SEARCH

SEARCH BY CITATION

Keywords:

  • CALPHAD;
  • crystalline phases;
  • first-principles;
  • thermodynamics

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

Progress in materials science through thermodynamic modelling may rest crucially on access to a database, such as that developed by Scientific Group Thermodata Europe (SGTE) around 1990. It gives the Gibbs energy inline image of the elements in the form of series as a function of temperature, i.e. essentially a curve fitting to experimental data. In the light of progress in theoretical understanding and first-principles calculation methods, the possibility for an improved database description of the thermodynamics of the elements has become evident. It is the purpose of this paper to provide a framework for such work. Lattice vibrations, which usually give the major contribution to inline image, are treated in some detail with a discussion of neutron scattering studies of anharmonicity in aluminium, first-principles calculations including ab initio molecular dynamics (AIMD), and the strength and weakness of analytic model representations of data. Similarly, electronic contributions to inline image are treated on the basis of the density of states inline image for metals, with emphasis on effects at high T. Further, we consider inline image below 300 K, which is not covered by SGTE. Other parts in the paper discuss metastable and dynamically unstable lattices, inline image in the region of superheated solids and the requirement on a database in the calculation of phase diagrams.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

A characteristic feature of today's technology is the development of tailor-made materials, which have complex composition, and often consist of several phases in a well-designed morphology, leading to a product with the desired properties. Most of these phases are in the crystalline state at the temperatures and pressures of utilization and processing. At the Ringberg Unary Workshop (RUW2013), one group therefore dealt with the thermodynamics of solid crystalline phases from 0 K to temperatures above the melting point for unary systems, thus exploring ways to improve the modelling approaches currently used in the SGTE (Scientific Group Thermodata Europe ([1])) unary database by firmly connecting them to the underlying physical phenomena. Since the SGTE unary database was adopted around 1990, there has been a steady progress, primarily regarding the possibility to let calculations replace laboratory experiments, but also regarding the description and understanding of basic physical phenomena. Two relevant concepts in this context are the density functional theory (DFT) and molecular dynamics (MD) simulations. Furthermore, it is highly desirable to include new experimental data, such as phonon frequencies, in the evaluation process leading to a new unary database. From a physical point of view, the thermodynamic properties of crystals have their origin in vibrational, electronic and magnetic excitations. The latter were discussed in another group during the RUW2013 and is dealt with in a separate paper of this series.

Over the past two decades many calculations of the enthalpy, H, of a crystalline phase have become routine, and are therefore not the main subject of the present paper. Obtaining the entropy, S, is usually a greater challenge today, especially at high temperatures. The time-honoured approach for determining S is the measurement of heat capacity, inline image, and integrating inline image from inline image to the temperature of interest. This approach accounts for all of the thermodynamic entropy from phonons, electrons and electron spins. For a CALPHAD-type approach, however, heat capacity measurements must be performed to high accuracy on each phase of interest. It is not realistic to expect such measurements to cover a large fraction of binary alloy phases, let alone ternary and quaternary alloys. In the future, we expect computational materials science to provide alloy entropies, free energies and phase diagrams. This has been a longstanding goal for many years, and it is an important part of this present paper to assess how close to this goal we are today.

Contributions to the entropy of an alloy phase can often be calculated individually and summed to give the total entropy. The largest contribution is the harmonic phonon entropy, for which we have the textbook models of Einstein and Debye. Although these models convey the important concept that the heat capacity of solids is dominated by the vibrations of atoms, their assumption about the shapes of phonon spectra and assumption of harmonic oscillations are usually inadequate for the entropies required by CALPHAD-type calculations. Fortunately, it is often straightforward to use first-principles methods to calculate more accurate harmonic phonon spectra, and these are usually reliable at low temperatures. At temperatures of order 1000 K, however, the crystalline solid has undergone thermal expansion, shifting its phonon frequencies. The ‘quasiharmonic’ approximation can be used to calculate the harmonic frequencies in an expanded crystal, typically adding a quasiharmonic contribution to the entropy. More difficult, both conceptually and for calculation, is the pure anharmonicity, where the vibrations cannot be modelled as harmonic normal modes, and the cubic and quartic parts of the potential are essential. Approaches to assess anharmonic vibrations with perturbation theory are described in this survey. Also described is the method of ab initio molecular dynamics (AIMD), which implicitly includes the detailed features of the interatomic potentials from DFT. Methods for determining thermodynamic functions from a first-principles simulation are described, such as thermodynamic integration of the free energy from a reference system having a known free energy. To these phonon contributions to the entropy or free energy, we must add the electron and spin contributions, and the spin contribution is significant in the example of nickel below.

The entropy of fcc aluminium metal includes only contributions from vibrations, electrons and some vacancies, offering a well-posed problem that is the main example in this paper. The calculated contributions to the vibrational entropy are compared to an entropy evaluated from experimental phonon spectra, which required a new analysis to account for the substantial broadening of phonon energies owing to shortened phonon lifetimes caused by anharmonicity. We find that the entropy of fcc aluminium determined from first-principles calculations agrees remarkably well with the entropy determined from inelastic neutron scattering, and these also agree well with the assessed entropy of fcc aluminium in the SGTE database. Aluminium is in some ways a simple case, and other unary systems need similar assessments. Nevertheless, the excellent agreement between thermodynamic functions obtained from theory, phonon measurements and heat capacity measurements gives hope that first-principles calculations of phase diagrams are now within reach for many materials.

The purpose of this paper is to give a basis for an improved version of a database that can be used in, e.g. CALPHAD-type work, and with a focus on the phonons and electrons in the crystalline state of the elements. We discuss experiments, first-principles calculations and representations of data, consider briefly additional high-temperature effects, lattice instabilities and superheated solids, and conclude with some remarks and recommendations.

2 Review of experimental data

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

2.1 Thermodynamic data

Thermodynamic data are needed for many applications in materials science and engineering and several databases are available. In most of them, some kind of assessment of experimental data has been carried out and summarized in a concise way in tables or functions. Examples are: JANAF thermochemical tables ([2]), SpringerMaterials Landolt Börnstein Database ([3]), ASM Handbook ([4]) and compilations by Barin ([5]). The SGTE has developed a widely used thermodynamic database for most solid and liquid phases of pure elements ([6]). Experimental information is thus the basis for our understanding of thermodynamic properties and of great importance in any evaluation of crystalline phases. It should be examined carefully and some knowledge of the experimental techniques is necessary to assess their reliability.

2.2 Experimental methods for thermodynamic properties

There are many techniques available to determine thermodynamic properties, as has been discussed carefully, e.g. in Refs. [7, 8]. These techniques have different application ranges and reliability, and it is not uncommon that they yield significantly different results. Performing additional key experiments to validate the reported experimental data is often desirable to obtain a good evaluation. Unfortunately, experiments require a significant effort in terms of time and cost and they are becoming rare. The paucity of new experimental data is however counterbalanced by the constant increase in state-of-the-art first-principles calculations, whose reliability has significantly increased in the past years and which can now be helpful in validating experimental data or in pointing to regions where new experimental data are necessary. Some groups of elements, e.g. the actinides, present challenges from both the experimental and the computational points of view.

image

Figure 1. Comparison of heat capacity (inline image) for pure Al from different sources. Red triangles are the recommended values evaluated by Desai ([9]) and the solid green line gives calculated values according to the SGTE unary database ([6]). The other symbols are experimental data compiled by Desai ([9]).

Download figure to PowerPoint

2.3 Example: Thermodynamic properties of pure aluminium and nickel

As examples, we consider aluminium and nickel, showing typical experimental data available and useful for thermodynamics. The experimental information on thermodynamic properties of pure Al up to 1984 was reviewed by Desai ([9]). This kind of work is a good starting point in a thermodynamic evaluation, but the original experimental sources should always be retrieved and analysed. In Ref. ([9]) recommended values for several thermodynamic properties are reported, including the heat capacity. Experimental values for the heat capacity of aluminium from the literature together with the calculated SGTE values (solid line) are shown in Fig. 1. As it can be seen, the description at low temperatures does not account correctly for the experimental data, but the SGTE was not developed for temperatures below 298 K (cf. Section 'Representation of thermodynamic quantities in current databases for crystalline phases'). For later reference, we also show in Fig. 2 the SGTE heat capacities of fcc and liquid aluminium. The experimental information on the thermodynamic properties of nickel has also been evaluated comprehensively by Desai ([10]). A comparison among experimental data, the recommended heat capacity by Desai, and recommended heat capacity by SGTE ([6]) is given in Fig.  3. The comparison shows the failure of the SGTE description below about 250 K. However, it is interesting to note that there is considerable scattering in the experimental heat capacity in the range where magnetic short-range ordering occurs. Magnetism can give an important contribution to the thermodynamic properties, in particular in the temperature range where a magnetic transition occurs. As an example, in a model currently adopted in the SGTE unary database [11-13], the magnetic moments and transition temperatures of each phase can be the key quantities for estimating the magnetic ordering energy. However, it should be noticed that in the SGTE unary database the magnetic moment and Curie temperature for (metastable) bcc Ni are given as 0.85inline image and 575 K, in disagreement with the experimental data 0.52 inline image and 456 K ([14]). See also the accompanying paper on lambda-transitions ([15]).

image

Figure 2. Heat capacity (inline image) of Al from the SGTE unary database showing the extrapolation of fcc Al above the melting temperature and of the liquid below the melting temperature.

