A First-Principles Study of the Structural and Thermo-Mechanical Properties of Tungsten-Based Plasma-Facing Materials

Abstract

Tungsten (W) and tungsten alloys are being considered as leading candidates for structural and functional materials in future fusion energy devices. The most attractive properties of tungsten for the design of magnetic and inertial fusion energy reactors are its high melting point, high thermal conductivity, low sputtering yield, and low long-term disposal radioactive footprint. Despite these relevant features, there is a lack of understanding of how the structural and mechanical properties of W-based alloys are affected by the temperature in fusion power plants. In this work, we present a study on the thermo-mechanical properties of five W-based plasma-facing materials. First-principles density functional theory (DFT) calculations are combined with the quasi-harmonic approximation (QHA) theory to investigate the electronic, structural, mechanical, and thermal properties of these W-based alloys as a function of temperature. The coefficient of thermal expansion, temperature-dependent elastic constants, and several elastic parameters, including bulk and Young’s modulus, are calculated. Our work advances the understanding of the structural and thermo-mechanical behavior of W-based materials, thus providing insights into the design and selection of candidate plasma-facing materials in fusion energy devices.

1. Introduction

Tungsten (W) is considered a leading candidate for structural plasma-facing materials (PFMs) in experimental fusion reactors given their high melting point, high thermal conductivity, low sputtering yield, and low long-term disposal radioactive footprint [1,2]. However, these attractive properties are unfortunately accompanied by a very low fracture toughness (mostly associated with inter-granular failure and bulk plasticity), low ductility at room temperature, and high ductile-to-brittle transition temperature (DBTT), which limit its applications [1,2,3,4,5].

To overcome such limitations, several families of W-based alloys have been explored in the past decades. High-entropy alloys (HEAs), originally conceived in the early 2000s as a blend of five or more elements with individual concentrations between 5 and 35 atom percent [6], have emerged as a promising class of materials given their remarkable properties [7,8,9,10]. In particular, tungsten-based HEAs (W-HEAs), designed to withstand the extreme environments of fusion energy applications [11,12], have shown superior mechanical properties at high temperatures, a superior melting point (above 2873 K), enhanced radiation resistance to heavy ion irradiation, and negligible radiation hardening when compared to pure tungsten [11,12,13,14,15,16]. Another potential alternative are the so-called tungsten-based “SMART alloys” (W-SAs), where SMART stands for Self-passivating Metal Alloys with Reduced Thermo-oxidation, which are capable of adjusting their properties to the environment [17,18,19,20,21]. For example, when exposed to the extreme temperatures achieved during a loss-of-coolant accident (LOCA) combined with air ingress, W-SAs containing small amounts of Ti or Y have shown the capability to create stable oxides that prevent the sublimation of W oxides.

A deep understanding of the response of these candidate PFMs to the extreme environments expected under fusion power plant conditions is a necessary step toward their consolidation as a viable option for this promising technology. Despite the numerous efforts in the literature to investigate the effects of alloying elements on various properties of W-based materials, such as phase stability [22,23,24,25,26], elastic properties [22,23,24,25,27,28,29], ideal tensile strength [25,30], ductility [31], radiation defects [11,23,32], neutron irradiation-induced precipitation [26,33,34,34], point defects [22,32,33,35,36,37,38], screw dislocation structures [39,40,41,42,43,44], grain boundaries [45], transmutation effects [46,47], etc., to the best of our knowledge, the challenge of understanding and predicting the temperature dependence of thermo-mechanical behavior of tungsten-based PFMs remains [48,49].

Given the difficulty and cost of performing experiments in such extreme environments, the use of computational modeling to provide insights and enrich the experimental knowledge of materials response has received much attention during the last several decades. Among the available techniques that allow for the prediction of mechanical properties across different temporal and spatial scales, the 2020 Roadmap of multiscale materials modeling [50] identifies density functional theory (DFT) as one of the most reliable methods for investigating the electronic structure of condensed matter systems [51]. While most DFT calculations focus on obtaining structural, mechanical, and vibrational properties of materials in their ground state, i.e., in the lowest-energy state at T = 0 K, previous experimental works have already reported changes in the elastic stiffness coefficients by over 20% as temperature increases [52,53,54,55,56]. Several first-principles approaches have also been formulated to account for the temperature dependence of the mechanical properties of materials. These include, for example, those combining DFT with Legendre transforms [57] or the quasiharmonic theory [29,58,59,60,61,62,63]. In this work, we integrate first-principles DFT calculations with the quasiharmonic approximation (QHA) to better understand the thermo-mechanical behavior of five W-based materials. In particular, we use direct information from our calculated phonon spectrum to investigate how the coefficient of thermal expansion (CTE) and elastic properties evolve with temperature.

