Temperature dependence of surface and grain boundary energies from first principles

In this study we systematically study the temperature dependence of interface energies in W using first principles. To that purpose, we compute interface free energies and consider different contributions, i.e., from lattice expansion, vibrational contribution from harmonic and quasiharmonic approximation with explicit phonon calculations, and electronic contribution. We find that interface energies decrease with temperature and that in most cases the harmonic approximation to the free energy is sufficient. The relative decrease of the interface energy at 2000 K for the surfaces varies from $-20$% to $-30$%, while for grain boundaries, the interface energies decrease from $-30$% to $-55$%. Our results are compared to models on temperature dependence of interface energies available in literature. We find that they are in qualitative agreement also exhibiting a general decrease with higher temperatures. However, most models neglect anisotropy effects and are not in quantitative agreement. The equilibrium crystal shape matches with experimentally observed particle and pore shapes.

Figure 1

Convergence of the contribution from harmonic approximation with increasing the calculation cells in $x$ and $y$ directions for (a) surfaces and (b) GBs. For the (100) surface no larger cells were investigated.

Figure 2

Model of the employed (a) (100), (b) the (110), (c) the (111), (d) the (211), and (e) the (310) surface. The green lines indicate the borders of the primitive surface cells.

Figure 3

Model of the employed (a) $\mathrm{\Sigma }3\left[110\right]\left(1\overline{1}1\right)$, (b) the $\mathrm{\Sigma }3\left[110\right]\left(1\overline{1}2\right)$, (c) the $\mathrm{\Sigma }5\left[100\right]\left(013\right)$, and (d) the $\mathrm{\Sigma }9\left[110\right]\left(1\overline{1}4\right)$ GBs. The green lines indicate the borders of the primitive GB cells.

Figure 4

Schematics for the computation of surface and GB energy.

Figure 5

The temperature dependence on lattice expansion from QHA with and without electronic contribution of the free energy is compared to measured lattice expansion from Refs. [45, 46, 47, 48, 49, 50] and lattice expansion computed from first principles within QHA [51].

Figure 6

The (100) surface of W is shown from top without reconstruction (a) and with reconstruction (b). The coloring that has been used for the different layers is apparent from (c), where the (100) surface is shown from the [010] direction. (d) Presentation of the vibrational DOS for the symmetric and reconstructed (100) surface.

Figure 7

The phonon density of states for the investigated (a) surfaces and (b) GBs. (c) The modes in the $\mathrm{\Sigma }3\left[110\right]\left(1\overline{1}1\right)$ GB that are responsible for the peak at high frequencies in the phonon density of states.

Figure 8

The temperature dependence of the investigated surfaces from QHA with a comparison to 0-K DFT results (Scheiber [11]), measured surface energies for solid W (Tyson [4]), and liquid W (Paradis [8]).

Figure 9

(a) The temperature dependence of the investigated GBs from QHA with a comparison to 0-K DFT results (Scheiber 2016 [11]), and measured GB energies by Hodkin et al. [9] and Allen [10]. (b) From the [012] direction the $\mathrm{\Sigma }5$(013) GB (solid frame) and its excited variant, the $\mathrm{\Sigma }{5}^{\prime }$(013) (dashed frame), which differ in their translation state along the [100] direction.

Figure 10

The temperature dependence of ${\gamma }_{\mathrm{GB}}$, ${\gamma }_{\mathrm{FS}}$, and the resulting $W_{\mathrm{sep}}$ for the $\mathrm{\Sigma }3\left(111\right)$, the $\mathrm{\Sigma }3\left(112\right)$, and the $\mathrm{\Sigma }5\left(013\right)$ GB from QHA with a comparison to 0-K DFT results (Scheiber 2016 [11]).

Figure 11

The contributions to the free interface energy for the (100) and the (110) surfaces and the $\mathrm{\Sigma }3\left(111\right)$ and the $\mathrm{\Sigma }3\left(112\right)$ GBs at 2000 K from lattice expansion $\left[U\right(V\left)\right]$, QHA $\left[F_{\mathrm{vib}}(V,T)\right]$, electronic contribution $\left[F_{\mathrm{el}}(V_0,T)\right]$, and harmonic approximation only $\left[F_{\mathrm{vib}}(V_0,T)\right]$.

Figure 12

Free interface energy for the (100) and the (110) surfaces for homologous temperatures: a comparison between W (this work) and Fe (Schönecker et al. [19]).

Figure 13

The plot in (a) compares our results for the temperature dependence of the investigated surfaces from QHA with two models (Tyson 1975 [4] and Cheng 2017 [60]) and an extrapolation (de Boer 1988 [6]). In (b), our results from QHA for the GB energies are compared to two models (Foiles 2010 [16] and Cheng 2018 [61]). For reasons of clarity, the models by Cheng et al. are only applied to one surface and one GB.

Figure 14

Wulff shapes constructed from free surface energies at different temperatures.

Figure 15

(a) and (c) Depiction of the SEM images of W powder with different magnification that has been treated at temperature of 1300 K and consists of small W particles. (b) and (d) Shown are different magnifications of pores of a fractured W sinter sample that has been treated at 2600 K. (e) and (f) An overlay of our calculated Wulff shapes with the experimental observations is presented.