Effect of Interstitial Hydrogen on the Mechanical and Thermal

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

pubs.acs.org/JPCC

Effect of Interstitial Hydrogen on the Mechanical and Thermal Properties of Tungsten: A First-Principles Study Min Luo,† Diyou Jiang,‡ Sanqiu Liu,† and Chuying Ouyang*,† †

Department of Physics, Laboratory of Computational Materials Physics, Jiangxi Normal University, Nanchang 330022, China Department of Mechanical Engineering, Jiangxi University of Technology, Nanchang 330098, China

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‡

ABSTRACT: Tungsten (W) was considered as one important candidate for the magnetically constrained plasma first wall materials. The H impurities have a strong influence on the mechanical and thermal properties of the W metal and seriously affect its service life. From firstprinciples calculations, we studied the influence of interstitial H on the mechanical and thermodynamic parameters such as elastic constant, elastic modulus, free energy, entropy, expansion coefficient, and phonon thermal conductivity of W. In particular, the temperature- and H concentration-dependent parameters are calculated, and the effect of H concentration on material’s performance is summarized. The results show that the mechanical strength of W metal decreases with the increase of H concentration. On the other hand, H impurity improves the ductility of the W metal, in agreement with previous reports in the literature. The thermal expansion coefficients of W increase with H impurity concentration. The phonon thermal conductivity of W is also strongly affected by H impurities, and it decreases with the increase of H concentration. These results suggest that H impurity is one important reason for the material degradation of W as a first wall material. As first wall materials, the mechanics and thermal properties are very important for the service of materials, and high mechanical strength and high thermal conductivity are good for material applications. The researchers conducted a series of studies on the thermodynamic and elastic properties of W and its alloys.14 The results showed that the isovolumic heat capacity (CV) of W, which is mainly contributed by the lattice vibration in a low temperature range (T < 500 K), increases rapidly with temperature. In a high temperature range (T > 500 K), electrons start to contribute to the CV, and the higher the temperature is, the contribution from electrons is more important. The elastic constants, bulk modulus, shear modulus, and Young’s modulus of W metal decrease approximately linearly with increasing temperature.15,16 Although hydrogen behavior in W is reported in some papers,17,18 the influence of the H impurities on the mechanic and thermal properties of the W metal is not well studied in the literature. Jiang et al.13 studied the influence of interstitial H impurities on the mechanical properties of metal W, and they found that the ductility of W was improved after H impurities were attached to W. However, they did not consider the effect of H impurities with different concentrations on the mechanical properties of W metal. On the other hand, as the first wall materials, the thermal conductivity of W metal is another important parameter. So far, however, the effect of H

1. INTRODUCTION At present, the major energy source used by humanity comes from fossil energy. However, limited storage of fossil energy cannot meet the growing demand of human society, which requires researchers to explore various possible new energy sources. So far, researchers have conducted extensive studies on energy sources such as wind power, hydroelectric power, solar power, and nuclear power. Among these new energy sources, controlled thermonuclear fusion energy is a clean and practical one that can possibly solve the human energy demand.1 First wall material for controlled thermonuclear fusion energy is one important engineering material that can bear the high-temperature and high-radiation conditions. W and its alloys are considered as the most promising first wall materials at present due to their high melting point, high thermal conductivity, high strength at high temperatures, low thermal expansion coefficient, low sputtering yield and high sputtering threshold, low vapor pressure, and low tritium retention.2−4 Plasma facing materials are subjected to the bombardment of high-beam hydrogen isotopes deuterium and tritium,5,6 which can result in a lot of defects and damage on the surface and bulk texture of tungsten materials, as well as the retention of hydrogen isotopes,7,8 indicating that part of H impurities may be retained in the W lattice.9 It is shown that H impurities tend to occupy the tetrahedral interstitial site (TIS) in W. In other words, when H occupies the tetrahedral interstitial site, it has the lowest energy and the most stable structure.10−13 © XXXX American Chemical Society

Received: November 1, 2018 Revised: December 5, 2018 Published: December 26, 2018 A

DOI: 10.1021/acs.jpcc.8b10659 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C impurities on the thermal conductivity of metal W has not been reported. In this paper, the effect of H impurities at different concentrations on the mechanical and thermal properties of metal W is investigated from first-principles calculations. By calculating the elastic constants, lattice dynamics, and other parameters of H impurities with different concentrations in the W lattice, the mechanical and thermal properties are analyzed, which provides a theoretical reference for the application of W metal as the first wall material.

