A first-principles model for anomalous segregation in dilute ternary tungsten-rhenium-vacancy alloys

We describe the state of a solid solute under irradiation by three concentrations: x_A, x_B, x_C that are, respectively, the solvent, solute and vacancy concentrations expressed in terms of the number per lattice site units. In our model, the three concentrations are subject to a constraint

$$\begin{eqnarray}&&{{x}_{A}}+{{x}_{B}}+{{x}_{C}}=1.\end{eqnarray} \tag{ 1 }$$

In cluster expansion (CE), the configuration enthalpy of mixing of a multi-component alloy is defined as [58]

$$\begin{eqnarray} &&\Delta {{H}_{\text{CE}}}(\vec{\sigma})=\underset{\omega}{\sum}{{m}_{\omega}}{{J}_{\omega}}{{\langle {{\Gamma}_{{{\omega}^{\prime}}}}(\vec{\sigma})\rangle}_{\omega}},\end{eqnarray} \tag{ 2 }$$

where summation is performed over all the clusters ω that are distinct under group symmetry operations applied to the underlying lattice, $m_{\omega}^{\text{lat}}$ are the multiplicities indicating the number of clusters equivalent to ω by symmetry, divided by the number of lattice sites, and $\langle {{\Gamma}_{{{\omega}^{\prime}}}}(\vec{\sigma})\rangle$ are the cluster functions defined as products of functions of occupation variables on a specific cluster ω averaged over all the clusters ${{\omega}^{\prime}}$ that are equivalent by symmetry to cluster ω. ${{J}_{\omega}}$ are the concentration-independent effective cluster interaction (ECI) parameters, derived from a set of ab initio calculations using the structure inversion method (SIM) [64]. A cluster ω is defined by its size (i.e. the number of lattice points) $|\omega |$ and relative positions of points. For clarity, each cluster ω is described by two parameters $(|\omega |,n)$, where $|\omega |$ is the cluster size and n is a label, see table S1 from the supplementary information (SI) for bcc lattice.

In CE developed for a K-component system, a cluster function is not a simple product of occupation variables, $\{{{\sigma}_{i}}\}$. Instead, it is defined as a product of orthogonal point functions ${{\gamma}_{{{j}_{i}},K}}({{\sigma}_{i}})$,

$$\begin{eqnarray} &&\Gamma _{\omega,n}^{(s)}(\vec{\sigma})={{\gamma}_{{{j}_{1}},K}}({{\sigma}_{1}}){{\gamma}_{{{j}_{2}},K}}({{\sigma}_{2}})\ldots {{\gamma}_{{{j}_{|\omega |}},K}}({{\sigma}_{|\omega |}}),\end{eqnarray} \tag{ 3 }$$

where sequence $(s)=(\,{{j}_{1}}{{j}_{2}}\ldots ~{{j}_{|\omega |}})$ is the decoration [65] of a cluster by point functions. We use the following definition of point functions for a K-component system

$$\begin{eqnarray}{{\gamma}_{j,K}}\left({{\sigma}_{i}}\right)=\left\{\begin{array}{*{35}{l}} 1 & \,\text{if}\,\,j=0, \\ -\cos \left(2\pi \left[\frac{j}{2}\right]\frac{{{\sigma}_{i}}}{K}\right) & \text{if}\,\,j>0\,\,\text{and}\,\text{odd}, \\ -\sin \left(2\pi \left[\frac{j}{2}\right]\frac{{{\sigma}_{i}}}{K}\right) & \text{if}\,\,j>0\,\,\text{and}\,\text{even}, \end{array}\right. \end{eqnarray} \tag{ 4 }$$

where ${{\sigma}_{i}}=0,1,2,\ldots,(K-1)$, j is the index of point functions ($j=0,1,2,\ldots,(K-1)$). In a ternary alloy, index K equals 3 and occupation variables are defined as $\sigma =0,1,2$, referring to the constituent components of the alloy A, B and C, which here correspond to W, Re and Vac.