This paper is organized as follows. After this introduction, we provide in Section 2 an overview of the computational methods employed. The results are given in Section 3, which includes (i) structural information about the random and minimum-energy configurations investigated; (ii) the phonon spectrum; and (iii) the evolution of the CTE, lattice parameter, and elastic properties with temperature. We finalize in Section 4 with a brief discussion and the conclusions in Section 5.

2. Computational Methods

The five W-based materials studied in this work are listed in

Table 1. Chemical composition of the five W-based alloys investigated in this work. The chemical composition of the two W-SAs was adjusted with respect to their reference value, given the number of atoms present in our bulk structures.

		Chemical Composition (at %) [No. Atoms in the Supercell]
		W	Cr	Ti	Y	Ta	V
W		100 [128]	-	-	-	-	-
W-HEA1	[64] and this work	25 [32]	25 [32]	25 [32]	-	25 [32]	-
W-HEA2	[64] and this work	25 [32]	-	25 [32]	-	25 [32]	25 [32]
W-SA1	[19]	67.16	26.98	5.86	-	-	-
	This work	67.19 [86]	26.56 [34]	6.26 [8]	-	-	-
W-SA2	[20]	67.93	31.11	-	0.958	-	-
	This work	67.19 [86]	31.25 [40]	-	1.56 [2]	-	-

. They include pure W, two W-based high-entropy alloys (W-HEAs), and two W-based smart alloys (W-SAs). A body-centered cubic (BCC) structure supercell containing 128 atoms was constructed for all materials, and the number of atoms was chosen to match the desired chemical composition.

2.1. Generation of Random and Minimum-Energy Configurations

Two types of atomistic configurations were investigated in this work: random solid solutions and minimum energy configurations (MECs).

Random solid solutions of the four W-based alloys listed in Table 1 are realized by generating special quasi-random structures (SQSs) [65] through the Alloy Theoretic Automated Toolkit (ATAT) [66,67]. The convergence criteria for selecting the final SQS structure were based on the Warren–Cowley short-range order (SRO) ${\alpha }_k^{ij}$ [68], defined as

$${\alpha }_k^{ij}=1-{\displaystyle \frac{{\rho }_k^{ij}}{c_i}}$$

(1)

where ${\rho }_k^{ij}$ is the possibility to find an atom i in the kth nearest-neighbor shell of atom j, and $c_i$ is the composition of atom i in the structure. As SQSs are generated, their SRO parameters are calculated. Once a specific SQS structure’s SRO = 0, no further SQS structures of that material system are created.

To find the ground-state configurations of W-based alloys at 0 K, an approach that couples DFT calculations with Monte Carlo simulations (denoted as MC-DFT hereinafter) was employed [69,70,71,72,73,74,75]. In essence, this MC-DFT method samples the phase space to find MECs by randomly swapping the chemical elements between atom locations. The probability of accepting or rejecting a swapped configuration depends on its energy and the Monte Carlo temperature. In this work, the SQS structures described above are considered as the starting points of the MC-DFT simulations. The MC temperature was set to 100 K, and the MC-DFT simulations were run for a total of 1000 steps. The reader is referred to the work by Qian et al. [75] for more details about the implementation and results of the MC-DFT approach in these W-based alloys. After obtaining the lowest-energy configuration from the MC-DFT simulations, the lattice parameter and internal atomic coordinates were further relaxed using VASP. The resulting configurations are considered as the ground-state structure of the W-based alloys at 0 K.

All DFT calculations were performed using the Vienna Ab initio Simulation Package (VASP) [76]. For the pseudopotential, the exchange–correlation functional with the Perdew, Burke, and Ernzerhof (PBE) [77] form under the generalized gradient approximation (GGA) [78] was chosen. The cutoff energy of the plane wave basis set was set to 300 eV. The Brillouin zone integration was performed on a 3 × 3 × 3 Monkhorst-Pack k-point mesh [79]. The conjugate gradient algorithm was used to relax the lattice structure, where the energy stopping criteria for the self-consistent electronic loop and ionic relaxation were set to 0.1 and 1 meV, respectively.

