Currently there are essentially two methods in use for the first-principles calculations of phonon frequencies: the linear response theory and the direct approach. The linear response theory evaluates the dynamical matrix through the density functional perturbation theory. In comparison, one advantage of the direct or mixed-space approach over the linear-response method is that it can be applied with the use of any code capable of computing forces. The direct approach is also referred to as the small displacement approach, the supercell method, or the frozen phonon approach.

However, none of the previous implementations of the supercell approach are able to accurately handle long-range dipole–dipole interactions when calculating phonon properties of polar materials. The problem has been solved by the parameter-free mixed-space approach, which makes full use of the accurate force constants from the supercell approach in real space and the dipole–dipole interactions from the linear response theory in reciprocal space. The mixed-space approach is the only existing method that can accurately calculate the phonon properties of polar materials within the framework of the supercell or small displacement approach.

The program YPHON is written in C++ and can be downloaded at http://cpc.cs.qub.ac.uk/summaries/AETS_v1_0.html. The precompiled executable binaries should work for most Linux and Windows systems. Recompiling YPHON requires the GNU Scientific Library (GSL), which is a numerical library for C and C++ programmers. If one is just interested in phonon dispersions, the phonon density-of-states (PDOS), or the neutron scattering cross-section weighted PDOS the so-called generalized phonon density-of-states (GPDOS), this is enough. YPHON also makes it a lot easier to plot phonon dispersions and PDOS. In this case, it is required that Gnuplot be installed.

The static energy and force constants from first-principles calculations need to be formatted to the YPHON input formats (text formats as detailed later). At present, YPHON works closely with VASP.5 or later. The mixed-space approach has built up a unique base of the supercell approach to polar materials and has been adopted in a number of software tools such as CRYSTAL14 by R. Dovesi, ShengBTE (a solver of the Boltzmann transport equation for phonons) by W. Li, J. Carrete, N. A. Katcho, and N. Mingo, the Phonopy package by Atsushi Togo, and the Phonon Transport Simulator (PhonTS) by Chernatynskiy and Phillpot.

The CALPHAD modeling of thermodynamics was pioneered by Kaufman and Bernstein [47] and has been reviewed in detail by Saunders and Miodownik [48] and Lukas, Fries, and Sundman [49]. Information on features of software tools for CALPHAD modeling can be found in two series of publications in the CALPHAD journal [50, 51]. The key feature of the CALPHAD method is the modeling of the Gibbs energy of individual phases using both thermodynamic and phase equilibrium data. The main significance of the CALPHAD method is the following.

1. i. It enabled the development of the concept of lattice stability, i.e. the energy difference between the stable and non-stable crystal structures of a pure element.

2. ii.The Gibbs energy expression of each phase covers the full temperature, pressure, and composition spaces including both stable and non-stable regions of the phase. This enables the evaluation of the Gibbs energy of a system as a function of non-equilibrium state, i.e. with ξ as an independent variable.

3. iii.Thermodynamic data are usually obtained by measurements of heat such as the enthalpy of transition and heat capacity, as discussed in Section 4.2, which have large uncertainties typically in the range of kilojoules per mole-of-atoms. On the other hand, phase equilibrium data as discussed in Section 4.1, though more accurate, only contain information on compositions of phases at equilibria, i.e., the relative Gibbs energy of phases at equilibrium. The combination of these two sets of data is foundational in CALPHAD modeling and allows for the accurate modeling of thermodynamic properties of individual phases and reliable calculations of phase stability and driving forces.

4. iv. CALPHAD provides a framework to model thermodynamic properties of multi-component systems of industrial importance, enabling computational materials design. It has also been extended to model a range of properties of individual phases in multi-component systems such as diffusion coefficients, elastic coefficients, and thermal expansion, supplying input data for computational simulations of phase transformations during materials processing.

In this chapter, the basics of CALPHAD modeling of the Gibbs energy of individual phases are presented. For detailed implementations in various software packages and modeling procedures, readers are referred to the references given above.

Importance of lattice stability

For modeling of the Gibbs energy of individual phases, it is necessary to define the values of °Gi in Eq. 2.48.

General condition for equilibrium

A system is heterogeneous if some properties have different values in different portions of the system when the system is at equilibrium. Two scenarios may exist, where variations of the properties can be either continuous or discontinuous. In the scenario of continuous variations, the gradients of the variations must be coupled so that the system remains at equilibrium. The number of independent variables is thus reduced. These gradients must also be constrained along the boundaries between the system and the surroundings. This type of constrained equilibrium is not discussed in the book as it involves heterogeneous boundary conditions between the system and the surroundings and depends on the morphology of the system.

In the second scenario, with discontinuous variations, the properties have different values in different portions of the system, but remain homogenous within each portion. The system is in equilibrium as each portion is in equilibrium with all other portions of the system. The homogeneous portions represent different phases in the system, with the properties in each phase being homogeneous at equilibrium. In the previous chapter, it was been shown that all potentials are homogeneous in a homogeneous system.