According to the derivation given in [59], the configuration enthalpy of mixing of a ternary alloy can be expressed analytically in terms of three-body interaction parameters as a function of concentrations, x_i, and the average pair and 3-body probabilities, $y_{n}^{ij}$ and $y_{n}^{ijk}$ as

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Delta {{H}_{\text{CE}}}(\vec{\sigma}) & = & J_{1}^{(0)}+J_{1}^{(1)}(1-3{{x}_{A}})+J_{1}^{(2)}\frac{\sqrt{3}}{2}({{x}_{C}}-{{x}_{B}}) \\ {} & \, & +\,\underset{n}{\overset{\text{pairs}}{\sum}}\left[\frac{1}{4}m_{2,n}^{(11)}J_{2,n}^{(11)}(1+3y_{n}^{AA}-6y_{n}^{AB}-6y_{n}^{AC}) \right. \\ {} && \left. \, +\,\frac{\sqrt{3}}{4}m_{2,n}^{(12)}J_{2,n}^{(12)}(-y_{n}^{BB}+y_{n}^{CC}+2y_{n}^{AB}-2y_{n}^{AC})+\frac{3}{4}m_{2,n}^{(22)}J_{2,n}^{(22)}(\,y_{n}^{BB}+y_{n}^{CC}-2y_{n}^{BC})\right] \\ {} & \, & +\,\underset{n}{\overset{\text{triples}}{\sum}}\left[\frac{1}{8}m_{3,n}^{(111)}J_{3,n}^{(111)}(-8y_{n}^{AAA}+12y_{n}^{AAB}+12y_{n}^{AAC} \right. \\ {} && \, -\,6y_{n}^{ABB}-6y_{n}^{ABC}-6y_{n}^{ACC}+y_{n}^{BBB}+3y_{n}^{BBC}+3y_{n}^{BCC}+y_{n}^{CCC}) \\ {} & \, & +\,\frac{\sqrt{3}}{8}(m_{3,n}^{(112)}J_{3,n}^{(112)}+m_{3,n}^{(121)}J_{3,n}^{(121)}+m_{3,n}^{(211)}J_{3,n}^{(211)}) \\ {} & \, & \cdot \,\,(-4y_{n}^{AAB}+4y_{n}^{AAC}+4y_{n}^{ABB}-4y_{n}^{ACC}-y_{n}^{BBB}-y_{n}^{BBC}+y_{n}^{BCC}+y_{n}^{CCC}) \\ {} & \, & +\,\frac{3}{8}(m_{3,n}^{(122)}J_{3,n}^{(122)}+m_{3,n}^{(212)}J_{3,n}^{(212)}+m_{3,n}^{(221)}J_{3,n}^{(221)}) \\ {} & \, & \cdot \,(-2y_{n}^{ABB}+2y_{n}^{ABC}-2y_{n}^{ACC}+y_{n}^{BBB}-y_{n}^{BBC}-y_{n}^{BCC}+y_{n}^{CCC}) \\ {} && \left. \, +\,\frac{3\sqrt{3}}{8}m_{3,n}^{(222)}J_{3,n}^{(222)}(-y_{n}^{BBB}+3y_{n}^{BBC}-3y_{n}^{BCC}+y_{n}^{CCC})\right] \\ {} & \, & +\,\underset{n}{\overset{\text{multibody}}{\sum}}\ldots \end{array}\end{eqnarray} \tag{ 5 }$$

It is worth mentioning here that equation (5) with C  =  V (vacancy) represents a generalized Hamiltonian of the so-called 'ABV' Ising model [66], the conventional form of which contains only the first and second nearest-neighbor effective pair interactions. The 'ABV' models have been applied to the investigation of non-equilibrium phenomena such as enhanced diffusion and segregation in irradiated materials [67]. We now apply the more accurate cluster-expansion based ABV model to the treatment of radiation-induced precipitation in bcc W-rich alloys, where anomalous segregation of Re atoms was observed experimentally [15, 20]. The enthalpy of mixing of a W–Re–Vac configuration is found by subtracting the total enthalpies corresponding to pure end terms as:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Delta {{H}_{\text{mix}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}}) & = & {{E}_{\text{tot}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}})-{{x}_{\text{W}}}{{E}_{\text{tot}}}(\text{W})-{{x}_{\text{Re}}}{{E}_{\text{tot}}}(\text{Re})-{{x}_{\text{Vac}}}{{E}_{\text{tot}}}(\text{Vac}) \\ {} & = & {{E}_{\text{tot}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}})-{{x}_{\text{W}}}{{E}_{\text{tot}}}(\text{W})-{{x}_{\text{Re}}}{{E}_{\text{tot}}}(\text{Re}), \end{array}\end{eqnarray} \tag{ 6 }$$

where the reference enthalpy of a vacancy is assumed to be zero, ${{E}_{\text{tot}}}(\text{Vac})=0$.