2.2. Quasiharmonic Approximation

To consider the temperature dependence of the lattice energy and structure, the QHA [80,81] is adopted. The QHA has been successfully employed in previous studies investigating temperature ranges below the melting point, where phonon–phonon interactions are weak. Let us denote the configuration of a general lattice by its primitive basis vectors $\mathit{a}=\{{\mathit{a}}_{\mathbf{1}},{\mathit{a}}_{\mathbf{2}},{\mathit{a}}_{\mathbf{3}}\}$. By minimizing the Helmholtz free energy $F(\mathit{a},T)$ with respect to the basis vectors $\mathit{a}$, the equilibrium configuration of the lattice at a specific temperature T can be obtained. In the QHA theory, the Helmholtz free energy is expressed as the sum of lattice internal energy $U\left(\mathit{a}\right)$, phonon vibrational energy $F_{\mathrm{ph}}(\mathit{a},T)$, and thermal electronic energy $F_{\mathrm{el}}(\mathit{a},T)$,

$$F(\mathit{a},T)=U\left(\mathit{a}\right)+F_{\mathrm{ph}}(\mathit{a},T)+F_{\mathrm{el}}(\mathit{a},T).$$

(2)

The vibrational energy of the lattice represents the contribution of thermal energy from phonons, which can be expressed as a summation of the phonon mode-dependent energy throughout the Brillouin zone [80,81,82],

$$F_{\mathrm{ph}}(\mathit{a},T)={\displaystyle \frac{1}{2}}\sum _{\mathit{q},j}{w}_{\mathit{q}}\hslash {\omega }_{\mathit{q},j}\left(\mathit{a}\right)-k_{B}T\sum _{\mathit{q},j}{w}_{\mathit{q}}ln(1-e^{-{\displaystyle \frac{\hslash {\omega }_{\mathit{q},j}\left(\mathit{a}\right)}{k_{B}T}}}),$$

(3)

where ${\omega }_{\mathit{q},j}$ is the frequency of the phonon branch indexed by a number j at wavevector $\mathit{q}$, ℏ is the Planck constant, and $w_{\mathit{q}}$ is the weight of wavevector $\mathit{q}$ in the summation.

The thermal electronic energy represents free energy contribution from electrons, in which the Fermi–Dirac distribution $f(ϵ,T)$ of an electron energy $ϵ$ is considered as temperature-dependent. Based on the assumption that the electron density of states (DOS) is temperature-independent, the free energy of electronic excitation is given by the entropy $S_{\mathrm{el}}$ and electron energy $E_{\mathrm{el}}$ as [83,84],

$$\begin{array}{cc}\hfill F_{\mathrm{el}}(\mathit{a},T)& =E_{\mathrm{el}}(\mathit{a},T)-T{S}_{\mathrm{el}}(\mathit{a},T)\hfill \\ \hfill \mathrm{with}{E}_{\mathrm{el}}(\mathit{a},T)& ={\int }_{-\infty }^{\infty }ϵnf,dϵ-{\int }_{-\infty }^{{ϵ}_F}ϵndϵ\hfill \\ \hfill S_{\mathrm{el}}(\mathit{a},T)& ={\int }_{-\infty }^{\infty }n\{flnf+(1-f)ln(1-f)\}dϵ,\hfill \end{array}$$

(4)

where the DOS $n=n(ϵ,T=0)$ at ground state is used at all temperatures, and ${ϵ}_F$ is the Fermi energy.

By inserting Equations (3) and (4) into Equation (2), the Helmholtz free energy is obtained as a function of basis vectors $\mathit{a}$ and temperature T. The lattice anharmonicity brought by temperature is considered through the lattice structural dependence of phonon frequencies ${\omega }_{\mathit{q},j}\left(\mathit{a}\right)$ and electron band structure $ϵ\left(\mathit{a}\right)$. Then, the equilibrium lattice configuration at any temperature T is obtained from ${\left.{\displaystyle \frac{\partial F}{\partial \mathit{a}}}\right|}_T=0$ by finding the one with the lowest free energy.