Table 1. Relationship between Total Energy Changes (ΔE) and the Strains (δ) of a Cubic Structure strain

2. THEORY AND METHODS 2.1. Computational Details. First-principles calculations were performed from the density-functional theory and the plane-wave pseudopotential method, which were implemented in the Vienna ab initio simulation package.19,20 The interaction between valence electrons and core ions is described by the projector augmented wave method,21,22 and the exchangecorrelation part is described with the generalized gradient approximation by Perdew−Burke−Ernzerhof.23 The Monkhorst−Pack24 scheme k-point sampling is used for the integration in the irreducible Brillouin zone, and the density of the k-point mesh is smaller than 0.03 Å−1. As different supercells are used in this study, we used different k-point meshes, namely, the k-point meshes for the cubic W unit cell, 1 × 1 × 2, 1 × 2 × 2, and 2 × 2 × 2 supercells are 13 × 13 × 13, 13 × 7 × 7, 11 × 7 × 7, and 7 × 7 × 7, respectively. The lattice constants and atom positions are fully relaxed. For cubic (unit cell and 2 × 2 × 2 supercell) and tetragonal (1 × 1 × 2 and 1 × 2 × 2 supercells) systems, the cubic and tetragonal symmetries are preserved during relaxation. The phonon dispersion spectrum and the vibrational contributions of free energies and entropies are calculated with the PHONOPY code,25 and the dynamic matrices of the lattice vibration are calculated by density-functional perturbation theory (DFPT).26 Since the phonon data calculation requires very high precision, the convergence criteria for the total energy and the atomic force are set to be 10−8 eV and 10−5 eV Å−1, respectively. The cutoff energy of the plane-wave function is set to be 600 eV. 2.2. Mechanical Property Calculations. The elastic constants can be expressed as a Taylor series expansion of the total energy (E) to strain (δ) ÅÄÅ ÑÉÑ ÅÅ ÑÑ 1 Å E(V , δ) = E(V0 , 0) + V0ÅÅÅ∑ τε Cijδiεiδj ÑÑÑÑ ∑ i iδi + ÅÅ ÑÑ 2 ij ÅÇÅ i ÑÖÑ (1)

change of total energy

e = (0, 0, 0, δ, δ, δ)

ΔE 3 = C44δ 2 V0 2

e = (δ, δ, 0, 0, 0, 0)

ΔE = (C11 + C12)δ 2 V0

e = (δ, δ, δ, 0, 0, 0)

ΔE 3 = (C11 + 2C12)δ 2 V0 2

Table 2. Relationship between Total Energy Changes (ΔE) and the Strains (δ) of a Tetragonal System strain

change of total energy

e = (δ, δ, 0, 0, 0, 0)

ΔE = (C11 + C12)δ 2 V0

e = (0, 0, 0, 0, 0, δ)

ΔE 1 = C66δ 2 V0 2

e = (0, 0, δ, 0, 0, 0)

ΔE 1 = C33δ 2 V0 2

e =(0, 0, 0, δ, δ, 0)

ΔE = C44δ 2 V0

e = (δ, δ, δ, 0, 0, 0) e = (0, δ, δ, 0, 0, 0)

C y i ΔE = jjjC11 + C12 + 2C13 + 33 zzzδ 2 V0 2 { k C y iC ΔE = jjj 11 + C13 + 33 zzzδ 2 V0 2 2 { k

Hill approximation.31−34 For a cubic crystal system, these parameters can be expressed as BH = B V = BR = (C11 + 2C12)/3 C − C12 + 3C44 G V + GR , G V = 11 , GR 2 5 5(C11 − C12)C44 = 4C44 + 3(C11 − C12)

(2)

GH =

9BH G H 3BH + G H

(4)

3BH − 2G H 2(3BH + G H)

(5)

EH = ν=

where E(V0,0) and V0 are the total energy and volume of the unstrained unit cell system, respectively, τi is the stress tensor element, εi is the Voigt index factor, Cij are the elastic constants, and the indices i and j run over from 1 to 6. The elastic constants of crystals are characterized by Cij and can be calculated from the continuous elastic theory.27,28 The supercells of W and WH0.0625 are cubic, whereas the supercells of WH0.25 and WH0.125 are tetragonal. For the cubic structure, there are three independent elastic constants, i.e., C11, C12, and C44.29 For the tetragonal structure, there are six independent elastic constants, i.e., C11, C12, C13, C33, C44, and C66.30 According to the continuous elastic theory, these elastic constants can be obtained by fitting the equations in Tables 1 and 2 for cubic and tetragonal systems, respectively. Bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (ν) can be obtained by Voigt−Reuss−