The values of ECIs ($J_{|\omega |,n}^{(s)}$) for ternary W–Re–Vac bcc alloys are derived by mapping DFT energies onto the CE energies computed for 224 structures, which are described in the next sections of this paper. According to the definition of cluster functions (equation (3)) and point functions (equation (4)) for ternary alloys, each cluster can be decorated in various ways for each nearest-neighbour shell of interactions. Here, we have used a set of 15 two-body, 12 three-body, and 6 four-body clusters on bcc lattice. Values of all the optimized ECIs for ternary W–Re–Vac alloys are shown in figure 1 and also in the last column of table S1 in the SI. Section 3 shows that the inclusion of five nearest neighbour shells in two-body interactions is essential for understanding the origin of rhenium–vacancy binding energy trends characterizing under-saturated solid solutions.

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To arrive at a suitable choice of Hamiltonian (equation (2)), the fitness of a cluster expansion can be, in general, quantified by means of leave-many-out cross-validation (LMO-CV) [68]. The set of structures $\vec{\sigma}$ for which we carry out first-principles (DFT) energy calculations, $\Delta {{E}_{\text{DFT}}}({{\vec{\sigma}}_{\text{input}}})$ is subdivided into a fitting set ${{\{\vec{\sigma}\}}_{\text{fit}}}$ and a prediction set ${{\{\vec{\sigma}\}}_{\text{pred}}}$. We determine ECIs ($J_{|\omega |,n}^{(s)}$) by fitting to $\Delta {{E}_{\text{DFT}}}(\vec{\sigma})$ in ${{\{\vec{\sigma}\}}_{\text{fit}}}$. The predicted values of $\Delta {{E}_{\text{CE}}}(\vec{\sigma})$ for each structure ${{\{\vec{\sigma}\}}_{\text{fit}}}$ are then compared with their known first-principles counterparts $\Delta {{E}_{\text{DFT}}}(\vec{\sigma})$. This process can be iterated M times for various subdivisions of ${{\{\vec{\sigma}\}}_{\text{input}}}$ into fitting and prediction sets ${{\{{{\vec{\sigma}}_{\text{fit}}}\}}^{(m)}}$ and ${{\{{{\vec{\sigma}}_{\text{pred}}}\}}^{(m)}}$ (m  =  1... N). The values of the LMO-CV score is defined as the average prediction error of this iterative procedure,

$$\begin{eqnarray}&&{{({{S}_{\text{CV}}})}^{2}}=\frac{1}{M}\frac{1}{{{N}_{\text{pred}}}}\underset{m=1}{\overset{M}{\sum}}\underset{\{\vec{\sigma}\}_{\text{pred}}^{(m)}}{\sum}{{[\Delta (E)_{\text{CE}}^{(m)}(\vec{\sigma})- \Delta {{E}_{\text{DFT}}}(\vec{\sigma})]}^{2}},\end{eqnarray} \tag{ 7 }$$

In this work, the CE Hamiltonian defined by equation (2) is applied to a defective vacancy-rich W–Re–Vac solid solution where, as opposed to a conventional ternary non-defective alloy [59], there are no further ground-state stable structures that can be found in comparison with the ones predicted from binary W–Re alloy calculations. Therefore, the LMO-CV score can be reduced to the leave-one-out cross validation (LOO-CV) score with M  =  1 in equation (7). The LOO-CV score characterizing the quality of agreement between DFT and CE has been assessed using 224 structures representing interactions in W–Re–Vac system. The value of the LOO-CV error is 4.12 meV/atom, confirming the very high accuracy of the set of ECIs used in the simulations.