For simple bcc lattice structures as in this work, the only variable in the lattice basis vectors is the lattice parameter a. To find the lattice parameter at ${\left.{\displaystyle \frac{\partial F}{\partial \mathit{a}}}\right|}_T=0$, we applied strains ranging from −0.5% to 0.5% with an increment of 0.1% to the ground-state lattice configuration. Then, the phonon dispersion and electron DOS of these strained lattices were calculated, providing inputs to calculate the Helmholtz free energy using Equations (2)–(4).

All phonon calculations were carried out using the finite difference method implemented in VASP [85]. A single k-point was used, which is sufficient for providing accurate energy and force values for a large system with 128 atoms in this work. Central difference displacements of ±0.015 Å were applied to each atom and the second-order force constants were calculated using the resulting forces and energy. The dynamical matrix was constructed using the obtained second-order force constants. Then, the phonon dispersion was obtained by Fourier interpolation of the dynamical matrix onto different q-points in the Brillouin zone. For electron calculations, the Methfessel–Paxton scheme [79] was used to determine the partial occupancies of electron orbitals, and the width of smearing was set at 0.05 eV.

2.3. Elastic Properties

The elastic constants $C_{ij}$ were obtained by evaluating the change in the total energy per unit volume ${\displaystyle \frac{\Delta U}{V}}$ of a system subjected to a general deformation, i.e.,

$${\displaystyle \frac{\Delta U}{V}}={\displaystyle \frac{1}{2}}{C}_{ij}{u}_i{u}_j\mathrm{with}{u}_i=\left\{\begin{array}{c}{ϵ}_i\mathrm{if}i=1,2,3\hfill \\ 2{ϵ}_i\mathrm{if}i=2,3,4\hfill \end{array}\right.,$$

(5)

where V is the volume of the crystalline cell, $C_{ij}$ is a tensor comprising the elastic constants, and $u_i$ are the engineering strain vectors. Since $C_{ij}$ constitutes the entire set of elastic constants, other elastic properties of interest for single crystals, such as the bulk modulus and the tetragonal shear elastic constant, will be extracted from these energy calculations. Additionally, mechanical properties of isotropic polycrystalline materials, including the shear modulus, Young’s modulus, and Poisson’s ratio, were also determined by using the Hill average [86] and the Voigt and Reuss bounds. The reader is referred to our previous works [46,47] for more details on how we extracted elastic properties from the energies calculated via DFT simulations.

3. Results

In this section, we present the results of the first-principles calculations that investigate the structure and the thermo-mechanical response of the five W-based materials listed in Table 1.

3.1. Structural Characteristics of the Lattice Configurations

Figure 1 shows, for each of the four W-based alloys, (i) the atomistic arrangement of the random SQS structures and (ii) the MECs that are obtained after relaxing the lowest-energy configuration found during the MC-DFT simulations (denoted as MC-DFT configurations hereinafter). In

Table 2. Structural parameters of SQS, MC-DFT, and MEC configurations. The first three rows are the lattice parameters of cells. $|{\mathbf{x}}_{\mathbf{MEC}}-{\mathbf{x}}_{\mathbf{MC}-\mathbf{DFT}}|$ is the average difference between atomic coordinates obtained from MECs and MC-DFT calculations. P is the hydrostatic pressure of the cell.

	HEA1	HEA2	SA1	SA2
$a_{\mathrm{SQS}}$(Å)	3.140	3.165	3.095	3.093
$a_{\mathrm{MC}-\mathrm{DFT}}$ (Å)	3.138	3.173	3.090	3.043
$a_{\mathrm{MEC}}$ (Å)	3.135	3.160	3.088	3.083
$|{\mathbf{x}}_{\mathbf{MEC}}-{\mathbf{x}}_{\mathbf{MC}-\mathbf{DFT}}|$ (Å)	0.01	0.16	0.12	0.15
$P_{MEC}$ (GPa)	−0.01	0.001	0.02	0.008

, we list the lattice parameters of three sets of configurations: SQS, MC-DFT, and MEC. The change in lattice parameters between these atomistic arrangements is very small, generally less than 0.1 Å. Moreover, a comparison between the MC-DFT and MEC atom coordinates shows that the atom displacement between these structures is less than 0.2 Å per atom for all four materials. Furthermore, for the MECs, we calculated the trace of their internal stress tensor, which can be seen as hydrostatic pressure experienced by the cell. Given the magnitude of the pressures observed in the MECs for all W-based alloys, we conclude that the MECs are the ground-state configurations at 0 K.