(3)

For a tetragonal crystal system, these parameters can be expressed as (2C11 + 2C12 + C33 + 4C13) BR + B V , BV = , 2 9 (C + C12)C33 − 2C132 BR = 11 C11+C12 + 2C33 − 4C13 (6)

BH =

GH =

(2C11 − C12 − 2C13 + C33 + 6C44 + 3C66) GR + G V , GV = 2 15

GR =

15 18B V /[(C11 + C12)C33 − 2C132] + 6/(C11 − C12) + 6/C44 + 3/C66

(7) B


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The Journal of Physical Chemistry C EH =

ν=

9BH G H 3BH + G H

γ= (8)

3BH − 2G H 2(3BH + G H)

1 2

(9)

q,ν

/kBT ] Svib =

where vm is the average acoustic velocity, h is the Planck constant, NA is the Avogadro constant, m is the relative atomic mass, n is the number of atoms in the system, ρ is the mass density, and k is the Boltzmann constant. The average acoustic velocity vm can be calculated by the longitudinal wave velocity vl and the transverse wave velocity vt as ÄÅ ÉÑ−1/3 ÅÅ 1 i 2 1 zyzÑÑÑÑ ÅÅ jj vm = ÅÅ jj 3 + 3 zzÑÑ ÅÅ 3 j v v l z{ÑÑÖÑ (17) ÇÅ k t

∑ ℏωq,ν(V ) + kBT ∑ ln[1−exp(−ℏωq,ν(V ) q,ν

i ℏωq , ν zy zz zz 2 k T B q,ν k { ÉÑ ÅÄÅ Ñ ÅÅ jij ℏωq , ν zyzÑÑÑ Å zÑÑ − kB ∑ lnÅÅ2 sinhjj z j 2kBT zÑÑ ÅÅ q,ν ÅÇ k {ÑÖ 1 2T

∑ ℏωq,ν cothjjjjj

(10)

where vl and vt can be obtained from the bulk and shear modulus by ij B + j v l = jjjj j ρ k

(11)

where ωq,ν is the frequency of phonon branch ν at wave vector q and the sum goes over the first Brillouin zone, ℏ and kB are the Planck constant and the Boltzmann constant, respectively, and T is the absolute temperature. Then, the Helmholtz free energy can be expressed as F(V , T ) = E0(V ) + Fvib(V , T )

iG y vt = jjjj zzzz k ρ{

1 d V (T ) V (T ) d T

(12)

zz zz {

(18)

(19)

3. RESULTS AND DISCUSSION 3.1. Crystal Structures and Mechanical Properties of WHx. First, we optimized the unit cell of the W metal in a body centered cubic (bcc) structure and the obtained lattice constant a is 3.171 Å, which is in good agreement with the experimental value of 3.165 Å.36−38 Different H concentrations are considered by adding one interstitial H atom in supercells of different sizes, namely, one interstitial H is added into supercells of 1 × 1 × 2 (WH0.25), 1 × 2 × 2 (WH0.125), and 2 × 2 × 2 (WH0.0625). In the bcc W metal structure, there are two types of interstitial sites, i.e., the tetrahedral interstitial site (TIS) and the octahedral interstitial site (OIS). So we compared the solution energy of an interstitial H atom in the TIS and OIS of the W lattice. The solution energy an H atom in the W lattice is defined as

(13)

The thermal conductivity of a metallic system is generally contributed by the transport of both electrons and phonons. Here, in this paper, the electron transport contribution to thermal conductivity is not considered. As a result, the obtained thermal conductivity in the following text of this paper represents only the contribution from phonons, and therefore they are much smaller than the experimentally measured ones. According to the theory of Slack,35 the contribution of the lattice vibration to thermal conductivity kL can be expressed as M̅ θD3δ κL = A 2 2/3 γ n T

4G y1/2 z 3 z z

1/2

Using the Helmholtz free energy expressed in eq 12, we can obtain the optimized volume (lowest Helmholtz free energy) at different temperatures. Then the bulk expansion coefficient (αv) can be calculated using the equation of state αν =