Download figure to PowerPoint

image

Figure 3. Comparison of heat capacity (inline image) for pure Ni from different sources at low (a) and high (b) temperatures. Red solid line with triangles is the recommended value evaluated by Desai ([10]), green dashed curve is calculated value according to SGTE unary database ([6]). Circles are experimental data compiled by Desai ([10]).

Download figure to PowerPoint

2.4 Notes on the re-evaluation of thermodynamic quantities

It is important to make sure that a re-evaluation of thermodynamic properties of crystalline phases provides at least the same level of accuracy as in the currently adopted databases, such as, for example the SGTE unary compilation. A comprehensive review of the original experimental reports regarding the properties of interest is always necessary, thus guaranteeing that the evaluation is not biased by the judgements and decisions made by previous assessors. In some cases, the currently available sources and databases may have inconsistencies or they may need to be corrected or improved. An illustrating example is related to the evaluation of the melting point of pure Cr, which has been discussed in the work for Fe–Cr by Xiong et al. ([16]). The evaluated melting point of pure Cr was considered to be 2180 K in the SGTE unary database ([6]), which is 44 K higher than the most recent reliable evaluation ([16]).

2.5 Notes on the evaluation of low-temperature heat capacity

The conventional way to analyse experimental low-temperature heat capacity inline image in non-magnetic materials is to plot inline image versus inline image. The result is a straight line consistent with

  • display math(1)

The parameter inline image is proportional to the electron density of states at the Fermi level, inline image, multiplied by an electron-phonon many body enhancement factor inline image; cf. Section 'Calculation of the electronic contribution'. When the parameter A is written as the leading low-temperature term in the Debye model of lattice vibrations, inline image, the experimental data yield the Debye temperature inline image that is often quoted in tables. It is important to note that this Debye temperature only refers to the low frequency part of the phonon spectrum, and only to the heat capacity. It may give results that are wrong by 20% or more if used in a Debye model to describe other vibrational properties, considered at other temperatures, cf. Section 'Model representations of harmonic phonons'.

3 Volume and thermal expansion

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

Volume is a thermodynamic variable and should be considered an integral part of thermodynamic modelling. Volume and its temperature dependence (thermal expansion) is also a prerequisite for modelling the pressure dependence. However, very few CALPHAD databases contain volume data at present. In the SGTE unary database ([6]), modelling of the PV term of the Gibbs energy was considered using a Murnaghan equation of state (EOS) ([17]), but it was applied only to very few elements such as Fe und C. The reason why not many elements have been modelled in this fashion is the demanding modelling effort and probably a lack of data for most elements. Here, we comment on modelling of the volume as a function of temperature (thermal expansion). Pressure dependence is dealt with in an accompanying paper ([18]).

image

Figure 4. Experimental linear thermal expansion of fcc Al, according to different data sets, compared with first-principles results ([22]) and two different CALPHAD assessments [19, 20]. Measurements have been done by dilatometry (volume expansion) and high temperature X-ray diffraction (lattice expansion).

Download figure to PowerPoint

Volume and thermal expansion were modelled in a CALPHAD context by Lu et al. ([19]) for a large number of elements and by Hallstedt ([20]) for Al, Si, Mg and Li. In both cases series of the same type as those used to model inline image (or inline image) in the SGTE unary database ([6]) were used to model the thermal expansion coefficient ([19]) or the volume ([20]). The temperature range 0 to about 200 K was not modelled. The volume at ambient conditions can be determined with high accuracy by measuring the lattice parameters with diffraction methods. The thermal expansion can be measured either by diffraction or by dilatometry. The achievable accuracy is high enough that the equilibrium vacancy concentration can be determined from the difference between the volume thermal expansion and the lattice thermal expansion. Experimental volume and thermal expansion data are available for most elements with sufficient accuracy. First-principles calculations using the quasiharmonic approximation (QHA) or AIMD can also provide thermal expansion data with good accuracy (cf. Section 'First-principles approaches'). However, there is typically a small constant shift in the lattice parameter so that at least experimental ambient data should be used to correct the first-principles data. Figure 4 shows a comparison between several experimental data sets for the linear thermal expansion for Al, state-of-the-art first-principles results and state-of-the-art CALPHAD assessments. The first-principles calculations include the contribution of lattice vibrations (quasiharmonic and anharmonic) and electronic excitations (cf. Section 'First-principles approaches') plus the contribution of vacancies (cf. the companion RUW paper on defects ([21])). We remark the excellent agreement among experiments, CALPHAD and first-principles results with differences which are hardly visible in the figure. Experimental data in the present case show limited scatter among different sets, but this could be significantly different for other elements. We also remark that the inclusion of all possible contributions in first-principles calculations was essential to achieve such an agreement with the experiments.

4 Entropy from phonon measurements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

4.1 General aspects

Inelastic neutron scattering is a proven method for measuring phonon dispersions in crystals, and for measuring the phonon density of states (DOS) in many materials. For pure elements, the measured spectral intensity can often be converted to an accurate phonon DOS, assuming the experiment was performed with care so there is only a small background from multiple scattering or from multiphonon scattering. After correcting for these effects and other experimental artefacts, the one-phonon scattering inline image is related to the phonon DOS inline image as

  • display math(2)

where E is the phonon energy. The details of using an inelastic neutron spectrometer to measure inline image are not provided here, but some points are noteworthy. The energy resolution of the data does not depend on temperature, but it does depend on the energy transfer from the neutron to the sample. For measurements with positive energy transfer with a Fermi chopper spectrometer, the resolution tends to be best at the highest phonon energies, degrading for lower phonon energies. The statistical quality of the measurements on polycrystalline samples is no longer an issue for modern instruments such as Angular-Range Chopper Spectrometer (ARCS) at the Oak Ridge National Laboratory ([23]), and the finer points of phonon quasi-harmonicity and anharmonicity (cf. Section 'First-principles approaches') are now accessible for experimental investigation.

The phonon DOS gives the energies and numbers of all normal modes of vibration, allowing the construction of all Boltzmann factors and their sum. From the phonon DOS, it is therefore possible to calculate the phonon partition function, and all quantities of phonon thermodynamics. The vibrational entropy, for example, is

  • display math(3)

where inline image is the Planck distribution for the temperature T. The only material parameter is the phonon DOS, inline image, which should be measured at the temperature of interest ([24]).

For measuring a phonon DOS of a compound, the problem is complicated by the effect of neutron weighting, which originates with differences in the scattering cross-sections and masses of the different elements. (For pure elements, all atoms scatter neutrons equally, so there is no neutron weighting of the phonon spectrum.) If a phonon mode is dominated by the motion of an element with a large cross-section and low mass, this phonon will be enhanced in the measured spectral intensity. In typical cases where the phonon spectrum contains many overlapping contributions from different elements, an accurate correction for neutron weighting can be difficult.

image

Figure 5. Phonon DOS curves of aluminium metal, obtained from inelastic neutron scattering measurements at the temperatures specified in the labels (units are K) ([25]).

Download figure to PowerPoint

image

Figure 6. Vibrational entropy calculated from the phonon spectra of Fig. 5 under various approximations described in the text, and the entropy in the SGTE unary database for aluminium. The curve labelled ‘Full Theory’ shows state-of-the-art first-principles calculations reproduced from Ref. ([22]).

Download figure to PowerPoint

4.2 Example: Phonon DOS of aluminium metal at elevated temperatures

A series of phonon DOS curves from fcc aluminium metal are shown in Fig.  5, obtained with ARCS and used for a study of phonon anharmonicity ([25]). With increasing temperature, the most prominent change in the phonon DOS is the broadening of the phonon spectrum. The shifts of phonon frequencies with temperature are modest, and appear to be consistent with quasiharmonic predictions. The phonon lifetime broadening was recently calculated from first-principles, using a cubic anharmonicity tensor obtained from DFT by a real space displacement method. The kinematics of all three-phonon processes that conserve energy and momentum was also calculated from the dispersion relations obtained from the harmonic part of the first-principles calculations. These first-principles calculations of phonon lifetime broadening using the cubic anharmonicity to second order were remarkably successful in reproducing the experimental results on phonon linewidths at elevated temperatures ([25]). From the experimental phonon DOS curves, it is straightforward to calculate the vibrational entropy with Eq. (3) at the temperatures where the phonon DOS curves were measured. The results from these eight DOS curves are presented in Fig. 6 as the points labelled Experimental. It is also straightforward to use a low-temperature phonon DOS with a Planck distribution at an elevated temperature to calculate the vibrational entropy at this elevated temperature. This is the harmonic model for vibrational entropy, presented as the curve Harmonic in Fig. 6. Shown for comparison is the total entropy curve from the SGTE unary database. We remark that the Harmonic curve correctly goes to zero in the low temperature limit, while the SGTE description was not meant to be correct in the low temperature range and it significantly off for Al. For accurate entropy at elevated temperatures, Eq. (3) requires a phonon DOS for the temperature of interest, giving the eight discrete points in Fig. 6. It may be reasonable to use each of these DOS curves over a range of temperatures, but the present analysis used a different approach that benefited from the fact that each spectrum was obtained with points at the same energy intervals separated by 1 meV. For each individual energy, the eight values of inline image were organized as a function of T, and a straight line was fit by a least-squares criterion to these eight points. The slope and intercept of this line could then be used to obtain the value of inline image for that same energy at any temperature. A total of 53 such lines were used to cover the energy range of 0–52 meV. Finally, the DOS calculated at the desired temperature was re-normalized to 1, as were all the original DOS curves. The curve labelled anharmonic in Fig. 6 was obtained this way, using Eq. (3) for each of the 1001 temperatures from 0 to 1000 K.

