Effects of PKA energies and H atoms on electronic thermal conductivity

The calculations in this section were performed at 800 K, which is the operating temperature for W-PFM. Fig. 2a illustrates the variation in κ_e values for pure tungsten systems during cascades, where the initial kinetic energy of PKA (represented as E_PKA) is 300 eV and 500 eV, respectively. It is evident that κ_e undergoes a significant decrease during the initial stage (approximately 0.3 ps), followed by a partial recovery at a slow pace. As E_PKA increases, κ_e further decreases during the initial stage, whereas the recovery rate remains almost unchanged during the recovery period. In addition, it should be noted that even after approximately 4 ps, κ_e values for cases with E_PKA values of 300 eV and 500 eV recover to only approximately 64% and 55% of the pristine crystal's value, respectively. We will provide a discussion of this phenomenon later.

Fig. 2Full-screen View

(Color online) The electronic thermal conductivity of pure tungsten systems (a) and tungsten systems with and without H atoms (b) at 800 K and different E_PKA as a function of time during cascades. (c) The number of Frenkel pairs and displaced atoms for the considered systems at 800 K as a function of time. The purple dashed line corresponds to 0.3 ps

Similar trends in κ_e variation were observed for systems containing H atoms, as shown in Fig. 2b. In the case of the system with 20 H atoms (H concentration: 0.88%), κe closely resembled that of pure tungsten systems, except for a reduction of approximately 10 W m^-1K^-1 at 0 ps. For systems with 60 H atoms (H concentration: 2.65%), κ_e in E_PKA=300 eV cascade exhibited a clear reduction compared to that of pure tungsten systems. In contrast, the reduction in κ_e in the E_PKA=500 eV cascade appeared to be less significant. This suggests that a high concentration of H atoms in tungsten can further reduce κ_e, particularly in low-energy cascades. Additionally, the rates of reduction and recovery in κ_e for H-containing systems are lower than those for pure tungsten systems, as reflected by the slopes of the curves in Fig. 2b.

To gain a deeper understanding of the variations in κe, we examined the numbers of displaced atoms and Frenkel pairs. We focused on the defects associated with W atoms for a clear comparison between systems with and without H atoms. As shown in Fig. 2c, the numbers of both defects first sharply increase and then reach local maxima at 0.3 ps. Subsequently, the number of Frenkel pairs gradually decreases and approaches zero, after which the number of displaced atoms gradually stabilizes. Furthermore, both defect numbers noticeably increase as E_PKA increases, whereas the introduction of H atoms has little effect on these quantities, indicating that the reduction in κe may be attributed to the H atom itself. The behavior of these defect variations roughly matches the κe behavior because an increase in the defect numbers enhances electron scattering and ultimately degrades the electronic transport performance. However, it is worth noting that although the number of Frenkel pairs approaches zero after 2 ps, the corresponding κe value remains close to its minimum and is significantly lower than the value at 0 ps. This suggests that despite the decrease in the number of vacancies and interstitials, the electrons in the system continue to experience severe scattering. In this scenario, we speculate that the severe electron scattering is due to the accumulation of slight structural distortions. These distortions likely arise from the incomplete restoration of displaced atoms to their initial thermal equilibrium state within a few picoseconds during the recombination of the point defects.

To validate this hypothesis, we extracted the centrosymmetry parameter (P) for each W atom in the systems. This parameter qualifies the deviation from centrosymmetry in the immediate vicinity of an atom. Table 1 presents representative data for both pure tungsten systems and systems containing 60 H atoms, where E_PKA is 500 eV. Initially (at 0 ps), approximately 2.12% of the W atoms in pure tungsten exhibit P values greater than 1, indicating significant deviations from centrosymmetry in the immediate vicinity of only a few atoms due to atomic thermal motion. As the cascade event progresses, the structural evolution leads to changes in the number of W atoms, with P values greater than 1. At approximately 4 ps, the number of W atoms with P values above 1 is slightly lower than that at approximately 0.3 ps but remains significantly higher than that at the initial thermal equilibrium state (0 ps). This indicates that a few picoseconds after the PKA incidents, the local environment of some W atoms still substantially deviates from the thermal equilibrium state. The accumulation of these deviations can result in strong electron scattering, as evidenced by a 1.66 nm decrease in MFP around 4 ps compared to the initial state (0 ps). When 60 H atoms are present in the system, two notable differences emerge in the analysis of centrosymmetry parameters. First, the presence of H atoms exacerbates the deviation from centrosymmetry near the W atoms. Second, the rate of change of Psum decreases when H atoms are present. These differences are consistent with the observed alterations in κe.