(15)

where CV is the isovolumic heat capacity, B is the bulk modulus, V is the volume of the unit cell, and αv is the bulk expansion coefficient. The Debye temperature can be obtained by the average acoustic velocity vm using the following relation Ä É1/3 h ÅÅÅÅ 3n ij NAρ yzÑÑÑÑ θD = ÅÅ jj zzÑ vm k ÅÅÇ 4π k m {ÑÑÑÖ (16)

where the subscripts V, R, and H refer to Voigt, Reuss, and Hill approximations, respectively. 2.3. Thermodynamic Property Calculations. The phonon spectral data are calculated based on the quasiharmonic approximation. Using the phonon data, we can calculate the vibrational contribution of free energy and entropy as Fvib =

3αvBV CV

E HS = En W + H − En W − μH

(20)

where EnW and EnW+H are the total energies of the W supercells without and with one interstitial H atom, μH is the H-chemical potential and is chosen to be −2.26 eV according to refs 12 and 39 for the comparison purpose. The results showed that H solution energies at the OIS and TIS of the W lattice are 0.22 and −0.23 eV, respectively, indicating that H is more stable in the TIS. The obtained solution energies are in good agreement with previous reports,10−13 which show that the models and calculation parameters in the present work can well describe the behavior of H in W. The calculated structural parameters, elastic constants, bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio in the ground state are listed in Table 3. As is seen in Table 3,

(14)

where n is the number of atoms contained in the system, δ is the cube root of the average atomic volume, θD is the Debye temperature, M is the average atomic mass in the unit cell, γ is the Grüneisen constant, and A is a constant. If the unit of thermal conductivity takes W m−1 K−1, M takes the atomic mass unit, and δ takes Å, then the value of A is 3.1 × 10−6. To calculate the thermal conductivity, we have to obtain the Grü neisen constant and Debye temperature first. The Grüneisen constant can be calculated using C


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The Journal of Physical Chemistry C

Table 3. Lattice Constants (in Å), Unit Cell Volumes V (in Å3), Bulk Modulus B (in Gpa), Shear Modulus G (in Gpa), Young’s Modulus E (in Gpa) and Poisson’s Ratio (ν) of WHx (x = 0, 0.0625, 0.125, 0.25) in the Ground States pure W this work a b c α V C11 C12 C13 C33 C44 C66 B G E B/G v

experiment

WH0.0625

WH0.125

WH0.25

3.195 6.390 6.390 90° 32.62 (×4) 463.54 221.10 230.85 478.57 132.63 126.97 297.36 127.43 334.51 2.33 0.31

3.217 3.217 6.433 90° 33.28 (×2) 453.38 235.72 261.55 463.26 134.88 132.58 309.14 121.05 321.23 2.55 0.33

37,38

3.171 3.171 3.171 90° 31.88 536.02 206.85

3.165 3.165 3.165 90° 31.70 533 205

6.366 6.366 6.366 90° 32.25 (×8) 504.84 216.90

144.54

163

139.19

316.58 152.25 393.64 2.08 0.29

314.33 163.40 417.80 1.92 0.28

312.88 141.09 367.95 2.22 0.30

Table 3, the B/G values of WHx (x = 0, 0.0625, 0.125, and 0.25) materials are all larger than 1.75, indicating that these materials are all ductile. However, the value of B/G increases with the H concentration, indicating that its ductility can be enhanced by H impurity. This is in agreement with reports in ref 13. The ductility of metallic materials can also be characterized by the Poisson’s ratio. As is seen in Table 3, Poisson’s ratio also increases with increasing H concentrations, showing that the ductility is enhanced by H. This seems to be contradictory to the hydrogen induced embrittlement in metal physics. In fact, the main reason for hydrogen induced embrittlement in W metal is related to the formation of H2 bubbles in the vacancies.12,13 For a perfect W lattice without vacancies, the ductility changed by the interstitial H can be explained by electronic structure analysis. Figure 1 shows a comparison of the total density of states (TDOS) of the W and WHx. It can be seen from Figure 1 that the metallic characteristic of the W metal is not changed by the interstitial H atoms. Generally speaking, the ductility of a metal is related to the metallic bonding nature, in which valence electrons are delocalized and shared between many atoms. The delocalized electrons act as