The curve labelled anharmonic is in good agreement with the discrete points from which the interpolated values were obtained for each temperature, T. The anharmonic curve lies below the total entropy of aluminium from the SGTE unary database. We believe there are two reasons for this. First, there is a contribution from electronic entropy. Aluminium is nearly a free-electron metal, so this contribution is fairly small, but it has been assessed. It is straightforward to obtain this contribution at 800 K from previous assessment of the electronic heat capacity ([26]) or from first-principles calculations (cf. Section 'Calculation of the electronic contribution'). This contribution of approximately 1 J mol−1 K−1 at 750 K is shown on Fig. 6.

The second reason for the discrepancy is less well understood, but the present assessment may point to a resolution. Equation (3) for the vibrational entropy was derived under the assumption of harmonic phonons. Fortunately, anharmonic phonon perturbation theory predicts that even if the phonons are anharmonic, Eq. (3) is correct to first order ([27]). It might then seem that the curve labelled anharmonic in Fig. 6 should be a best estimate. However, the large lifetime broadening of the phonons in aluminium raises a concern that the anharmonicity is outside the range of validity of perturbation theory.

Approximately, each phonon frequency is broadened significantly by the lineshape of a damped harmonic oscillator

  • display math(4)

where the central energy of a phonon, inline image, is spread over a range of energies E if the phonon is damped with a quality factor inline image. This function D becomes similar to a Lorentzian function when inline image is large, but unlike a Lorentzian it decreases to zero at E = 0.

The inline image in Eq. (4) is the number of vibrational periods for a decay of the phonon spectral weight to inline image of its initial value. This decay occurs by transfer of energy to other phonons through the cubic anharmonicity, and does not represent energy dissipation in the conventional sense. During the decay of one phonon, the vibrational amplitude of an atom is coming under the influence of other phonons, so its thermal excursions are not diminishing as for classical damping. For obtaining vibrational entropy from experimental DOS curves, is it more appropriate to consider the central phonon frequency without the lifetime broadening when using expressions such as Eq. (3)?

4.3 The role of phonon lifetime broadening on vibrational entropy

We assessed the effect of lifetime broadening with Eq. (4) on the vibrational entropy. Figure 7a shows a set of damped harmonic oscillator functions for different values of inline image. For smaller values of inline image, the damped harmonic oscillator function shifts more spectral weight to higher energies. When convoluted with a single phonon frequency in Eq. (3) (i.e. a phonon DOS, inline image that is a Dirac inline image-function), the calculated entropy is lower than for the unbroadened phonon frequency. For the high temperature limit of vibrational entropy, we calculated how the vibrational entropy of a phonon frequency is altered by broadening with the different values of inline image shown in Fig. 7a. The results are shown in Fig. 7b. The quality factor for the phonon damping in fcc aluminium was assessed previously [28, 29]. At 750 K a good average value of inline image is approximately 4. From Fig. 7b, this corresponds to approximately 1.8 J mol−1 K−1, which means that to remove the effects of phonon lifetime broadening from the phonon entropy, we need to add this contribution to the curve labelled anharmonic in Fig. 6. Together with the electronic contribution to the entropy, this correction for the lifetime broadening brings the entropy derived from the phonon DOS curves into much better agreement with the SGTE value. The agreement is, in fact, remarkably good considering the entirely different origin of the entropy curves.

image

Figure 7. (a) Spectral functions for damped harmonic oscillators with different quality factors, inline image. (b) Decrease in phonon entropy caused by the broadened oscillator function, compared to a single frequency.

Download figure to PowerPoint

It is physically reasonable that the anharmonic phonon DOS should be calculated from the phonon central frequencies, rather than from the lifetime-broadened phonon spectrum. Consider the essential origin of entropy in terms of the phase space explored by the atoms during their vibrations. For each pair of position and momentum coordinates inline image referenced to an equilibrium position, we have a contribution to the entropy of

  • display math(5)

where inline image is 1 if the coordinate pair inline image is accessible for a system at energy E, and 0 otherwise. If we consider the coordinates as normal coordinates of phonons, we realize that the damping of an individual phonon will move inline image to smaller values of inline image and inline image, reducing inline image. A reduction of the position and momentum coordinates is unavoidable from classical damping. However, we also need to consider the gain of vibrational amplitude from other phonons that become excited when the original ones decay. For the damping of a classical harmonic oscillator, the energy goes into heat. Phonons are the heat, however. The transfer of energy between phonon modes by anharmonic interactions allows energy to distribute among the different modes, allowing thermal equilibration.

From Hamiltonian mechanics, we know that the peak kinetic energy of an atom vibration is unaffected by anharmonicity in the potential. Likewise, the range of excursions of atoms in an anharmonic potential well is not altered by damping if the atom retains the same thermal energy. (The range of excursions is altered by cubic and quartic components that alter the shape of the potential that cause shifts of phonon energies.) The spectral broadening caused by phonon damping may therefore not be important for consideration in the vibrational entropy. Further tests of this hypothesis may be appropriate, but in the case of aluminium it seems that the large damping of phonons does not alter the vibrational entropy, and is in fact a characteristic of the phonon DOS that needs correction before thermodynamic analysis is performed with experimental data.

Finally, we added a quasiharmonic correction to the vibrational entropy, allowing for how the thermal expansion contributes to the heat capacity through the relation inline image. Using a prior assessment of the heat capacity ([26]), we find that this contribution gives a decrease in phonon frequency of 15% at 800 K, corresponding to an increase in vibrational entropy of 3.4 J mol−1 K−1. If this and 1 J mol−1 K−1 from the electronic entropy is added to the curve marked harmonic, at 800 K, the agreement with the SGTE value is excellent. The quasiharmonic correction obtained from state-of-the-art first-principles calculations (from Ref. ([22]), cf. Section 'First-principles approaches') is 3.4 J mol−1 K−1, while the electronic contribution is 0.86 J mol−1 K−1 in almost perfect agreement with the above estimations. Although aluminium is highly anharmonic, its average phonon frequencies evidently follow the temperature dependence expected from a quasiharmonic model. This is not expected in general, of course. It is a consequence of the cancellation of phonon shifts from the cubic and quartic anharmonicities in aluminium, as confirmed by state-of-the-art first-principles calculations (cf. Section 'Insights from first-principles calculations'). At 800 K, this first-principles anharmonic contribution is inline imageJ mol−1 K−1, while a small contribution is given by vacancies 0.02 J mol−1 K−1. We report in Fig. 6 state-of-the-art first-principles calculations for Al, including the harmonic, quasiharmonic, anharmonic, electronic and vacancies contributions (from Ref. ([22])). These results are consistent with the thermal expansion calculations reported in Fig. 4. Figure 6 shows the impressive agreement of three entropy curves –SGTE, Full Theory and the results from neutron scattering corrected for electronic and lifetime broadening contributions. These three results were obtained from completely different methods.

5 First-principles approaches

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

5.1 Calculation of the vibrational contribution

Vibrational excitations (phonons) provide the main contribution to the thermodynamics of crystalline solid phases. In addition to the increased computational power available, recent progress in theory and computer codes has made it possible to perform calculations of phonon frequencies and related properties for several systems [22][30-33]. Phonon frequencies can be computed by either density-functional perturbation theory (DFPT) ([34]) or supercell (or small displacements) approaches [35, 36]. The former method is an elegant one that considers vibrational displacements of atoms with respect to their equilibrium positions (at 0 K) as perturbations, and iteratively solves the Kohn–Sham equations modified for a perturbative expansion. The method therefore must to be coded within a DFT code. Computer codes into which DFPT has been implemented are, e.g. Quantum Espresso ([37]) and ABINIT ([38]). By contrast, in the supercell (or small displacements) method, atomic forces are evaluated by displacing the atoms from their equilibrium positions (at 0 K) and performing a standard DFT calculation. A supercell must have sufficient size to limit effects from the periodic boundaries. From the forces acting on the atoms after displacement, the dynamical matrix of the system can be derived, and phonon frequencies computed. Software packages are available that have this method implemented (PHON ([35]), PHONON ([39]), PHONOPY ([40])), and can be coupled with most standard DFT codes to calculate the forces.

