is peculiarly based on thermal expansivity describing compression. The difference method (blue triangles) provides a cluster of points, and so was not fit. Red = direct calorimetry measurements. Green dashed line = 1:1 correspondence, for reference. Circles = materials for which storativity was not discernably affected by compression. Open cross = metals, by calorimetry. Black line = fit to the scattered dynamic measurements. Data sources listed in Figure 11.

**Figure 14.**Dependence of α/c

_{P}on ρ/(ΞN/Z). Literature sources of data on elements are in Figure 13. For the insulators, tables of [54] were used, where Co

_{2}SiO

_{4}was omitted because α was estimated. Fits are least squares and are labeled with the number of solids in each category: (

**a**) insulators and cubic fcc metals. Lead strongly influences the slope due to its softness, as shown by the two fits. Iridium has little influence as it is near a cluster of points. Orthorhombic Fe

_{2}SiO

_{4}has a shearing transition whereas α for orthorhombic Mn

_{2}SiO

_{4}is unconfirmed; (

**b**) cubic bcc and hexagonal hcp metals. Outliers Li and Be have very small cations and few valance electrons.

**Figure 15.**Measures of discrepancy of the data from (46) as a function of Poisson’s ratio. Data on μ from [41,82]; see Figure 14. Fine line = ideal match. Dotted line and circles = bcc. Thick line and squares = fcc. Diamonds = hcp: (

**a**) difference = {ρ/(ΞN/Z) − α/c

_{P}}/(α/c

_{P}) in percent; (

**b**) ratio of α/c

_{P}divided by ρ/(ΞN/Z).

**Figure 16.**Evaluation of Equations (43) and (46) at high T for well-studied solids: (

**a**) dependence of α/c

_{P}on temperature. See Appendix A and Figure 13 and Figure 14 for data sources. Jumps in Ta curve result from data-combining studies. The graph begins at 200 K as cryogenic data were previously shown to closely correspond [27,28,29,30]; (

**b**) dependence of ρ/Ξ with the structural factor on T. Constant ambient ρ was used due to uncertainties in Young’s modulus. Measured data on Ξ from [85,86,87,88,89,90,91]. For Au, Fe, MgO, NaCl, and KCl, we used T derivatives near and above 298 K for B and G from [41] to compute dΞ/dT.

**Figure 17.**Comparison of the temperature dependence of the RHS and LHS of Equation (43). The effect of structure is not included. Data for these solids are described in Figure 16 and Appendix A.

New Formula | Theory | Experimental Confirmation |
---|---|---|

B_{T} = B from elasticity measurements | Section 2.3.2 and Section 2.3.6 | Section 3.1 (ambient and elevated T) |

${\frac{1}{{c}_{P}}\frac{\partial {c}_{P}}{\partial P}|}_{T}\simeq -\frac{1}{{B}_{T}}\equiv \frac{1}{V}{\frac{\partial V}{\partial P}|}_{T}$ | Section 2.4.2 | Section 3.2 (ambient T) |

$\frac{\alpha}{\rho {c}_{P}}\propto \frac{1}{\mathrm{Young}\u2019\mathrm{s}\text{}\mathrm{modulus}}$ | Section 2.4.6 | Section 3.3 (ambient and elevated T) |

**Table 2.**Dependence of energy reservoirs on the state of matter and the complexity of its atomic constituents.

Type | Motion | Solids | Gases | ||
---|---|---|---|---|---|

Manifestation | Energy Storage | Manifestation | Storage | ||

monatomic | Displacements parallel to path | Longitudinal acoustic mode | Longitudinal stress/strain ^{1} | Translational K.E. | Heat |

Displacements perpendicular to path | Transverse acoustic modes | Transverse stress/strain ^{1} | n/a | n/a | |

Electron-cation dipoles | Optical continuum | Heat | Collisions | n/a ^{3} | |

polyatomic | Longitudinal | Longitudinal acoustic mode | Longitudinal stress/strain ^{1} | Translational KE | Heat |

Transverse | Transverse acoustic modes | Transverse stress/strain ^{1} | n/a | n/a | |

Electron-cation dipoles | Optical continuum | Heat | Collisions | n/a ^{3} | |

Cyclical, tiny ^{2} | Optical modes | Additional heat | Internal modes | Heat |

^{1}For solids, these together compose elastic storage of energy in tension-compression and shear, respectively.

^{2}These internal motions and energies are in addition to those described for monatomics above, but are also found in certain monatomic structures such as Raman modes (diamond and hcp metals). Although Raman modes do not directly absorb light, their overtone/combinations do.

^{3}Presumed to be brief and conservative in the historical model.

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**MDPI and ACS Style**

Hofmeister, A.M.; Criss, E.M.; Criss, R.E.
Thermodynamic Relationships for Perfectly Elastic Solids Undergoing Steady-State Heat Flow. *Materials* **2022**, *15*, 2638.
https://doi.org/10.3390/ma15072638

**AMA Style**

Hofmeister AM, Criss EM, Criss RE.
Thermodynamic Relationships for Perfectly Elastic Solids Undergoing Steady-State Heat Flow. *Materials*. 2022; 15(7):2638.
https://doi.org/10.3390/ma15072638

**Chicago/Turabian Style**

Hofmeister, Anne M., Everett M. Criss, and Robert E. Criss.
2022. "Thermodynamic Relationships for Perfectly Elastic Solids Undergoing Steady-State Heat Flow" *Materials* 15, no. 7: 2638.
https://doi.org/10.3390/ma15072638