the calculated elastic constants and elastic modulus of pure W metal reproduced the experimental results well, showing that the calculation method is reliable. It is also found that the volume of the unit cell is expanded slightly after the H atom is added into the interstitial site of the W lattice. It increased from 31.88 Å3 for pure W to 32.25, 32.62, and 33.28 Å3 for WH0.0625, WH0.125, and WH0.25, respectively. The expanded lattice volume has a direct influence on the mechanical properties of WHx, namely, the moduli of W are decreased by interstitial H. Although the changes in the moduli by interstitial H are small, they have strong correlations between the concentration of the H and the value of the moduli changed by the H, as shown in Table 3. The mechanical stability can be characterized by elastic constants. Cubic systems can be mechanically stable if the elastic constants conform to the following conditions (C11 − C12) > 0, C11 > 0, C44 > 0, (C11 + 2C12) > 0

whereas the mechanical stability of a tetragonal system can be characterized by (C11 − C12) > 0, (C11 + C33 − 2C13) > 0, C11 > 0, C33 > 0, C44 > 0, C66 > 0, (2C11 + C33 + 2C12 + 4C13) >0

From the elastic constant data presented in Table 3, we found that all of the WHx systems are mechanically stable. From Table 3, we also notice that C11 shows a decreasing trend with an increase of H concentration, C12 shows an increasing trend, and C44 shows a decreasing trend. The elastic constant C11 is a very important mechanical parameter that is related to the hardness of crystals, whereas C12 represents the ability of crystal resistance to lateral deformation. It indicates that the strength of the material decreases with increasing H concentration, and the ability of the material to resist lateral deformation is enhanced. According to Pugh’s theory,40 the ratio of bulk modulus to shear modulus (B/G ratio) can characterize the ductility of metallic materials. When the B/G value is larger than 1.75, the material is considered to be ductile, otherwise it is considered to be brittle. As shown in

Figure 1. TDOS of WHx (x = 0, 0.0625, 0.125, and 0.25), the energy is with respect to the Fermi level. D


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Figure 2. Phonon dispersion curves of W (a), WH0.0625 (b), WH0.125 (c), and WH0.25 (d) based on DFPT calculations.

“electron glues”, which attract the metal atoms (nucleus and core electrons) together and at the same time allow them to slide past one another. As the electronic states near the Fermi level are denser for WHx materials, the metallic bonding nature becomes more dominated. As a result, interstitial H atoms can actually improve the ductility of the W metal. 3.2. Lattice Dynamics Properties of WHx. 3.2.1. Lattice Vibration Calculation. The thermodynamic properties of solid state materials are strongly related to their lattice dynamics or phonon dispersion characteristics. In general, the phonon dispersion spectrum provides detailed information concerning the lattice vibration. If all phonon frequencies are positive, the crystal is kinetically stable, otherwise if negative phonon frequency (virtual frequency) occurs at some k-points in the Brillouin zone, the structure is considered to be kinetically unstable. Figure 2 shows the phonon dispersion curves of the WHx (x = 0, 0.0625, 0.125, and 0.25) supercell along the high symmetry points in the first Brillouin zone. As is seen, all phonon frequencies of WHx (x = 0, 0.0625, 0.125, and 0.25) materials are positive, indicating that H intercalated W lattices are kinetically stable, even when the H concentration x is as high as 25%. There are six dispersion curves can be found for pure W (two W atoms in the unit cell), with the highest vibration frequency (6.4 THz) observed at the M-point. The vibration frequencies are relatively low, due to the very large atomic weight of the W atom. For the dispersion curves of WHx (x = 0.0625, 0.125, and 0.25), in addition to the low frequency from W atom vibrations, which are similar to the phonon dispersion curves of the pure W metal, we can also find three vibration branches with particularly high frequencies. These high frequency vibrations originated from the interstitial H, which