The vibrational Helmholtz energy F is obtained from the phonon frequencies inline image of modes inline image and at wavevectors inline image as

  • display math(6)

The first term in Eq. (6) is the zero-point energy (ZPE) and the second term is the phonon contribution at finite temperature. The sums are taken over the phonon modes inline image (all branches) and over all vectors inline image in the Brillouin zone. Equation (6) and its associated equations for entropy and heat capacity have the correct behaviour in the low temperature limit. The (vibrational) Helmholtz energy F goes to the ZPE at 0 K, while both the entropy and heat capacity go to zero. As already discussed, this is in contrast to the current SGTE description of unary crystalline phases.

5.2 The quasiharmonic approximation

A relatively simple approach to account for anharmonicity is the QHA ([41]). In the QHA, phonon calculations are performed for different volumes (lattice parameters) and thus Eq. (6) becomes

  • display math(7)

At a given temperature, the Helmholtz energy (from Eq. (7) plus the 0 K total energy) is then fitted by using a Murnaghan EOS, thus determining the equilibrium lattice parameter inline image, bulk modulus inline image and its pressure derivative inline image. Other properties such as the thermal expansion are then obtained from the temperature dependence of inline image and classical thermodynamic relationships [27][41]. Despite its simplicity, the QHA can describe the main anharmonic effects in many systems. However, there are many systems in which this approximation fails [32, 33] and in general at high temperatures it becomes less reliable than other more advanced techniques based on AIMD.

5.3 Beyond the quasiharmonic approximation

Going beyond the QHA can be accomplished in two conceptually different ways. First, one can continue the perturbation expansion of the total internal energy to higher orders in atomic displacements. In this perturbation series, the QHA corresponds to the second order derivative of the total internal energy with respect to atomic displacements (e.g. ([42])). The next two terms in the expansion correspond to the third and fourth order tensor, which contain the full information about the third and fourth order derivative of the electronic energy with respect to atomic displacements. From these two orders, it is possible to derive an explicit anharmonic broadening and an explicit anharmonic shift of the quasiharmonic frequencies (inline image and inline image in Eq. (27)). In principle, an infinite number of orders follows but practically even the third and fourth order are at the edge of computational feasibility. The reason is that the number of tensor elements scales as inline image where O is the order and N the number of atoms, hence even a rather small 100 atom cell requires the handling of several billion of tensor elements in fourth order. Exploiting symmetries can reduce the effort, but in general the task of computing higher order tensors remains impractical.

The second (and in practice typically used) possibility to take into account effects beyond the QHA is based on MD simulations. An advantage of MD over the perturbation expansion is that all orders up to infinity are implicitly included, thus eliminating problems associated with convergence of the series. A conceptual disadvantage is that typical MD simulations are classical and as such they do not include quantum mechanical effects. Thus, MD simulations results are not reliable at low temperatures and cannot be used to compute the zero-point energy (ZPE), which is a merely quantum effect. This disadvantage is believed to be of little concern since the QHA—a fully quantum mechanical approach—supposedly captures the relevant quantum mechanical effects. Quantitatively, this remains to be shown in the future by simulations going beyond standard MD such as path integral methods. A technical disadvantage of MD is that corresponding simulations need to be performed explicitly at different temperatures providing only a discrete mesh of anharmonic free energy points while perturbation theory allows to derive analytical expressions for the full temperature dependence of the free energy ([27]). The latter allows for well-defined derivatives enabling a direct determination of highly important quantities such as entropy and heat capacity. The discrete mesh of free energies obtained from MD cannot be directly used for computing derivatives and therefore work has been invested to find optimized fitting functions, a task that has now been successfully accomplished ([22]).

Focussing in the following on the MD based approach to explicit anharmonicity, an important additional concept needs to be introduced. A single MD simulation at a specific temperature can provide only the internal energy of the system while no practical way exists to accurately determine the entropy from such a single calculation. The reason is that the entropy is directly related to the partition function and as such it entails information from the complete phase space necessitating infinite simulation times for achieving statistical convergence. To overcome this problem, the state-of-the-art approach is to introduce a reference system for which the free energy can be calculated analytically and to perform a thermodynamic integration from this system to the fully anharmonic one. A convenient and by now well approved reference system is the quasiharmonic system, exactly the one as used for the QHA and with the analytical free energy given by Eq. (7). This reference system is coupled to the full anharmonic DFT system for which purpose, in principle, an arbitrary coupling function can be used. In practice, this freedom is not further exploited as a simple linear coupling works sufficiently well. One therefore introduces a parameter inline image to couple the reference quasiharmonic potential energy surface, inline image, to the full anharmonic DFT surface, inline image, as:

  • display math(8)

With this definition and using standard thermodynamic concepts ([42]), the explicitly anharmonic free energy, inline image can be obtained by

  • display math(9)

where T is the simulation temperature. Equation (9) shows that in order to obtain the anharmonic free energy we need to perform an integral over inline image. In practice this integral needs to be discretized (five equidistant inline image points are reasonable) and it is beneficial to have sound fitting functions at hand to achieve a faster convergence of the integral. Reference ([43]) shows that, instead of a simple polynomial expansion, a cotangent fit provides a quicker convergence. Once inline image is obtained from Eq. (9), it is added to the quasiharmonic free energy from Eq. (7), inline image, to obtain the total free energy due to atomic vibrations:

  • display math(10)

To obtain the final thermodynamic quantities such as Gibbs energies, entropies and heat capacities, it is necessary to perform MD calculations for different inline image values, different temperatures and different volumes. It must be stressed that each of the MD calculations is fully based on ab initio (DFT) energetics which gives rise to the notation AIMD. As such, corresponding investigations quickly become computationally prohibitive and only recent methodological developments allow accurate assessments of the anharmonic free energy contributions of different elements.

One such methodological development is the upsampled thermodynamic integration using Langevin dynamics (UP-TILD) approach ([22]). Its main idea is that low precision DFT convergence parameters (e.g. the electronic inline image-point sampling) can be efficiently used to obtain a phase-space trajectory (termed inline image in the following) that closely resembles the phase-space trajectory inline image that would be obtained from parameters yielding highly converged results. In this way, it is possible to sample various inline image, temperatures, and volumes with modest computational resources. However, the resulting quantity inline image, which is required as input for the thermodynamic integration, needs to be corrected in a second step. For this purpose, perturbation theory is employed: a small set of n uncorrelated structures is extracted from inline image and the upsampling average inline image is calculated as:

  • display math(11)

Here, inline image (inline image) refers to the electronic energy of the i’th uncorrelated structure calculated using low (high) DFT convergence parameters and inline image and inline image are the corresponding ground state energies at T=0 K (i.e. for the equilibrium structures without atomic displacements). The inline image dependence of inline image is hidden in the trajectory inline image, which is additionally dependent on the volume and temperature. In the last step, the quantity of interest inline image, which corresponds to high convergence parameters, is obtained from:

  • display math

and thus the anharmonic free energy reads:

  • display math(12)

The efficiency of this method is exemplified by the fact that in practice less than 100 uncorrelated configurations have to be calculated with high convergence parameters to get statistical error bars below 1 meV, whereas a full thermodynamic integration includes many thousands of configurations ([22]).

5.4 Calculation of the electronic contribution

The electronic states are usually described in the single-particle approximation, i.e. each electron state is regarded as a quantum-mechanical eigenstate with a sharp energy. The distribution of electron energies is given by the electronic DOS inline image. In this model, the total electronic energy at temperature T is given by ([44])

  • display math(13)

Here inline image is the Fermi–Dirac distribution function. The prefactor 2 accounts for the two spin directions. The corresponding expressions for the entropy and the heat capacity are

  • display math(14)
  • display math(15)

In the limit of low T we get the standard textbook result

  • display math(16)

where inline image is the DOS at the Fermi energy inline image (i.e. the chemical potential of the electrons inline image at T= 0 K).

The expressions (13)(16) rest on approximations. At low temperatures, inline image and corresponding quantities are enhanced by electron–phonon many body interactions that are not included in the single-particle picture. The enhancement, which gives a significant correction in the measured electronic heat capacity at very low temperatures, varies with T and becomes very small above the Debye temperature of the material. It has no practical importance in CALPHAD type calculations of phase diagrams, and we will ignore it.