DFT calculations were performed using Vienna ab initio simulation package (VASP) with the interaction between ions and electrons described using the projector augmented waves (PAW) method [69, 70]. Exchange and correlation were treated in the generalized gradient approximation GGA-PBE [71], with PAW potentials containing semi-core p electron contributions. Supercell calculations were performed considering vacancy clusters interacting with Re atoms in bcc W lattice under constant pressure conditions, with structures optimized by relaxing both atomic positions as well as the shape and volume of the supercell. To treat clusters containing from 2 to 47 sites with Re atoms and vacancies, orthogonal supercells containing 128 and 250 atoms were used. Total energies were evaluated using the Monkhorst-Pack mesh [72] of k-points in the Brillouin zone, with the k-mesh spacing of 0.15 ${{{\mathring{\text{A}}}}^{-1}}$. This corresponds to $4\times 4\times 4$ or $3\times 3\times 3$ k-point meshes for a bcc supercell of $4\times 4\times 4$ or $5\times 5\times 5$ bcc structural units, respectively. The plane wave cut-off energy was 400 eV. The total energy convergence criterion was set to 10⁻⁶ eV/cell, and force components were relaxed to 10⁻³ eV ${{{\mathring{\text{A}}}}^{-1}}$. Vacancy and self-interstitial-atom (crowdion) formation energies evaluated using the above conditions for pure bcc W were 3.307 eV and 10.917 eV, respectively.

Mapping of DFT energies to CE was performed using the ATAT package [73]. For binary bcc alloys we used 58 structures from [74], and the corresponding results for the enthalpy of mixing in the binary W–Re are shown in figure 2 and table S2 in the SI. We find that negative values of the enthalpy of mixing characterizing bcc W–Re alloys were smaller than those obtained using similar DFT/CE calculations performed for binary W–Ta and W–V alloys [39]. The most stable configurations predicted for W–Re binary alloys remain the same as in the extended treatment of ternary W–Re–Vac alloys as well as in the simplified CE scheme discussed in the last section of the paper (see table S4 in the SI), where in the first nearest neighbour (1NN) shell the effective chemical interaction between W and Re is attractive but is more than 5 times smaller than that between Re and a vacancy. In the second nearest neighbour (2NN) coordination shell the effective W–Re interaction is weak and repulsive whereas Re–Vac interaction is strong and attractive. Hence we conclude that clustering of Re in bcc W is driven primarily by binding between Re solute atoms and the vacancies.

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The DFT database of structures used in this study uses data not only for the W–Re binary system but also data on vacancy clusters in W (W–Vac) [39], as well as data on Re-vacancy cluster interactions in W.

Quasi-canonical MC simulations were carried out using the ATAT package [73]. All the simulations described in section 4 used a $30\times 30\times 30$ bcc supercell containing 54 000 lattice sites. To evaluate the formation free energy for steady state configurations using thermodynamic integration, simulations were performed starting from a disordered high-temperature state, corresponding to T  =  3000 K. Configurations were then cooled down with the temperature step of $\Delta T=10$ K, with 3000 MC steps per atom at both thermalization and accumulation stages. It was found that it was not necessary to use a higher number of MC steps to achieve convergence.

Calculations of binding energies of Re atoms with a mono-vacancy at zero Kelvin as a function of the ratio of the number of Re atoms to vacancies were performed by replacing W atoms by Re atoms in the first and then in the second coordination shells of a vacancy, see blue and pink spheres in figure 3(a). Results are given in figure 4 and table S3 in the SI. The binding energy of a vacancy with a Re atom is 0.183 eV, which is slightly smaller than the value of approximately 0.2 eV found earlier [40]. Since the enthalpy of mixing of W and Re atoms is small, see figure 2, the strength of binding in vacancy-Re clusters depends primarily on the number of Re atoms in the neighbourhood of a vacancy, and does not depend significantly on the specific configuration of Re atoms in the first or the second shells around a vacancy. As a result, the binding energy increases almost linearly as a function of Re atoms around a vacancy, up to the value of 1.304 eV corresponding to the case where Re atoms occupy eight positions in the first nearest neighbour coordination shell. Binding energy decreases once Re atoms start occupying positions in the second nearest neighbour coordination shell, see figure 4.