might have a strong influence on the thermal properties of the W metal. 3.2.2. Thermal Properties. With the above phonon dispersion data, we can now turn to discuss the thermal properties of the W and how much interstitial H atoms change them in WHx. The thermal expansion coefficient and thermal conductivity are very important factors for designing W-based plasma-facing materials. Although the thermal conductivity is also contributed by electrons, as mentioned above in the method section, here, we only consider the phonon contribution. Before calculating the thermal expansion coefficient and the thermal conductivity, the vibrational contribution to the Helmholtz free energy of the system must be calculated in advance. Figure 3 presents the vibrational free energy and entropy in H intercalated systems WHx (x = 0, 0.0625, 0.125, and 0.25) as a function of temperature. It can be seen from Figure 3a that the interstitial H atom in the W lattice increases the vibrational free energy of the W slightly, and the amount of increase has a positive correlation with the H concentration x, namely, the higher H concentration results in higher vibrational free energy. The reason for this vibrational free energy change can be found from the vibrational entropy data given in Figure 3b, from which we can see that the entropies of the WHx materials decrease with the increase of H concentration. The decreased entropy by H is related to the influence of the H atom in the TIS on the vibration of its neighboring W atoms, which experience a compressive stress that results in a locally stiffer region. Namely, the amplitude of the vibration of these W atoms is reduced, and therefore, the system samples less number of states from the W vibrations. The decreased vibration entropy makes the −TS term in the Helmholtz free energy higher at high H concentrations. E


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also increases with the H concentration x. According to the relationship between the bulk expansion coefficient and the linear expansion coefficient (α), i.e., αv = 3α, the linear expansion coefficient at room temperature can be obtained, as shown in Table 4. The linear expansion coefficient of pure W metal obtained in this work is 4.71 × 10−6 K−1, which is in good agreement with the theoretical value of 4.6 × 10−6 K−1.41 Generally speaking, solid materials can be classified into three types according to the linear expansion coefficient, namely, low expansion materials (0 < α < 2 × 10−6 K−1), medium expansion materials (2 × 10−6 K−1 < α < 8 × 10−6 K−1), and high expansion materials (α > 8 × 10−6 K−1).42 According to this classification, W metal belongs to a medium thermal expansion material. When the concentration of H impurity increases to a certain value, the W material becomes a high expansion material. To calculate the thermal conductivity of WHx, the Debye temperatures and the Grüneisen constants of the WHx systems are calculated and presented in Table 5 and Figure 6. The parameters used for calculating the Debye temperature, such as the mass density, transverse wave velocity, longitudinal wave velocity, and average acoustic velocity, are also presented in Table 5. As is seen, the acoustic velocity of W metal decreases after interstitial H is intercalated into the W lattice. As a result, the Debye temperature also decreases slightly. The Debye temperature of pure W metal is obtained to be about 372.39 K, in good agreement with other theoretical values. The Grüneisen constants range from ∼1.7 to 2.2, which are typically in the scope of metal materials. After obtaining the Debye temperatures and the Grüneisen constants, the phonon thermal conductivity of the WHx (x = 0, 0.0625, 0.125, and 0.25) materials can be calculated according to eq 14. Figure 7 shows the phonon thermal conductivity of the WHx (x = 0, 0.0625, 0.125, and 0.25) materials as a function of temperature. From Figure 7, we can see that the phonon thermal conductivity of W is very sensitive to the temperature. At room temperature (300 K), the thermal conductivity of W metal contributed by the lattice vibration is about 52.75 W m−1 K−1, which is about 30.3% of the total thermal conductivity (174) W m−1 K−1,43 showing that the lattice vibration has a substantial contribution to the total thermal conductivity. However, the lattice vibration contribution decreases substantially with increasing temperatures, and it becomes only 6.74 W m−1 K−1 at 2000 K. Meanwhile, the thermal conductivity of W metal contributed from lattice vibration becomes unimportant at high temperatures. The situation becomes even worse when interstitial H atoms are intercalated into the W lattice. Even at room temperature (300 K), the thermal conductivities contributed from lattice vibration are 13.5, 13.3, and 9.9 W m−1 K−1 for WH0.0625, WH0.125, and WH0.25, respectively. This implies that the interstitial H atom decreases the thermal conductivity substantially, which is obviously unwanted in the application. At high temperatures like 2000 K, the thermal conductivity from lattice vibration even becomes negligibly small (∼1 W m−1 K−1). Considering the fact that thermal conductivity of a metal contributed by electrons also decreases substantially with increased temperature, the thermal conductivity of W with interstitial H will be further lowered at high temperatures. This is obviously not needed for W metal as a plasma facing material.

Figure 3. Vibrational free energy (a) and entropy (b) of each atom in the WHx (x = 0, 0.0625, 0.125, and 0.25) system as a function of temperature.