If the energy E in the integrand in Eq. (13) is replaced by 1, the integral gives the total number of electrons, which does not vary with T. From that relation, we can calculate the temperature dependence of the chemical potential inline image that appears in the Fermi–Dirac function. For convenience of discussion, in Eq. (15) we have taken inline image. This is a reasonable approximation since, as we will now see, there are other and more important effects at high temperatures. According to standard textbooks the electron heat capacity samples the electron states in a narrow interval of approximate width inline image. However, it turns out that the correct weight function, inline image, samples an approximate energy range inline image. This will cause large deviations from the linear law in Eq. (16) if the DOS varies significantly within the energy interval inline image. This is illustrated in Fig. 8, which is based on a single-particle calculation of inline image at fixed volume and inline imageK. inline image may bend either upwards or downwards as a function T, a result easily explained by the variation of inline image around inline image that leads to an increase or decrease in the average inline image within the energy window probed by the electronic heat capacity.

image

Figure 8. Electronic heat capacity for several elements obtained from the inline image calculated using DFT. The temperature axis has been normalized to the melting temperature of the elements from Ref. ([6]).

Download figure to PowerPoint

A serious deficiency in the single-particle picture is that the electron states are calculated for an infinite perfect lattice. In a real system at high temperatures, the electrons see a rather disordered lattice. Anharmonicity would also affect the electronic DOS at high temperatures, but little is known about the importance of this effect. That will wash out sharp structures in inline image. Such effect has been shown in AIMD calculations performed on bcc and fcc Mo ([45]), but as it was already pointed out these calculations are computational demanding. We may still use such a smoothed inline image in the single-particle expressions but the accuracy of this procedure is not well known. We conclude that first-principles calculations performed at low temperatures (in practice, inline imageK) now have reached a high sophistication, with an accurate corresponding inline image. However, that information can be of limited value at high temperatures, where effects arise which are not included in the low-temperature approach.

image

Figure 9. Different contributions to the heat capacity of fcc Al according to different approximations calculated using DFT (adapted from Ref. ([22])). Label ‘h’: vibrational contribution as in the harmonic approximation (cf. Section 'Calculation of the vibrational contribution'); label ‘q’: vibrational contribution as in the quasiharmonic approximation (cf. Section 'The quasiharmonic approximation'); the enlargement on the right shows the decomposition of the residual contributions (label ‘sum’) as from electronic excitations (label ‘el’, cf. Section 'Calculation of the electronic contribution'); the anharmonic contribution beyond the QHA (label ‘ah’, cf. Section 'Beyond the quasiharmonic approximation') and the vacancies contribution (label ‘vac’, cf. Ref. ([21])).

Download figure to PowerPoint

5.5 Insights from first-principles calculations

Sections 'Calculation of the vibrational contribution''Calculation of the electronic contribution' showed how first-principles calculations can give the phonon and electron eigenvalues in different approximations. Considering fcc Al as an example, we report in Fig. 9 all different contributions to the heat capacity computed using DFT-based methods. As expected, the harmonic contribution (label ‘h’) is the most important one over the whole temperature range up to melting point and the heat capacity goes to the Dulong–Petit limit (inline image) at high temperature. The QHA (label ‘q’) provides the second major contribution to the inline image of fcc Al, while a minor contribution is given by electronic excitations. Remarkably, the anharmonic contribution (beyond the QHA) is negative in this case and essentially cancels the effect of electronic excitations (cf. Section 'The role of phonon lifetime broadening on vibrational entropy'.) The vacancy contribution is negligible apart for temperatures close to the melting point. As seen in Fig. 9, first-principles calculations are invaluable for assessing the different contributions to thermodynamic functions.

With two examples, we test how well the first-principles approach stands up in a comparison with experimental data. A key quantity to assess the low-temperature thermodynamics of a solid crystalline element is the standard entropy, i.e. the entropy of the element at 298 K and inline image Pa (inline image). Experimentally such a quantity is obtained by integration of heat capacity data from temperatures of a few kelvin to 298 K. Below 4 K (boiling point of liquid helium), experiments are extremely difficult and thus some kind of extrapolation is needed in this region. In the SGTE unary database, the standard entropies for all elements are reported as evaluated from experimental data. Table 1 gives our computed standard entropy for several elements in the QHA, both excluding and including the electronic contribution, compared with the assessed values from the SGTE description ([6]). Anharmonic effects at this temperature are negligible. For some elements, calculations using different first-principles codes and methods are reported for comparison. The difference between the first-principles and the assessed values for the standard entropy is less than 1 J mol−1 K−1 for all the elements in Table 1. From 0 K up to 298 K, first-principles calculations provide reliable results for the thermodynamic properties such as the Helmholtz energy, the entropy and the heat capacity already in the QHA.

Table 1. Comparison between first-principles and assessed standard entropy (inline image) for several fcc and bcc elements. Assessed values are from Ref. ([6]). QE are values computed using Quantum Espresso and density functional perturbation theory, VASP are values computed using VASP and the supercell method. QHA refers to the quasiharmonic approximation, QHA + el to the quasiharmonic approximation including the electronic contribution
elementmethodassessed inline imagecalc. inline imagecalc. inline image
  (J mol−1 K−1)QHAQHA + el
   (J mol−1 K−1)(J mol−1 K−1)
Al fccQE28.3027.4227.70
Cr bccQE23.542923.0723.72
Cr bccVASP23.542923.0323.50
Cu fccQE33.1533.5833.79
Fe bccQE27.279726.5827.41
Ni fccQE29.795528.0230.47
Ni fccVASP29.795528.3629.65
V bccQE30.8930.3731.66
W bccQE32.617633.1033.39

As a second example, we discuss the high temperature behaviour where the situation is more complex. The QHA can still provide reliable results in some cases ([41]), but it can break down in some other cases ([32]). It is not possible to predict a priori when this would happen. The physical reasons for the failure of the QHA approach are related to the greater importance of anharmonic contributions that are not properly described in this approximation, such as higher order phonon-phonon interactions. Anharmonic effects beyond the QHA can give either a positive or a negative contribution (as in the case of fcc Al in Fig. 9). Some issues related to electronic excitations and anharmonic effects at high temperatures have already been discussed, and it was pointed out that a full evaluation of these effects in a large temperature range is more complex and requires a significantly higher computational effort. This research field is indeed very active and new approaches may become available in the future to overcome present limitations. At present, however, an approach based only on first-principles at high temperatures may not have the accuracy required for thermodynamic calculations and experimental validation is still necessary. An example is an element with an allotropic phase transition at a high temperature. The exact prediction of such a temperature from first-principles methods with an accuracy of inline image10 to 20 K is still a daunting task.

Besides the high temperature limitations of first-principles-based approaches, it must be remarked that some quantities can be difficult to compute using first-principles methods at low temperatures. For example, although DFT is based on a formally correct theory, the exact form of the exchange-correlation functional is unknown. It has been proposed to use the difference between the results obtained using two of the most popular approximations of the exchange-correlation functional as an ‘error bar’ for DFT-based results [22][31]. Some quantities such as the bulk modulus and the equilibrium volume (lattice parameters) may significantly differ from experiments when computed with different exchange-correlation functionals, even for 0 K DFT calculations. In this respect, input from key experiment data, such as room temperature volume and bulk modulus, may prove very useful to correct the DFT results, thus restoring the predictive power of the DFT + QHA approach [46-48].

A final note must be made on the form in which first-principles results are usually stored. Contrary to what is generally done in CALPHAD, thermodynamic properties computed using first-principles methods are typically stored as discrete points for a defined grid of independent variables, such as T and V. For this reason, the incorporation of first-principles results into CALPHAD and current thermodynamic software is not trivial, and requires either a proper parameterization based on some models or a deeper revision of current codes. It is desirable to further investigate the possibility to store quantities such as phonon frequencies and the electronic DOS and derive thermodynamic quantities at any temperature, pressure and volume directly from these physical meaningful quantities. These quantities can be obtained from first-principles, but experimental data can also be used as described in Section 'Entropy from phonon measurements'.

6 Representation of data

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

6.1 First-principles data

The quantum-mechanical eigenvalues obtained from first-principles calculations of electrons and phonons are labelled by a wave vector in the first Brillouin zone and a band (mode) index. In thermodynamic applications one is usually not interested in the electron band structure per se, but only in the electron DOS inline image. It could be displayed as a graph, but the underlying data are discrete values for a histogram-type representation.

The phonon frequencies inline image (energies inline image) are given in an analogous manner, with a DOS g(E) that is obtained as discrete values for a histogram representation. However, in many thermodynamic applications it is also of interest to know how individual atoms or groups of atoms move relative to the lattice. Such information is given in the phonon dispersion curves. They may be plotted as smooth graphs but the underlying data are discrete frequency values inline image. However, by Fourier interpolation phonon frequencies can be obtained virtually in any point of the first Brillouin zone.

Thus, the first-principles results for single-particle electron states and quasiharmonic phonon frequencies are summarized in tables giving discrete numbers as a function of the electron and phonon energy (frequency). The energy resolution is mainly a matter of computing effort.

6.2 Model representations of harmonic phonons

In science and technology, there is much to gain if discrete data are represented by simple and explicit mathematical models, and that is of course the case also for the phonon and electron states. Much of this is standard material in textbooks. Here, we will focus on some aspects that are usually not dealt with in the textbooks; see Ref. ([44]) for details.