3.2. Phonons

Phonon calculations were performed on the MECs for all four W-based alloys. The converged phonon dispersion and DOS are shown in Figure 2. For HEA1 and HEA2, only very few phonons near the $\Gamma$ point exhibit negative frequencies in the acoustic branches. For SA1 and SA2, negative frequency phonons in the three acoustic branches appear near P, N, and $\Gamma$. Most of the phonons in the Brillouin zone have positive frequencies. And, those imaginary frequency phonons, while present, have a DOS of phonons all around zero, suggesting their reduced impact on the free energy and thermal properties.

A further analysis of the phonon DOS shows that the maximum phonon frequency of HEAs is lower than 300 cm⁻¹, whereas that of SAs is higher than 300 cm⁻¹. This might suggest that SAs have larger elastic constants and higher stiffness than HEAs. Furthermore, comparing the phonon DOS of alloys with that of pure W shown in Figure 2, an upward shift of the DOS peaks as well as the entire spectrum is observed. The phonon DOS of W exhibits two peaks, a feature that is preserved in the DOS of HEA1, HEA2, and SA1. Only in SA2 does the second peak at around 270 cm⁻¹ disappear. Consequently, these differences in the phonon spectrum will affect thermal properties since they are decisive in determining the Helmholtz free energy, as described in Equation (3).

3.3. Thermo-Mechanical Behavior

In this section, we investigate the temperature dependence of the structural and mechanical properties of the five W-based materials listed in Table 1 using the QHA, with inputs from the phonon spectrum and electronic band structure.

As a preliminary step in our calculations, the CTE of pure tungsten was obtained and compared with previous experimental works in the literature. DFT calculations were performed with the PBE + GGA functional as they have already been applied to calculate the energy, structural, and thermal properties of W [87,88,89]. The results, shown in Figure 3a, reveal a good agreement between our work and the experimental measurements.

With the confidence conferred by the benchmarking exercise of pure tungsten, next, we proceed to calculate the CTE of the four W-based alloys. Figure 3b shows such calculations for both the SQS and MEC configurations of all the alloys. The figures reveal several interesting trends that are consistent with previous works in the field. Firstly, the CTE increases rapidly with temperature up to room temperature. Secondly, the CTE of all W-based alloys is higher than its counterpart for pure W over the entire temperature domain. There are also significant differences between the two families of W-based alloys considered here: HEAs have a larger CTE than SAs, and the disparity between the two HEAs is larger than the one observer between the two SAs, whose behavior is almost identical. Lastly, the CTE of the SQS and MEC of each alloy follows similar trends for all four alloys.

Figure 4 and Figure 5 show the temperature dependence of the equilibrium lattice constant, single-crystal elastic constants, and polycrystalline elastic properties for the five W-based materials. While it was not possible to find experimental nor computational works in the literature comparing the temperature dependence for the exact W-based alloys studied in this work, the observed linear behavior of pure W matches well with previous experimental studies [90,91]. In advance of discussing these results and their implications in detail in the following section, we note the following features from the figures: (i) the lattice constant is the only measurement increasing for all five W-based PFMs as temperature increases. $C_{44}$, G, and E are relatively flat in HEA1. All other properties are seen to decrease with temperature for all five W-based PFMs. Such a decrease is monotonic for HEA1 and both SAs, while it is more pronounced for HEA2, particularly at higher temperatures; (ii) the lattice parameter and the elastic properties of W are higher than the four W-based alloys over the entire temperature domain. The elastic properties obtained for SAs, whose chemical compositions still have W as the primary element (>67 at.%), are in some cases closer to those from pure W, especially when compared with the four-component HEA1 and HEA2 equiatomic alloys; (iii) for all the alloys, the differences between the SQS and MEC are almost negligible at all temperatures. Only HEA2 exhibits slightly different elastic constants of the SQS and MECs. Therefore, only the thermo-mechanical properties of the MECs are presented in the rest of this section. The reader is referred to Appendix A for more details about the thermo-mechanical behavior of the SQS for all W-based alloys.