With the temperature-dependent Helmholtz free energy data, we can figure out the optimized volume at different temperatures. Figure 4 presents the optimized volume of WHx (x = 0, 0.0625, 0.125, and 0.25) at different temperatures. The solid lines of different colors are the fitting curves of Helmholtz free energy as a function of volume for temperatures below 3000 K. For a certain temperature, the minimum Helmholtz free energy corresponds to the equilibrium volume at that temperature. It can be seen that the equilibrium volume of the WHx (x = 0, 0.0625, 0.125, and 0.25) lattice increases with temperature, showing normal thermal expansion behaviors. The thermal expansion coefficient is of great significance to materials that are serving under high temperatures. The thermal expansion coefficient of the W metal should be considered carefully for the design of the plasma devices. With the above temperature-dependent volume data, here, we calculate the thermal expansion coefficient of the WHx (x = 0.0625, 0.125, and 0.25) materials, and the results are compared with that of the pure W metal, as shown in Figure 5. As is seen, the bulk expansion coefficient of WHx (x = 0, 0.0625, 0.125, and 0.25) materials increases with increasing temperature. On the other hand, the bulk expansion coefficient F


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Figure 4. Helmholtz free energy of W (a), WH0.0625 (b), WH0.125 (c) and WH0.25 (d) as a function of temperature and volume.

Table 4. Bulk (αv) and Linear (α) Expansion Coefficient of WHx (x = 0, 0.0625, 0.125, and 0.25) Materials at Room Temperature (T = 300 K) αv (×10−6 K−1) pure W WH0.0625 WH0.125 WH0.25

α (×10−6 K−1)

14.14 14.86 15.64 17.42

this work

ref 41

4.71 4.95 5.21 5.81

4.6

Table 5. Mass Density (ρ), Transverse Wave Velocity (vt), Longitudinal Wave Velocity (vl), Average Acoustic Velocity (vm), and Debye Temperature (θD) of WHx (x = 0, 0.0625, 0.125, and 0.25) Materials Figure 5. Bulk expansion coefficient of WHx (x = 0, 0.0625, 0.125, and 0.25) materials as a function of temperature.

acoustic velocity (m s−1) −3

pure W WH0.0625 WH0.125 WH0.25

4. CONCLUSIONS In summary, we studied the influences of interstitial H on the mechanical and thermal properties of W metal, which is an G

θD (K)

ρ (g m )

vt

vl

vm

this work

ref 41

19.15 18.95 18.74 18.38

2819 2729 2608 2567

5208 5142 4994 5056

3146 3050 2917 2877

372.39 366.96 369.11 361.60

333.4


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substantial negative influences on the thermal performance of the W metal.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Diyou Jiang: 0000-0002-9665-5049 Chuying Ouyang: 0000-0001-8891-1682 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China under Grant No. 11564016. The computations were partly performed on TianHe-1(A) at the National Supercomputer Center in Tianjin.

Figure 6. Grüneisen constants of WHx (x = 0, 0.0625, 0.125, and 0.25) materials as a function of temperature.

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REFERENCES

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Figure 7. Thermal conductivity of WHx (x = 0, 0.0625, 0.125, and 0.25) materials as a function of temperature contributed from lattice vibrations.

important candidate for plasma facing materials. The mechanical parameters are calculated from density-functional theory, whereas the lattice vibrational data are obtained from density-functional perturbation theory. Using these data, the temperature-dependent thermal expansion coefficient and vibration-contributed thermal conductivity of WHx (x = 0.0625, 0.125, and 0.25) are calculated and compared with those of pure W metal. The results confirmed that the interstitial H atom in W is both thermodynamically and dynamically stable, even at very high H concentrations. Although interstitial H decreases slightly the strength of the W metal, it is beneficial to improve the ductility of the W metal if the vacancy defects are not considered. However, the thermal expansion coefficient of the W metal is increased substantially by interstitial H, particularly at high temperatures. When the H concentration increases to a certain extent, W material becomes a high-expansion material from a medium expansion material. The phonon contribution to the thermal conductivity of W metal is calculated to be about 52.8 W m−1 K−1 at 300 K, which is about 30.3% of the total thermal conductivity. However, it decreases substantially to 13.5, 13.3, and 9.9 W m−1 K−1 for cases of WH0.0625, WH0.125, and WH0.25, respectively. At high temperatures, the phonon-contributed thermal conductivity of the W with interstitial H becomes negligibly small. These results suggest that interstitial H has H


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Effect of Interstitial Hydrogen on the Mechanical and Thermal

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