Phonons usually give the dominating contribution to the heat capacity and related properties of solids. Within the QHA, the full thermodynamic information is contained in the DOS inline image describing the inline image vibrational degrees of freedom. For a macroscopic piece of material, N typically is at least of the order of inline image (cf. Avogadro's constant). Two standard textbook models, the Einstein model and the Debye model, reduce this enormous number to just one. The Einstein model describes inline image by a single peak at a typical phonon frequency given as inline image. Then the Helmholtz energy becomes

  • display math(17)

where inline image is the Einstein temperature defined by

  • display math(18)

and R is the gas constant (inline image). Other thermodynamic functions such as the entropy and the heat capacity are easily derived:

  • display math(19)
  • display math(20)

Despite its simplicity, the Einstein model provides a reasonable description of the heat capacity of a crystalline solid at intermediate temperatures.

The Debye model assumes that the crystal is an isotropic elastic medium. Elastic waves are considered to be quantized (phonons) in the volume V of the crystal and they always have two transverse and one longitudinal mode. For each mode, the wave velocity is independent of the direction inline image

  • display math(21)

where inline image are the wave velocities.

In the Debye model one can define the Debye temperature according to

  • display math(22)

where inline image is a cut-off frequency, defined such that the total number of allowed states is inline image for a crystal containing N atoms (the number of states up to inline image for each normal mode inline image is inline image and thus the number of allowed states up to inline image is inline image).

Under these assumptions one obtains for the heat capacity

  • display math(23)

In many mathematical tables and textbooks, numerically evaluated values of the following integral can be found

  • display math(24)

From this integral, the entropy and the Helmholtz energy in the Debye model can be obtained according to the following equations:

  • display math(25)
  • display math(26)

Both the Einstein and Debye models can satisfactorily describe thermodynamic functions such as the free energy, entropy and heat capacity below room temperature, better representing these quantities with respect to SGTE polynomials. In comparison to the Einstein model, the Debye approach reproduces the correct inline image behaviour at low temperatures. Apart from this, both Einstein and Debye models can account well for the vibrational contribution at intermediate temperatures while at high temperature they provide a common baseline for the other contributions.

It is obvious that reducing the available information from about inline image numbers to just one, as in the Einstein and Debye models, means a loss of accuracy in the description of thermophysical phenomena.

We now point out some facts that are often overlooked. From heat capacity data (experimental or accurate first-principles calculations) we can obtain an Einstein or Debye temperature by fitting at a certain temperature. We could also choose a temperature interval and fit, e.g. in a root-mean-square deviation sense. If we instead fit another property, e.g. the vibrational entropy, and at another temperature, we will not get the same inline image or inline image as in the first mentioned fit. Thus there are infinitely many Einstein and Debye temperatures for a given substance, depending on which property is fitted and at which temperature. Typically they may differ by 10–20%, as shown in several graphs in ([44]). This is because, e.g. the heat capacity, the entropy, the Helmholtz energy and the vibrational displacement amplitude all weight the individual phonon frequencies in different, and temperature dependent, ways. It is often asked what is the Debye temperature of this substance. It is clear from our discussion that there is no single and unique Debye temperature. Tables often give the value obtained from a fit to the heat capacity at very low T (cf. Eq. (1)). This means a fit to an extreme part of the DOS inline image (the low-energy limit) and at an extreme temperature.

6.3 Model representations of anharmonic phonons

Since the late 1960s, there is a well-established general theory for how lattice vibrations are affected by anharmonicity. The phonon frequency of a mode inline image can be written

  • display math(27)

Here inline image is the phonon frequency in the harmonic approximation, at the crystal equilibrium volume inline image. The term inline image is the shift in frequency caused by thermal expansion, when the vibrations are still treated in the harmonic approximation but at the new volume V. This is the QHA. The terms inline image and inline image represent anharmonic shifts obtained in low-order quantum mechanical perturbation theory. Let u be the displacement amplitude and expand the potential energy in u, keeping the inline image and inline image terms. To lowest order in perturbation theory only the inline image term enters since inline image is an odd function in u. Therefore, we should also keep the second-order perturbation contribution containing the inline image term. The corresponding shifts in the phonon frequencies, inline image and inline image, are present even if the crystal is kept at fixed volume (i.e. they are present in the heat capacity inline image).

Neglecting further corrections in Eq. (27), this is the frequency measured, e.g. in neutron scattering. Further, it has been shown that anharmonic corrections to the vibrational entropy are correctly accounted for (within the low-order perturbation theory) if inline image from Eq. (27) are used as frequencies in the entropy expressions for harmonic vibrations. In this respect, the entropy differs from other thermodynamic expressions for harmonic vibrations. For instance, Eq. (27) used as frequencies in the expression for the heat capacity will obviously not give a correct result, since the harmonic heat capacity is limited to inline image/atom at high T.

All three terms inline image, inline image and inline image increase linearly in T at high T (typically inline image). The shift inline image has already been discussed within the QHA. It can be more important than inline image and inline image, thus accounting for the major part of inline image. This has sometimes led to the incorrect conclusion that inline image must have the high-temperature limit 3 inline image/atom.

The explicitly anharmonic shifts inline image and inline image are difficult to calculate (cf. Section 'Beyond the quasiharmonic approximation'). A further complication is that they are only the first terms in a series of corrections that become increasingly important at high T. It is not known to what extent it is still a good approximation to represent these high-order anharmonic effects by shifted frequencies used in the harmonic vibrational entropy expression. MD calculations offer a possibility to obtain the anharmonic contributions to thermodynamic properties beyond the QHA (cf. Section 'Beyond the quasiharmonic approximation'). The result may be represented as a power series in T,

  • display math(28)

This is valid at high T (typically inline image) and would overestimate the anharmonicity at lower T. Here, we have only given a brief account of how anharmonicity affects thermodynamic properties. Details can be found, e.g. in [22][44].

6.4 Grüneisen parameters

Anharmonic phonon frequency shifts are accounted for in the QHA as the volume dependence of the frequencies. From a mathematical point of view, in a system with strictly harmonic vibrations the frequencies cannot depend on the volume (lattice parameter), and there is no thermal expansion. The fact that there is thermal expansion shows that the phonons are not entirely harmonic. A simple explanation goes like this. When atoms are displaced from their symmetry positions in a lattice, they are forced back by interatomic forces. If the displacements are sufficiently small, we can describe the vibrations of the atoms under the influence of these forces as harmonic. If now the crystal volume V (the lattice parameter) has a new value (due to thermal expansion or external strain), the restoring forces, and hence the vibrational frequencies, will change. The frequency shift is usually represented by a Grüneisen parameter

  • display math(29)

Similarly, we can define Grüneisen parameters inline image and inline image for the Einstein and Debye temperatures. This approach has the same drastic reduction in information as was discussed above when the full DOS was represented by Einstein and Debye temperatures. There is no single Grüneisen parameter. One must specify the property of interest and the temperature. Different Grüneisen parameters may differ by a factor of two, i.e. a larger spread than is normally found for the Debye temperature; see, e.g. Fig. 8.1 in Ref. ([44]). Nevertheless, there is in fact a unique quantity called the ‘thermodynamic Grüneisen parameter’. It is defined as

  • display math(30)

Here, inline image is the thermal expansion coefficient, inline image the isothermal bulk modulus and inline image the heat capacity at constant volume. The thermodynamic Grüneisen parameter is consistent with the difference in phonon entropy obtained from the difference inline image. If the thermal expansion is entirely described by the quasiharmonic model, we get inline image directly related to the average of the phonon mode Grüneisen parameters inline image. See Refs. [27][44] for further details about the QHA and the Grüneisen parameters.

6.5 Representation of electronic states

The most important quantity regarding the thermodynamics of electrons is the DOS at the Fermi energy, inline image. In textbooks, it is common to introduce the Fermi temperature inline image as inline image. This may be a useful concept for systems that are well described by the free-electron model but cannot be used, e.g. for transition metals. We remark that there is no unique reference level from which to calculate energies, unlike entropy where the 3rd law says that inline image. The Fermi energy inline image in metals is just a label referring to the highest occupied level of the electron states at inline imageK. We may take inline image or any other value following a certain energy convention.

Given the electron DOS inline image for sharp electron states, we can easily calculate the thermodynamic functions even at high temperatures, as shown in Section 'Calculation of the electronic contribution'. However, at high temperatures the electron states are increasingly damped (i.e. broadened in energy). This would result in a smearing in inline image. Very little is known about this effect, but first-principles work using AIMD by Asker et al. ([45]) suggests that it can be significant.