Effect of temperature on electronic thermal conductivity

It is widely acknowledged that the influence of temperature on defect generation during a cascade is relatively minor [43]. However, the effect of temperature on κ_e values of defective systems is complex [44]. To further investigate the impact of temperature on κ_e during cascades, we studied the κ_e values of tungsten systems with varying E_PKA and different numbers of H atoms under a high temperature of 1300 K.

As shown in Fig. 3a, an increase in temperature leads to a noticeable improvement in κ_e values throughout the considered cascades. Specifically, at 0.3 ps, the corresponding minimum κ_e values for all examined systems exhibit a relatively high value of approximately 50 W m^-1 K^-1. Subsequently, κ_e values are consistently significantly higher than those observed at 800 K. In addition, the disparity in κ_e between the pure tungsten system and the tungsten system containing H atoms remains nearly unchanged with increasing temperature, except at 0 ps. At 0 ps, κ_e of the pure tungsten system remains almost constant, whereas that of the system with H atoms shows a significant improvement.

Fig. 3Full-screen View

(Color online) The electronic thermal conductivity (a) and number of Frenkel pairs and displaced atoms (b) of tungsten systems with and without H atoms at 1300 K and different E_PKA as a function of time during cascades. The purple dashed line corresponds to 0.3 ps. For comparison, the blue and red dashed lines in (a) correspond to the electronic thermal conductivities of pure tungsten systems at 800 K

A count of typical defects in the cascades at 1300 K was performed, as illustrated in Fig. 3b. It was observed that the local maxima for both types of defects slightly increase compared to those at 800 K (Fig. 2c. After approximately 2 ps, the defect counts at both temperatures become similar. Moreover, Table 2 provides representative data on the CSP and MFP for the pure tungsten system and the system containing 60 H atoms at 1300 K with an E_PKA value of 500 eV. The major distinction between the two temperature cases lies in the significant increase in the number of W atoms with P values greater than 1 and the corresponding Psum throughout the entire process with increasing temperature. This indicates that an elevated temperature causes a greater deviation from centrosymmetry in the immediate vicinity of the W atoms, leading to enhanced electron scattering. However, the thermal transport properties of the electrons are enhanced at 1300 K compared to those at 800 K. This phenomenon arises from the complex effect of temperature on κ_e. On the one hand, a temperature rise strengthens the thermal motion of atoms, thereby augmenting the electron scattering, as supported by the CSP and MFP analyses; on the other hand, the energy carried by electrons also increases with a higher temperature, directly facilitating heat transport. The complex effect of temperature is also reflected in the Wiedemann-Franz law [45], as discussed in our previous work [44].

In summary, although an increase in temperature has a slight effect on defect production and even enhances electron scattering, the accompanying increase in the energy carried by electrons significantly improves electronic thermal transport.

Effect of grain boundary on electronic thermal conductivity

Previous studies [16, 20] have demonstrated that GBs can serve as effective sinks for point defects, particularly interstitials, leading to self-healing of irradiation damage. Additionally, the extent of overlap between the cascade center and the GB is known to influence the absorption effect of the GB for point defects [17, 19, 22], which may affect the electron transport properties. Hence, we examined the influence of different levels of overlap between the cascade center and the GB on κ_e during cascades. We employed a common symmetric tilt GB in W, ${\displaystyle \sum }5\left[001\right]\left(130\right)$ (Fig. 1(b)). In our treatment, E_PKA was 500 eV and the temperature was 800 K. When a PKA (green sphere in Fig. 1(b)) was situated near the GB, the majority of the cascades occurred within the GB plane. Conversely, when the PKA (yellow sphere in Fig. 2(b)) was far from the GB, only a small portion of the cascade edge overlapped with the GB plane.

Then, we calculated the κ_e values for systems with a specific GB during the cascades. Because the GB exhibited anisotropy, we extracted κe values separately along the X, Y, and Z directions. Figure 4a shows a substantial reduction in κ_e values in all three directions compared to the cases of the pure tungsten systems throughout the cascades. In particular, κ_e in the Y direction experienced the most pronounced decrease, primarily due to the strong electron scattering at the GB interface. Throughout the cascades, the presence of the GB itself reduced the electronic transport properties (0 ps), and the reduction (0 ps–0.3 ps) and subsequent recovery (0 ps-4 ps) of κ_e were relatively slow compared to those in the pure tungsten systems. Notably, the degree of overlap between the cascade center and GB does not significantly affect κ_e for the considered systems.