Figure 3. (a) CTE of pure W. The results of ME configurations are compared with experimental data from the literature, Exp.a [92], Exp.b [93], and Exp.c [94]; (b) CTE of both SQS and ME configurations of the W-based alloys in comparison to those of pure W.

Figure 3. (a) CTE of pure W. The results of ME configurations are compared with experimental data from the literature, Exp.a [92], Exp.b [93], and Exp.c [94]; (b) CTE of both SQS and ME configurations of the W-based alloys in comparison to those of pure W.

Figure 4. Evolution of (a) the lattice parameter a and the elastic constants (b) $C_{11}$, (c) $C_{12}$, and (d) $C_{44}$ with temperature of W-based alloys and pure W. The figure includes the values for each material’s SQS and ME configurations.

Figure 4. Evolution of (a) the lattice parameter a and the elastic constants (b) $C_{11}$, (c) $C_{12}$, and (d) $C_{44}$ with temperature of W-based alloys and pure W. The figure includes the values for each material’s SQS and ME configurations.

Figure 5. Evolution of (a) the bulk modulus B, (b) the tetragonal shear elastic constant $C^{\prime }$, (c) the shear modulus G, and (d) the Young’s modulus E with temperature of W-based alloys and W. In Figure (a), our bulk modulus data are compared with those obtained from experiments (Exp.a [90] and Exp.b [91]).

Figure 5. Evolution of (a) the bulk modulus B, (b) the tetragonal shear elastic constant $C^{\prime }$, (c) the shear modulus G, and (d) the Young’s modulus E with temperature of W-based alloys and W. In Figure (a), our bulk modulus data are compared with those obtained from experiments (Exp.a [90] and Exp.b [91]).

Notice that the changes in these properties with temperature are less pronounced than the existing differences between the families of W-based alloys at T = 0 K. As such, we decided to further investigate the temperature effect in terms of the relative difference ${\delta }_X$, defined as

$${\delta }_X={\displaystyle \frac{X_t-X_0}{X_0}}$$

(6)

where $X_0$ is the value of the property of interest X at T = 0 K and $X_T$ is the value of that same property at a specific temperature T.

Following this notation, Figure 6 and Figure 7 show the temperature dependence of the equilibrium lattice constant, single-crystal elastic constants, and polycrystalline elastic properties for the five W-based materials, expressed in terms of the relative difference ${\delta }_X$. It can be seen from these results that the elastic properties of the W-based alloys are more sensitive to temperature than those of pure W. And, among all four alloys, HEA2 exhibits the largest decrease in elastic properties as temperature increases. In particular, in HEA2, B and $C^{\prime }$ are reduced by 20–25% and G and E are reduced by 50–60% over 2000 K. This indicates that the resistance to volume and shear deformation in HEA2, as well as the stiffness, are significantly weakened as temperature increases, whereas it is only moderately weakened for the other three alloys. For SA1 and SA2, all elastic parameters show a similar softening degree as temperature increases. At 2000 K, B, $C^{\prime }$, G, and E are all reduced by around 1–0–15%. The smallest reduction in elastic properties among all four alloys occurs for HEA1, where properties drop in the range of 10–15% over 2000 K.

Next, we analyze the impact of temperature on ductility, as it is known that ductility is one of the main limiting factors of W-based materials for their application in fusion energy devices. Despite the scalability limitations of ab initio methods in directly characterizing dislocation glide, grain boundaries, hardening, and other mesoscopic phenomena governing the plasticity of bcc metals, we investigated two indicators of ductility that are based on its relationship with material parameters that can be determined via DFT simulations: Pugh’s ratio [95], defined as the ratio between the bulk modulus B and the shear modulus G; and Poisson’s ratio $\nu$, which is found to be proportional to the intrinsic ductility of crystals [96]. Our results of these two empirical metrics are shown in Figure 8. Both Pugh’s and Poisson’s ratios indicate that W is more brittle than the two HEAs and more ductile than the two SAs over the entire temperature domain. For its part, our calculated Poisson’s ratio at 0 K is 0.297, which compares well with previous DFT calculations and experimental measurements [49,52,90,97,98]. Furthermore, all materials expect from HEA2 present a slightly decreasing ductility with temperature. HEA2 is unique as both metrics increase with temperature, the rise being even more significant at temperatures above 1000 K.