Thus, if inline image calculated at inline imageK is used to obtain the electronic heat capacity inline image and related properties at high T, there are two simultaneous and related complications when inline image is rapidly varying near the Fermi energy inline image. First, inline image depends on inline image in an energy window of increasing width, and, second, inline image itself becomes increasingly smeared because of the thermal disorder in the lattice. The two effects cannot be dealt with separately, but since the former involves a standard and accurate calculation, in contrast to the latter, we suggest that the electronic heat capacity is represented as

  • display math(31)

Here inline image is the heat capacity calculated as in Section 'Calculation of the electronic contribution' and inline image is a broadening or smearing function that might be written as a power series inline image The motivation for this separation is that inline image already captures some smearing effects, and the remaining correction inline image could be small. It remains for future research to see if this gives a reasonable picture.

6.6 Representation of thermodynamic quantities in current databases for crystalline phases

Within the SGTE unary database the Gibbs energy of each structure of an element is described relative to the so called stable element reference (SER), which means that the Gibbs energy is taken relative to the enthalpy at 298.15 K and entropy at 0 K for the structure which is stable at 298.15 K and inline image Pa. The Gibbs energy is given as consecutive series in temperature, nominally starting at 298.15 K and ending at some temperature considered to be sufficiently high above the melting temperature. Temperatures below 298.15 K were not considered important for CALPHAD purposes, which were mostly directed towards high temperatures where phase equilibria can be achieved. The form of the series is

  • display math(32)

The corresponding heat capacity is

  • display math(33)

The Gibbs energies of non-stable structures were still described relative to the stable structure using either a constant term or a linear temperature dependence just as with the original lattice stabilities, but with changes in the actual numbers used. The inline image term in Eq. (33) leads to a rapid increase in the heat capacity at high temperature, in particular above the stable melting point. If this extrapolation is continued, it often leads a restabilization of the solid phase below 6000 K. This temperature is a standard maximum temperature above which the properties of condensed phases are considered to be of no interest. To avoid this, the polynomial (Eq. 32) was cut at the stable melting point and a further polynomial added for the ‘metastable’ range above the melting point. This polynomial has the same form as that of the liquid phase and contains an additional inline image term, which smoothly brings the heat capacity of the solid phase towards that of the liquid phase. This is illustrated by the heat capacity for Al from the SGTE unary database in Fig. 2. The corresponding series used in the SGTE database are given below:

  • display math(34)

These expressions give the Gibbs energy in joule/mole when T is the numerical value of the temperature in kelvin. We note that the SGTE description divides the temperature in intervals with different assigned series. This was necessary for a proper description over the whole temperature range. Above the melting point (933.47 K), the inline image term guarantees that the heat capacity of fcc Al approaches that of the liquid phase.

The SGTE unary database was intended to provide reliable results above 298.15 K. Extrapolation down to 0 K can give completely non-physical results. The recent progress in first-principles calculations, which in most cases are performed at 0 K, has made it important to extend the SGTE unary database below room temperature. Finally, we note that it is difficult to establish a physical interpretation of the coefficients in the SGTE polynomial description, since it is just a fitting to data.

7 Metastable and unstable phases

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

The majority of the elements have a thermodynamically stable low-temperature phase with fcc, hcp or bcc crystal structure. It is often believed that when a close packed phase (fcc or hcp) is stable, the bcc phase is a possible metastable phase, and vice versa. However, this is not the case. On the contrary, when a transition metal has a stable close packed crystal structure, the corresponding bcc phase usually is dynamically (mechanically) unstable, i.e. it has some vibrational modes with imaginary frequencies. Likewise, if the bcc structure is stable, the two close packed structures tend to be dynamically unstable. There are some exceptions, notably Fe, but dynamical instability is very common, not only in elements but in alloys and compounds; see ([49]) for a detailed review.

The dynamical instability may be present at T = 0 K, or develop gradually as the temperature is increased. More common is that a structure is stable at high T but becomes unstable when T decreases (bcc Ti, Zr, Hf). This has been proved by AIMD calculations, for example for Mo ([45]). In other cases, the dynamical instability persists from 0 K to high T, as for fcc W ([50]). Under increasing pressure, a phase may become unstable or be stabilized (bcc Mg stabilized at high pressure). See also the accompanying paper on pressure effects ([18]). A change in composition may result in an analogous behaviour. For instance, the solid solution of W in fcc Pt is the thermodynamically stable phase in a wide range of compositions, but fcc W is dynamically unstable.

There is no other simple way to decide if a certain lattice structure is stable, but to investigate in a theoretical calculation if imaginary phonon frequencies occur. A first check should also be made for the elastic constants, since many structures are unstable only for some long-wavelength phonons.

In a dynamically unstable phase, the thermodynamic functions like S, F, G and inline image have no meaning. Consider the Gibbs energy, separated in the two parts

  • display math(35)

We could obtain the enthalpy H (or energy U) in a first-principles calculation at inline imageK even if the lattice is unstable, because H and U are then calculated for a rigid lattice. But such information cannot even give inline imageK), because the entropy S has no physical meaning for an unstable system. It follows that enthalpy differences inline image, although possible to calculate by first-principles, have a physical meaning only if both inline image and inline image are (meta)stable phases. The consequences of instabilities for the CALPHAD-type calculation of phase diagrams is dealt with in Ref. ([49]).

The CALPHAD models for solid solution phases such as the fcc, bcc and hcp phases are defined across the complete composition range from pure element A to pure element B. This requires that the Gibbs energy is defined for all elements that are included in the solid solution model. It also requires that the Gibbs energy varies smoothly from pure element A to pure element B. More specifically, this means that Gibbs energies must be assigned also to phases that are mechanically unstable, such as fcc Cr, although this can be questioned from a theoretical point of view. These Gibbs energies should be selected to give a smooth continuation of the Gibbs energy through the unstable range, which is current CALPHAD practice (see beginning of Section 'Representation of thermodynamic quantities in current databases for crystalline phases'). This does not necessarily mean that the unstable range is considered to be metastable. It is merely a convenient reference that makes it possible to describe multicomponent solutions in a consistent way.

8 Thermodynamic functions in the superheated regime

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

The heat capacity C (=inline image or inline image) or other thermodynamic properties of a solid are well defined also in the metastable superheated regime, until the lattice eventually becomes unstable. There are several possible instability mechanisms, entering at different temperatures ([49]). Although not much reported in detail in the literature, both QHA ([51]) and AIMD calculations show clearly that some phonon frequencies become imaginary at some high T for some elements and compounds. Vacancies also play a role at temperatures close to and above the melting point (see accompanying paper on defects ([21])). Finally, the concept of entropy catastrophe may be useful, although controversial, to set an upper boundary to the maximum temperature where a crystalline solid may exist ([52]). The temperature where the extrapolated entropy of the overheated crystalline solid will be higher than the entropy of the corresponding liquid phase is accordingly considered as a higher limit to the stability range of the crystal, whatever instability mechanism is actually taking place.

In the superheated regime, anharmonic effects may become particularly important and thus the temperature where such instabilities appear will be better evaluated by AIMD rather than in the QHA. However, as it was already pointed out, the computational effort involved in such calculations is (at present) too large to be systematically extended to all the elements in a dense temperature grid of points.

Let inline image be the lowest temperature where an instability arises in the phase inline image. Above inline image the heat capacity and all other thermodynamic quantities have no physical meaning. This fact affects how C, S and G should be modelled at high temperatures. In particular, if the heat capacity of the solid phase inline image continues indefinitely to increase with T, while the liquid heat capacity remains fairly constant, the solid would re-enter as the thermodynamically most stable phase at very high T. To avoid that behaviour, the SGTE unary database has adopted the ad hoc solution of introducing additional inline image and inline image terms in the Gibbs energy (Eq. 34, cf. Section 'Representation of thermodynamic quantities in current databases for crystalline phases' and Ref. ([6])), to make the heat capacity approach that of the liquid at high T. Here we will introduce a physically more transparent approach. Rather than trying to model the inline image phase correctly until it suddenly ceases to exist as a metastable phase at inline image, we imagine a fictitious phase inline image. Its thermodynamic properties agree with those of the inline image phase up to a temperature inline image close to inline image, where they change smoothly but rather rapidly towards values consistent with a temperature independent heat capacity. The implementation of this procedure leaves us with a number of decisions to be taken regarding the definition of inline image, the high-temperature properties of inline image, the mathematical form of the variation near inline image and the thermodynamic quantity (C, S or G) that is primarily modelled. The instability temperature inline image, and the corresponding instability mechanism, are still unknown for the elements. This is not so important for the modelling of an equilibrium phase diagram since the inline image phase is far from being the most stable phase at T near inline image. Nevertheless, we need a prescription for the choice of a reasonable inline image of the fictitious phase and propose that inline image is the temperature at which the entropy of the extrapolated inline image phase would become equal to the entropy of the liquid phase. It makes inline image equal to inline image in the ‘entropy catastrophe’ model ([52]), but this fact should not be given a deep significance since the relevance of the ‘entropy catastrophe’ model is not well established. Since the phase inline image is introduced in order to prevent superheated solid phases from re-entering above the melting temperature, inline image should have a Gibbs energy inline image lower than or equal to that of the liquid at high T. We choose to let the thermodynamic functions of inline image be equal to those of the liquid in the limit of high T. In order to model the transition from the low temperature to the high temperature regime, we need a weighting function f that is similar to a step function, but smoothly varying so that it gives weight 1 to the properties of the inline image phase and weight 0 to those of the liquid at T well below inline image, with the reversed weights at T well above inline image. A natural mathematical form would be analogous to the Fermi–Dirac function. We take the weight f for the inline image phase and inline image for the liquid phase, with