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Binding energies of di-vacancies as functions of Re/vacancy ratio for di-vacancies in the first, second, third, fourth and fifth nearest neighbourhood indicated as 1NN, 2NN, 3NN, 4NN and 5NN, respectively, are shown in figure 4(a) and table S3 in the SI. In pure W matrix, the binding energies of all the configurations apart from 4NN di-vacancy configuration, are negative, meaning that vacancies repel each other. The most negative is the binding energy of a 2NN di-vacancy configuration that is equal to  −0.177 eV per vacancy, which is slightly less negative than  −0.190 eV per vacancy reported earlier [36, 39]. The binding energy of the 1NN di-vacancy configuration equals to  −0.012 eV per vacancy, and is small but still negative.

In order to investigate the effect of Re atoms on the binding energies of various di-vacancy configurations, W atoms in the 1NN and 2NN positions were systematically replaced by Re atoms. Since the binding energy primarily depends on the number of Re atoms in the 1NN and 2NN positions with respect to vacancies, the largest contribution to the binding energy of a cluster comes from Re atoms that have two vacancies in their first nearest-neighbour coordination shells. Hence, a systematic investigation of the effect of Re atoms on binding energies of the 3NN di-vacancy configuration starts with a replacement of W atoms by Re atoms in the first nearest neighbourhood of two vacancies, see red spheres in figure 3(b). Afterwards the replacement of W atoms by Re atoms was performed in the following sequence: atoms that have one vacancy in the 1NN shell, two vacancies in the 2NN shell and one vacancy in the 2NN shell indicated by blue, green and pink spheres in figure 3(b), respectively. The same scheme was applied to the replacement of W atoms by Re atoms in other di-vacancy and multi-vacancy configurations, see figures 3(c) and (d) for the case of four-vacancy cluster and a 15-vacancy void, respectively.

Binding energies of all the di-vacancy configurations apart from the 2NN di-vacancy configuration (vacancies ordered in the (1 0 0) direction) are positive, meaning that vacancies in the neighbourhood of a Re atom attract each other. In other words, a Re atom provides a centre of nucleation for vacancies. The largest binding energy that is equal to 0.157 eV per vacancy characterizes the 3NN di-vacancy configuration (vacancies ordered in the (1 1 0) direction) bound by a Re atom, despite the fact that the binding energy of a 3NN di-vacancy configuration without Re atoms is only slightly more negative that that of the 1NN and 4NN di-vacancy configurations. The 3NN di-vacancy configuration is the most preferable energetically up to the Re/vacancy ratio approaching 7. For the ratios above 7, all the configurations of vacancies apart from the 1NN di-vacancy configuration, have quite similar binding energies. Moreover, they usually have smaller binding energies than those characterizing Re atoms and a mono-vacancy, see the black line in figure 4(a). It means that for the Re/vacancy ratios higher than 7, vacancies may prefer to occupy configurations other than the third nearest neighbour sites. The binding energies of tri-vacancy configurations are shown in figure 4(b) and table S3 in the SI as functions of the Re/vacancy ratio. In a pure W matrix the most stable tri-vacancy configuration is the one with vacancies forming the most compact cluster, see figure 5(f). This configuration remains the most stable in the presence of one Re atom and two Re atoms. For higher Re/vacancy ratios the most stable configuration is the one where vacancies are in the 3NN coordination shell, see figure 5(h).

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Binding energies of quarto-vacancy configurations, forming a (1 0 0) surface, see figure 5(g), and with vacancies in the 3NN coordination shell to each other, see figure 5(h), are shown in figure 4(c) whereas those corresponding to the most compact vacancy cluster (void) configurations, see figure 5(i), are shown in figure 4(c). The values of binding energies are given in table S3 in the SI.

Void configurations, starting from three-vacancy clusters in the 1NN and 2NN configurations, to a 15-vacancy cluster, are stable in a pure W matrix, according to previous ab initio calculations [39, 76]. When the Re/vacancy ratio increases, configurations with vacancies in the 3NN coordination shell become more stable, similarly to how it was found in the tri-vacancy case with the same relative positions of vacancies. Similarly to the case of a tri-vacancy configuration, where the Re/vacancy ratio is higher than 7, binding between Re atoms and a mono-vacancy is stronger than in quarto-vacancy configurations.

Figure 4(d) and table S3 in the SI show that the binding energy of the most compact vacancy clusters (voids) without Re atoms increases as a function of the number of vacancies in a void. This agrees with the earlier work [39]. Results obtained with W atoms replaced by Re atoms show that binding energies increase to over 1.5 eV when the voids are surrounded by Re atoms. Overall, the Re-void configurations containing a larger number of vacancies are more stable than those with smaller number of vacancies over the entire range of Re/vacancy ratios.