4. Discussion

4.1. Effects of the Chemical Composition

The chemical composition of the W-based alloys plays a central role in determining their thermo-mechanical properties. Previous studies have already shown that variations in chemical compositions of HEAs could cause large differences in their elastic moduli [99,100]. As listed in Table 1, in the W-based alloys, W atoms are replaced by lighter elements (Cr, Ti, Y, Ta, and V). Such replacements lead to a reduction in the effective mass of the lattice, thus resulting in a hardening of the phonon spectrum of the alloys compared to pure W shown in Figure 2. The similarity of chemical compositions between two HEAs and SAs also determines that they have similar elastic constants and CTEs. However, it is worth noting that, despite the very similar chemical composition between HEA2 and HEA1, HEA2 exhibits a very different temperature dependence of elastic constants (Figure 6) and parameters B, C, G, and E (Figure 7). Such a distinct temperature dependence behavior of HEA2 is also seen in the ductility study in Figure 8. The unique temperature behavior of HEA2 is because the phonon spectrum of HEA2 is the lowest among the W-based alloys, shown in Figure 2. Indeed, HEA2 is the softest material and the most sensitive to temperature changes. Its elastic properties show the most significant change when the temperature increases. It can be inferred that the substitution of W by V elements introduces a significant change to the chemical bonds, potentially with the W-V bonds possessing different properties compared to W-Cr, Ti, Y, and Ta in the other W-based alloys. A deeper discussion of the chemical environment is beyond the scope of this work. Nevertheless, it is concluded that small variations in the chemical composition could potentially lead to large differences in the temperature-dependent thermo-mechanical properties of W-based alloys.

The temperature dependence of elastic constants and elastic moduli is critical for the high-temperature performance of HEAs. From Figure 6, the variation in $C_{11}$ and $C_{44}$ with temperature is similar for all four W-based alloys. This means that the resistance to axial compression and shear deformation is reduced by a similar extent as temperature increases. In fact, except for HEA2, the other three alloys exhibit small variations in the elastic constants with temperature, indicating good thermal stability and weak elastic softening behavior at high temperature [101]. This is also shown in Figure 7, where the decreases in B, $C^{\prime }$, G, and E are all small from 0 K to 2000 K. Only HEA2 shows strong elastic softening behavior at high temperature, where the decrease in $C_{11}$, $C_{44}$ is around 25% and G, E is around 60%. The large variation might be attributed to the bonds evolution accompanied by the bonds expansion [101], where the W-V and V-V bonds of HEA2 behave differently from bonds in the other three W-based alloys.

4.2. Phonon and Electron Contributions to the Thermo-Mechanical Behavior

In this section, we present our investigations about the impact of phonons and electrons on the thermo-mechanical behavior of the five W-based materials listed in Table 1.

As shown in Equation (3), the temperature dependence of Helmholtz free energy consists of phonon ($F_{\mathrm{ph}}(\mathit{a},T)$) and electron ($F_{\mathrm{el}}(\mathit{a},T)$) terms. To quantify the individual contributions from $F_{\mathrm{ph}}$ and $F_{\mathrm{el}}$ to the thermal properties, we calculated the CTE by only including $F_{\mathrm{ph}}$ in the Helmholtz free energy and comparing it with the CTE obtained when both $F_{\mathrm{ph}}$ and $F_{\mathrm{el}}$ are included in the free energy. The results are plotted in Figure 9a, where it can be seen that ${\alpha }_{\mathrm{ph}+\mathrm{el}}$ are only slightly larger than ${\alpha }_{\mathrm{ph}}$ at all temperatures, indicating that the CTEs are dominated by the phonon contributions.

Quantification of the phonon contributions toward the CTEs can be obtained from ${\displaystyle \frac{{\alpha }_{\mathrm{ph}}}{{\alpha }_{\mathrm{ph}+\mathrm{el}}}}$, which are plotted in Figure 9b. For all the HEAs, at low temperatures, the phonons almost contribute 100% to the CTE, which indicates that electrons have negligible impact on the CTE. As the temperature increases, the electron contribution generally increases. Still, even at high temperatures, such as 2000 K, the phonon contributions are all higher than 85%. Given the importance of the CTE defining the lattice structure and, therefore, determining the thermo-mechanical response of materials, these results suggest that the phonons dominate the thermo-mechanical behavior of the five W-based materials investigated here.