For a heterogeneous system, the same conclusion can be obtained. If the temperature is inhomogeneous, heat can be conducted from high temperature locations to low temperature locations, and this process is irreversible based on the second law of thermodynamics because it increases the internal entropy of the system. If the pressure is inhomogeneous, the amounts of lower molar volume phases will increase to reduce the internal energy of the system. If the chemical potential of a component is inhomogeneous, the chemical potential difference of the component will drive that component to locations with a lower chemical potential in order to decrease the internal energy of the system. Therefore, it can be concluded that all potentials are homogeneous in a heterogeneous system at equilibrium, and the variables that are not homogeneous are thus their conjugate molar quantities. Under certain special circumstances, to be discussed later in this book, some molar quantities may also have the same values in difference phases.

In the previous chapter, the experimental techniques used to obtain the thermochemical and phase equilibrium data that were the inputs for the thermodynamic modeling of a system were summarized. However, experimental data are not always available. This is due to the fact that (i) the experiments are expensive, especially when they involve developing new materials, and (ii) the experiments cannot reliably access the non-stable phases in most cases. The alternative approach is to predict the thermochemical data by first-principles calculations. The prediction of material properties, without using phenomenological parameters, is the basic spirit of first-principles calculations. In particular, the steady increase of both computer power and the efficiency of computational methods have made the first-principles predictions of most thermodynamic properties possible, including both enthalpy and entropy as a function of temperature, volume, and/or pressure.

By definition, the term “first-principles” represents a philosophy that the prediction is to be based on a basic, fundamental proposition or assumption that cannot be deduced from any other proposition or assumption. This implies that the computational formulations are based on the most fundamental theory of quantum mechanics, the Schrödinger equation or density functional theory, and the inputs to the calculations must be based on well-defined physical constants – the nuclear and electronic masses and charges. In other words, once the atomic species of an assigned material are known, the theory should predict the energies of all possible crystalline structures, without invoking any phenomenological fitting parameters.

This chapter is organized in sequence from thermodynamic calculations to fundamental theory, to help those readers who are more interested in realistic calculations using existing computer codes. Detailed theoretical discussions follow the subsections on thermodynamic calculations for those readers who are also interested in the derivation of the formulations used in the thermodynamic calculations. The subsections are arranged accordingly in the order: (i) examples of the commonly adopted calculation procedures for thermodynamic properties using the elemental metal nickel as the main prototype; (ii) derivation of the Helmholtz energy expression under the first-principles framework; (iii) introduction of the solution to the electronic Schrödinger equation within two well-developed frameworks – the quantum chemistry approach and the density functional theory; (iv) detailed description of the procedure on how to solve the Schrödinger equation for the motions of atomic nuclei by means of lattice dynamics; and (v) First-principles approaches to disordered alloys.

The most widely used thermodynamic modeling technique is the CALPHAD (CALculation of PHAse Diagram) method to be discussed in detail in Chapter 6. The input data in the evaluations of thermodynamic model parameters came primarily from experiments and estimations until first-principles calculations based on the density functional theory [8] became a user tool in the later 1990s. Experimental data include both thermodynamic and phase equilibrium data; the first-principles calculations, which provide thermodynamic data for individual phases, are discussed more extensively in Chapter 5.

Three fairly recently published books summarize the methods commonly used for experimental measurements of the thermodynamic properties of single [9] and multiple phases [10] and phase diagrams [11]. The methods are briefly discussed here, and readers are referred to these books for details. The main techniques for crystal structure analysis include X-ray diffraction, electron backscatter diffraction (EBSD), electron diffraction in transmission electron microscopy, neutron scattering, and synchrotron scattering, which are not discussed in this book.

Phase equilibrium data

Equilibrated materials

The most common method to determine phase equilibria is to use equilibrated materials. This method typically involves material preparation through high temperature melting or powder metallurgy, homogenization heat treatment, isothermal or cooling/heating procedures, and identification of crystal structures and phase compositions. It is important to avoid macro-inhomogeneity as it can be difficult to remove the inhomogeneity in subsequent treatments. It is also important to use starting materials of the highest purity and to minimize the loss and contamination of materials during the entire experiment using a protective atmosphere of inert gas or vacuum. Typical melting techniques include high temperature furnaces with crucibles, arc melting, and induction melting. Attention needs to be paid to possible reactions between materials and crucibles/containers, which can be avoided by levitating the materials by electromagnetic fields or other means. In addition to using pure elements as raw materials, master alloys with well-controlled compositions are often utilized because the compositions and melting properties of master alloys are usually much closer to those of the final materials than the pure elements. For materials with very high melting temperature or volatile components, the powder metallurgy method can be used where compacts are made, capsulated, and sintered.

Homogenization during subsequent heat treatment is achieved through diffusion, in which time and temperature are two important parameters.