Fig. 4Full-screen View

(Color online) The electronic thermal conductivity (a) and number of vacancies and interstitials in grain interior (b) of GB systems with different overlap degrees of the GB and cascade center at 800 K as a function of time during cascades. The green symbols correspond to the case in which the PKA is near the GB, and the yellow symbols correspond to the case in which the PKA is far from the GB. The gray vertical dashed line corresponds to 0.3 ps

To better understand the variation of κ_e, we counted the numbers of vacancies and interstitials outside the GB. Throughout the cascades, the vacancies outnumbered the interstitials for both degrees of overlap between the cascade center and GB. Moreover, increasing the degree of overlap decreased the number of vacancies and interstitials outside the GB. These results were consistent with those of previous studies [17, 19, 22]. However, the enhancement in the absorption effect of the GB does not improve κ_e but rather slightly weakens it at approximately 0.3 ps in the X and Z directions. This indicates that point defects absorbed by the GB continue to negatively impact electronic transport in the short term, despite reducing the defects within the grain interior. The corresponding data in Table 3 for both cases support this observation. Specifically, similar CSP data and comparable MFP for the two cases indicate similar deviations from centrosymmetry in the immediate vicinity of the atoms and comparable electron scattering intensities.

Discussion on using TB model for calculation

First, we highlight the reliability of the WHTB potential model for calculating the electronic thermal conductivities of tungsten-based systems. In this work, electronic thermal conductivity was obtained based on electrical conductivity through the Kubo-Greenwood method; therefore, an accurate calculation of electrical conductivity is crucial. Reference [46] compared the electrical conductivities of pristine tungsten calculated using the WTB potential model (on which the WHTB potential is based) and measured experimentally at different temperatures. Notably, a favorable alignment between the theoretical and experimental values at high temperatures ($T\ge 800K$) was observed for pristine tungsten. However, at low temperatures, the TB model underestimates electrical conductivity. This is because the impact of long-range electron interactions increases as the temperature decreases, and the TB model cannot deal with these interactions well.

A similar underestimation is evident in the calculations of the electronic thermal conductivity of pristine tungsten systems, and this underestimation also intensifies with decreasing temperature. However, it is noted that the referenced work compared the electronic thermal conductivity calculated by the Kubo-Greenwood method and the total thermal conductivity measured directly in experiments, with the latter including not only the electronic thermal conductivity but also the lattice thermal conductivity. Hence, some of the underestimation stems from the absence of the lattice thermal conductivity. In experiments, electronic thermal conductivity can be indirectly determined by measuring the electrical conductivity and applying the Wiedemann-Franz law [47]. In our previous work [44], we conducted a comparative analysis of the electronic thermal conductivity of pristine tungsten at 800 K obtained by TB calculations and experiments. The respective values were 95.92 W m^-1 K^-1 and 102.36 W m^-1 K^-1, which are close to each other. Therefore, the electronic thermal conductivity data for tungsten-based systems at high temperatures ($T\ge 800K$) obtained using the WHTB potential model can be considered reliable.

Next, we discuss the significance of our investigation using the TB model. In this study, we focused on the dynamic evolution of the electronic thermal conductivity of tungsten during the displacement cascade process under diverse conditions, which is difficult to directly measure experimentally. This process corresponds to the early stage of neutron irradiation, leading to the generation of a significant number of vacancies and interstitials in the materials within a brief timeframe at the picosecond level. The resulting alterations in the thermal transport properties are substantial, potentially affecting the normal service of PFM. Hence, the electronic thermal conductivity variation in tungsten during the cascade process must be investigated. Currently, research on the electronic thermal conductivity of tungsten during cascade processes is limited because of the difficulty of handling electronic structures in large and complex systems. Although the WHTB model can be used to determine the electronic thermal conductivity of tungsten under low-energy cascades, it is difficult to directly compare the results with the conductivities of materials experiencing high-energy and sustained neutron irradiation at the macroscopic level. Nevertheless, our research contributes to this field in two ways. First, the results reveal the influence of diverse cascade conditions on the electronic thermal conductivity of tungsten; second, our study can provide foundational data for simulations using the phase-field model, facilitating simulations that more accurately reflect the actual evolution of thermal transport properties of tungsten under neutron irradiation.