  • display math(36)

The quantity inline image gives the width of the transition region. In order to get a definite value, we somewhat arbitrarily choose inline image. It remains to decide which of C, S and G is taken as the primary quantity in the description of the inline image phase. We will illustrate the advantages and disadvantages with the three choices

  • display math(37)
  • display math(38)
  • display math(39)

where inline image, inline image, inline image, inline image, inline image and inline image are defined at all T, when necessary through extrapolation. Equation (37) has the advantage that the right hand side refers to directly measurable quantities. The disadvantage is that the weight factors in Eq. (38) no longer are as simple as inline image and f but involves an integration that must be performed numerically, with a further integration to get inline image. Weighting as in Eq. (38) has the advantage that information on anharmonicity and positional disorder can be included in the entropy in a natural way. Moreover, the standard entropy at 298.15 K is a quantity of interest, which may be estimated without detailed heat capacity data using first-principles methods. A drawback is that the weight factors become mathematically complicated in Eq. (37), and must by calculated numerically in Eq. (39). Using Eq. (39) with its simple weight factors inline image and f has the advantage that inline image retains an explicit and simple mathematical form when inline image and inline image are given as series expansions like in Eq. (34). The drawback is that the second derivate of inline image and f with respect to T changes sign at inline image, which leads to unphysical negative and positive narrow spikes in the heat capacity inline image around inline image.

9 Conclusions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

We have outlined a scheme for obtaining and representing the heat capacity inline image and related thermodynamic quantities for elements in their crystalline state. The focus was on contributions from phonons and electrons. Magnetism, lattice defects, liquids and high-pressure effects are dealt with in companion papers. In the ideal case accurate measurements of inline image from 0 K to the melting temperature would be available. While this is true for some elements, e.g. Al, Cu and Fe, the results for other elements are lacking in some temperature intervals and show large differences between different measurements. It seems likely that these problems with direct experimental heat capacity data will remain for a considerable time. Although not yet included in many thermodynamic databases, volume and thermal expansion data are also important and in the future, we suggest including lattice parameter data directly in the database. The database then also must provide crystal structure information. For cubic structures volume and lattice parameter data are equivalent, but for other symmetries lattice parameter data provide additional information. The lattice parameter data can be used not only to calculate volumes, but also to calculate lattice mismatch, which may serve as input for calculation of interfacial energies and strains (and stresses) caused by coherent precipitates, for example. For details, see the accompanying paper on the effects of pressure and stress ([18]). Since many thermodynamic quantities are connected through exact mathematical relations, information on, e.g. thermal expansion, elasticity and heat of formation can be simultaneously assessed to make the best use of all available information.

Heat capacity measurements have, until recently, been the only ones for practical use. However, thermodynamic information can sometimes be extracted from other kinds of experiments. In particular, neutron scattering provides data on the phonon frequencies, from which the corresponding contribution to the heat capacity can be derived. Although the phonon DOS is measured at discrete temperatures, interpolation methods can work well for obtaining DOS curves at temperatures between the measurements. A data structure that parametrizes the intensity at each phonon energy as a straight line in T worked well for aluminium, but a higher-order polynomial may be appropriate in other cases. Phonon spectra measured at elevated temperature include effects from both quasi harmonicity and anharmonicity. In the case of aluminium, it seems that the temperature dependence of the phonon DOS is predicted well from a quasiharmonic model, although this seems to involve a cancellation of cubic and quartic anharmonic effects (cf. Eq. (27)) because we see that the cubic anharmonicity is large, and causes a significant lifetime broadening of phonons at elevated temperatures. By comparison with thermodynamic data, the present work showed that the central frequencies of the phonons, not their full spectral profile, is the appropriate information to use in standard expressions for the vibrational entropy. Lifetime broadening, which shifts intensity to higher energies, suppresses the apparent vibrational entropy of aluminium calculated from phonon DOS curves. A method for its correction was proposed. Although the method proved successful, and can be justified on physical grounds, further investigations may be appropriate. It should also be stressed that our purpose has been to discuss general aspects of anharmonicity in connection with neutron scattering experiments. Our example (aluminium) is unusual because it has large cancellations of the cubic and quartic parts of the anharmonicity. Neutron scattering experiments on Cr at high temperatures ([53]), and an analysis of thermodynamic data for Mo and W using Eq. (3) ([54]), indicate that these metals have large anharmonic effects on the central frequencies of the phonons, and large anharmonic contributions to the vibrational entropy. Experimental measurements of phonon energies can thus be helpful for evaluating models of non-harmonic behaviour.

In addition to conventional thermodynamic experiments and new measurements on phonon frequencies, first-principles calculations now have reached such a high accuracy that they may be a substitute for experiments in many cases. Moreover, they are much faster and cheaper than experimental work. Several examples in this paper have shown what can be achieved. However, we also remarked that in the high-temperature regime when anharmonic effects beyond the QHA may become important, calculations are still difficult and time consuming. Furthermore, first-principles results in the most common implementation using DFT suffer from the unknown exact form of the exchange-correlation functional, even at 0 K. Key input experimental data, such as volume and bulk modulus data at room temperature, can be used to further enhance the accuracy of a DFT-based approach.

Phonons give the dominant contribution to inline image. At low temperatures (typically up to some characteristic Debye temperature), anharmonic effects are small. Nevertheless, inline image should be calculated from the detailed shape of the phonon DOS inline image, given as numerical values in a table. Replacing inline image with a Debye or Einstein spectrum introduces an unnecessary loss of information. At high temperatures, anharmonic effects become important. Sometimes they can be dealt with in the QHA and by incorporation into inline image, but near the melting temperature explicit and high-order anharmonicity may give a significant contribution. What is here referred to as anharmonicity may not be just a direct consequence of increasing vibrational displacement in an effectively anharmonic potential, but an indirect result of changes in the electronic structure when the electron scattering becomes strong in the thermally disordered lattice. The only realistic approach to the phonon terms beyond the QHA is to parametrize their effect on inline image, e.g. as a power series in T.

The contribution to inline image from the electrons has a simple form at low temperatures, with inline image. This is an excellent approximation up to temperatures where inline image starts to deviate significantly from the value inline image within the characteristic energy window inline image. Then the calculation must use the full shape of inline image. Further, there is a smearing effect in inline image, as discussed in Section 'Calculation of the electronic contribution'. We conclude that an assessment of inline image should first rely on numerical tables of the phonon DOS inline image in the QHA and on the single-particle electron DOS inline image. At high temperatures, corrections should be added for explicit anharmonicity and sometimes also smearing in inline image. Those corrections may be poorly understood. Since they vary slowly, a power series in T can be an adequate representation. If nothing is known about the high-temperature corrections, the quasiharmonic and the single-particle electron descriptions may still suffice to give a useful description.

The primary data in our approach, inline image, are given as entries in tables. Then, secondary data are derived for inline image and related quantities, and given in tables as a function of temperature. If the secondary data are also fitted to simple mathematical expressions is a matter of taste and practicality.

Finally, our approach assumes that it is possible to identify the phonon and electron contributions to the heat capacity as separate terms. This is an excellent approximation at low and intermediate temperatures, but may not be fully adequate at very high temperature. A self-consistent treatment of electrons and lattice vibrations, which combines MD and DFT could cast light on the limitations of the scheme used in this paper, but a revision of the SGTE-type database cannot wait for such future development. Thus, experimental validation at high-temperatures is still essential.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References

We are grateful to all participants of the Ringberg workshop 2013 for discussions, particularly B. Sundman and S. G. Fries. We also thank F. Körmann for providing some of the first-principles results reported in this work. M.P. acknowledges financial support through ThyssenKrupp AG, Bayer MaterialScience AG, Salzgitter Mannesmann Forschung GmbH, Robert Bosch GmbH, Benteler Stahl/Rohr GmbH, Bayer Technology Services GmbH, the state of North-Rhine Westphalia, the European Commission in the framework of the ERDF and the Deutsche Forschungsgemeinschaft (DFG) through projects C6 of the collaborative research center SFB/TR 103. B.H. acknowledges financial support from Deutsche Forschungsgemeinschaft (DFG) through the bundled project PAK 461 (HA 5382/3-2). B.F. acknowledges financial support from the U.S. DOE BES under contract DE-FG02-03ER46055.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Review of experimental data
  5. 3 Volume and thermal expansion
  6. 4 Entropy from phonon measurements
  7. 5 First-principles approaches
  8. 6 Representation of data
  9. 7 Metastable and unstable phases
  10. 8 Thermodynamic functions in the superheated regime
  11. 9 Conclusions
  12. Acknowledgments
  13. References