In order to compare the relative stability of the most compact vacancy configurations (small voids) with other vacancy cluster configurations, a structure containing 8 vacancies, derived from MC simulations of W–Re alloys with 2% at.Re and 0.05% at. vacancies quenched down from high temperatures (see the section 4.2), is also included in table S3 of SI. The binding energy characterizing this structure, where each vacancy is in the third nearest-neighbour position with respect to others, is smaller than those of the 8-vacancy void configuration for the low ratio of Re/vacancy, but becomes larger when this ratio is $\geqslant$4. In particular, we find that the highest value (1.566 eV) of binding energy between Re atoms and 8-vacancy clusters corresponds to the Re/vacancy ratio of 6.63.

Concluding this section, we note that binding energies at T  =  0 K for various vacancy cluster configurations show consistent trends in terms of their stability as functions of Re concentrations. In the limit where the Re/vacancy ratio is small, the more compact configurations are preferred. As this ratio increases, configurations with vacancies being further apart become more stable. Configurations with vacancies being in the 3NN coordination shell with respect to each other have particularly high stability over a broad interval of Re/vacancy ratios.

Within the framework of a constrained model for phase stability of alloys containing high concentration of vacancies, the thermodynamic properties of W–Re–Vac system were analyzed using quasi-canonical MC simulations and ECIs derived from DFT calculations. The dependence of the configuration entropy on temperature is given by equation

$$\begin{eqnarray}&&{{S}_{\text{conf}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}},T)={\int}_{0}^{T}\frac{{{C}_{\text{conf}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}},{{T}^{\prime}})}{{{T}^{\prime}}}\text{d}{{T}^{\prime}},\end{eqnarray} \tag{ 9 }$$

where the configurational contribution to the specific heat C_conf is related to fluctuations of the enthalpy of mixing at a given temperature [77, 78] through

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}{{C}_{\text{conf}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}},T)\\ \quad =\frac{\langle {{({{H}_{\text{mix}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}},T))}^{2}}\rangle -{{\langle {{H}_{\text{mix}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}},T)\rangle}^{2}}}{{{T}^{2}}},\end{array}\end{eqnarray} \tag{ 10 }$$

where $\langle {{H}_{\text{mix}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}},T)\rangle$ and $\langle {{({{H}_{\text{mix}}}({{x}_{\text{W}}},{{x}_{\text{Re}}},{{x}_{\text{Vac}}},T))}^{2}}\rangle$ are the mean and mean square average enthalpies of mixing, respectively, computed by averaging over all the MC steps at the accumulation stage for a given temperature and for a given composition of the ternary W–Re–Vac system.

The accuracy of evaluation of configuration entropy depends on the temperature integration step in equation (9) and on the number of MC steps performed at the accumulation stage. Test simulations show that choosing a sufficiently small temperature integration step is particularly significant. Calculations of configuration entropy for all the alloy compositions considered below were performed with 3000 MC steps per atom at thermalization and accumulation stages, and with the temperature step of $\Delta T=10$ K. Figure 6 shows the dependence of configuration entropy as a function of temperature for W-2%Re and W-5%Re with 0.1$\%$ of vacancy concentration. Transitions from ideal random solid solutions into the Re-precipitation regime can be seen clearly as steps in the configuration entropy at T  =  1700 K and T  =  1950 K for the alloys containing 5 and 2 at.$\%$Re, respectively.

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Table 1 describes structures of W-Re-vacancy alloys at 100 K containing 54 000 sites derived from MC simulations by quenching down from 2500 K with the temperature step of 100 K and shown as functions of Re concentrations of 1, 2, 5 and 10% and vacancy concentrations of 0.05, 0.1 and 0.2%. The high fraction of vacancy defects considered in this study is consistent with experimental observations of radiation induced precipitates in W–Re alloys forming under self-ion irradiation [20, 79]. The most stable configuration of Re-vacancy clusters at 100 K depends on the concentration of Re atoms. For each vacancy concentration, including 0.05%, 0.1% and 0.2%, for low concentrations of Re atoms, 1% or 2% at.Re, the most stable configurations are the voids surrounded by Re atoms. For higher Re concentrations, 5% or 10% at.Re, vacancies prefer to be further apart from each other, and the most stable configurations are sponge-like Re-vacancy clusters. Those results are in agreement with the analysis of binding energies of Re-vacancy configurations at 0 K treated as a function of Re/vacancy ratio described in section 4.