4.3. Impact of Phonons on the Thermo-Mechanical Properties

The thermo-mechanical properties of the W-based alloys can be revealed by the phonon spectrum since the phonon free energy $F_{\mathrm{ph}}$ is an integration of phonon-energy-related terms over the Brillouin zone according to Equation (3). It has already been shown that mass and force constants are the key components dominating phonon behavior, thus influencing the thermodynamic and elastic properties of HEAs [102,103]. From Table 1, the mass of the W-based alloys is $m_{\mathrm{HEA}1}=93.15$, $m_{\mathrm{HEA}2}=115.90$, $m_{\mathrm{SA}1}=140.30$, and $m_{\mathrm{SA}2}=141.91$, respectively, compared to $m_{\mathrm{W}}=183.84$ of pure W (all weights are in atomic units). In a dynamic system, the vibrational frequency $\omega \sim \sqrt{k/m}$, where k and m represent effective stiffness and mass, respectively. As shown in Figure 2, the highest phonon frequencies of all four W-based alloys are larger than that of W by around 100 cm⁻¹. And, the phonon spectrum of the alloys have a decent portion above the highest energy level of the W phonon spectrum, which is slightly larger than 200 cm⁻¹. Such a comparison between the phonon spectrum of the W-based alloy and W translates into Figure 4 and Figure 5, which shows that the elastic properties and effective stiffness k of all the alloys are smaller than those of W. With the effective mass m of all W-based alloys being smaller than pure W due to the replacement of lighter elements, the frequencies of optic phonons in alloys become larger than those of pure W. Indeed, given that the W-based alloys have a lower effective stiffness, all alloys experience a larger lattice thermal expansion than W as temperature increases. Thus, the CTE of the alloys is larger than that of W, which is shown in Figure 3b. Moreover, the phonon spectrum of two SAs reach a higher frequency range than those of the two HEAs. With the mass of W-SAs being larger than the W-HEAs, it can be inferred that the elastic properties of the SAs are larger than those of the HEAs, which is also confirmed by Figure 4 and Figure 5. And, the CTE of SAs is smaller than those of the HEAs, as can be seen from Figure 3b.

5. Conclusions

To summarize, we have integrated first-principles DFT calculations with the QHA to better understand the thermo-mechanical behavior of five W-based materials, including pure W, two HEAs, and two SAs. In particular, we used direct information from our calculated phonon spectrum to unravel how the CTE and elastic properties evolve with temperature.

Our first conclusion from these investigations is that the chemical composition of the candidate W-based alloys plays a central role in determining their phonon spectrum and, consequently, their thermo-mechanical properties. While the phonon spectra of the five W-based materials show good structural stability, with the majority of the phonons having positive frequencies, there are still significant differences between them. For example, the highest phonon frequencies of the four W-based alloys are larger than those of pure W, and a portion of their spectrum is above the highest energy level found in W. This translates into their thermo-mechanical response, where W-alloys present a higher CTE and lower elastic properties than W over the entire temperature range. A comparison of the phonon spectrum of the two families of W-alloys considered here shows that the two W-SAs have higher frequencies in their phonon spectrum when compared to the two W-HEAs. Such a difference in the phonon spectrum determines the Helmholtz energy landscape and, consequently, the temperature behavior. From the frequency–mass–effective stiffness relation, it is concluded that W-SAs have larger elastic constants and thus a smaller CTE than the W-HEAs.Furthermore, the elastic properties of HEA2 present the largest sensitivity to temperature among all the W-based materials investigated here, which is attributed to the lowest frequency value of its highest optic phonons.

Our results also suggest that the phonons dominate the thermo-mechanical behavior of the five W-based materials investigated here. In particular, we report that the individual contributions of the phonon term in the Helmholtz free energy are more significant than those of its counterpart from the electron term over the entire temperature range, and particularly at lower temperatures. This observation applies to the five W-based materials investigated here. Interestingly, HEA2, the W-based alloy whose behavior is the most sensitive to temperature change, presents the smallest ratio between the contributions of phonon and electron terms.

Our current and future efforts are directed toward studying the effect of each alloying element added to W, the presence of defects in these materials, and the suitability of other DFT-based approaches [104].