Table 1. Monte Carlo results as functions of Re and vacancy concentration for alloys quenched down from high temperatures. MC simulations were performed starting from 2500 K. Alloys were cooled down with the temperature step of 100 K to the temperature of 100 K with 3000 MC steps per atom performed both at thermalization and accumulation stages.

	1% at. Re	2% at. Re	5% at. Re	10% at. Re
0.05% at. Vac
0.1% at. Vac
0.2% at. Vac

In this section, to understand the anomalous segregation of rhenium occurring in W–Re alloys under irradiation condition, where alloys are not quenched from high temperatures but are irradiated at a constant temperature, we performed MC simulations at various temperatures assuming a certain concentration of vacancies and Re atoms.

Table 2 shows structures of W–Re alloys with 2% at.Re obtained from MC simulations performed assuming T  =  300 K, 800 K, 1600 K and 2500 K and four different concentrations of vacancies: 0.05%, 0.1%, 0.2% and 0.5%, which can be related to various values of irradiation doses. At the lowest concentration of vacancies of 0.05%, clustering of vacancies in the form of sponge-like Re-vacancy clusters is observed only at 800 K. At 300 K Re atoms are bound to vacancies but do not aggregate. Formation of Re-vacancy clusters becomes significantly more pronounced as the concentration of vacancies increases. If it exceeds 0.1%, at 1600 K vacancies form voids, whereas at 300 K they aggregate into sponge-like Re-vacancy clusters. In W-2%Re alloy at 800 K we observe either the formation of voids surrounded by Re atoms or sponge-like Re-vacancy clusters. The former ones are observed in alloys where vacancy concentration is equal to 0.2% and 0.5%, whereas the latter ones form in alloys where the concentration of vacancies is lower, in the range from 0.05% to 0.1%. This can be explained by the fact that a high concentration of vacancies corresponds to low Re/vacancy ratio, which is the key parameter controlling the stability of Re-vacancy configurations, as described in section 3. Another important conclusion is that at the border between regions of stability of voids and sponge-like Re-vacancy clusters, it is possible to find both types of configurations. At 2500 K both vacancies and Re atoms are fully dissolved.

Table 2. Results of Monte Carlo simulations as functions of vacancy concentration and temperature. Concentration of Re atoms is fixed and equal to 2%. MC simulations were performed at various temperatures and screen shots were taken after 20 000 MC steps per atom.

	0.05% at. Vac	0.1% at. Vac	0.2% at. Vac	0.5% at. Vac
300 K
800 K
1600 K
2500 K

Table 3 shows structures of W–Re alloys with vacancy concentration of 0.1% obtained from MC simulations performed for temperatures of 300 K, 800 K, 1600 K and 2500 K for four different concentrations of Re atoms: 1%, 2%, 5% and 10% at.Re. Similarly to the case of W-2%Re alloys with various concentrations of vacancies, clustering of vacancies for each concentration of Re atoms is only observed at 800 K. At 2500 K all the vacancies and Re atoms are fully dissolved. For Re concentrations of 5% and 10%, vacancies do not aggregate at 300 K and 1600 K. In W–Re alloys containing 1% and 2% Re, sponge-like Re-vacancy clusters form at 300 K and voids form at 1600 K. Stability of Re-vacancy configurations at 800 K also depends on the Re/vacancy ratio. At low Re concentration close to 1% the Re/vacancy ratio is small enough to favour the formation of a void surrounded by Re atoms. At higher Re concentrations and higher values of the Re/vacancy ratio, sponge-like Re-vacancy clusters are more stable.

Table 3. Results of Monte Carlo simulations shown as functions of Re concentration and temperature. Concentration of vacancies is fixed and equal to 0.1%. MC simulations were performed at fixed temperatures and screen shots were taken after 20 000 MC steps per atom.

	1% at. Re	2% at. Re	5% at. Re	10% at. Re
300 K
800 K
1600 K
2500 K