Designing order-disorder transformation in high-entropy ferritic steels

Order-disorder transformations hold an essential place in chemically complex high-entropy ferritic-steels (HEFSs) due to their critical technological application. The chemical inhomogeneity arising from mixing of multi-principal elements of varying chemistry can drive property altering changes at the atomic scale, in particular short-range order. Using density-functional theory based linear-response theory, we predict the effect of compositional tuning on the order-disorder transformation in ferritic steels -focusing on Cr-Ni-Al-Ti-Fe HEFSs. We show that Ti content in Cr-Ni-Al-Ti-Fe solid solutions can be tuned to modify short-range order that changes the order-disorder path from BCC-B2 (Ti atomic-fraction = 0) to BCC-B2-L21 (Ti atomic-faction $>$ 0) consistent with existing experiments. Our study suggests that tuning degree of SRO through compositional variation can be used as an effective means to optimize phase selection in technologically useful alloys.


Introduction
High-entropy alloys, including metals and ceramics with near-equiatomic compositions with four and more elements [1][2][3][4][5][6], continue to gain significant interest due to the unprecedented opportunity to explore large materials design space and uncover potentially remarkable compositions with outstanding structural and functional properties [4][5][6][7][8][9][10][11]. The design strategy in high-entropy alloys has been to use the concept of entropy to stabilize the single-phase solid-solution (e.g., face-centered cubic (FCC phase) or body-centered cubic (BCC phase)) [12] with an attempt to find specific electronic, thermodynamic, and microstructural properties [4,[13][14][15]. While the progress over the last decade towards first-generation high-entropy alloys is remarkable, the critical thermodynamic behavior of these alloys indicates that only a little is known on the effects of SRO [16] and the associated lattice deformations [17] on electronic and/or mechanical response [18].
High-entropy ferritic steels (HEFSs) are one important class of multi-principal element alloys due to their cost efficiency, low thermal expansion, and good thermal conductivity compared to Ni-based superalloys and austenitic steels [19][20][21]. Similar to conventional alloys, HEFSs show chemistry and temperature dependent ordering that may undergo one or more phase transitions into less ordered phases. Precipitation hardening due to presence of ordered phases in HEFSs gives excellent creep and oxidation behavior [22], which is analogous to the presence of ′ phase in Ni-based superalloys. Unlike L12 phases in austenitic steels, presence of ordered B2 or B2/L21 phases in body-centered cubic matrix [23,24] may provide similar mechanical effects [25]. Therefore, a detailed understanding of order-disorder transformations and precipitate formation along with compositional control in HEFSs can be of fundamental importance.
Here we present a systematic study on the effect of compositional tuning of Ti on order-disorder behavior in Cr-Ni-Al-Ti-Fe HEFSs using density-functional theory (DFT) methods in combination with configurational averaging [26,27]. The linear response theory was used for calculating short-range order in the disordered Cr-Ni-Al-Ti-Fe HEFSs [25]. We show that degree of SRO can be controlled using Ti content, which modifies order-disorder pathway from BCC-B2 to BCC-B2-L21 [28]. Our findings are in good agreement with recent observations of Wolf-Goodrich et al. [29], who report combinations of BCC/B2 and BCC/B2/L21 phases depending on Ti composition (at.%Ti) in Cr-Ni-Al-Ti-Fe HEFSs. The linear-response theory for SRO analyzed by concentration wave method was used to seed the fully self-consistent KKR-CPA calculation in the broken symmetry case, which, unlike Monte-Carlo methods [30], does not rely on fitted interactions. We also discussed the phase stability (formation energy) and electronic-structure origin of disordered and (partial) ordered phases for selected HEFS compositions. We found that SRO can be a key structural feature for optimizing phase selection and mechanical response.

Computational details:
Density-functional based linear-response theory: Phase stability and electronicstructure were addressed using an all-electron, Green's function based Korringa-Kohn-Rostoker (KKR) electronic-structure method [26]. The configurational averaging to tackle chemical disorder is handled using the coherent-potential approximation (CPA) [27], and the screened-CPA was used to address Friedel-type charge screening [31].
Valence electrons and shallow lying core electrons affected by alloying are addressed via a scalar-relativistic approximation (where spin-orbit terms only are ignored) [26,27,31], whereas deep lying core are address using the full Dirac solutions.
Electronic density of states (DOS) and Bloch-spectral function (BSF) were calculated within the atomic sphere approximation (ASA) with periodic boundary conditions. The interstitial electron contributions to Coulomb energy are incorporated using Voronoi polyhedra. The generalized gradient approximation to DFT exchange-correlation was included using the libXC opensource code [32]. Brillouin-zone integrations for selfconsistent charge iterations were performed using a Monkhorst-Pack k-point mesh [33].
Each BSF was calculated for 300 k-points along high-symmetry lines in an irreducible Brillouin zone.
Dominant pairs driving SRO are identified from the chemical pair-interchange energies (2) (k;T) (a thermodynamically averaged quantitynot a pair interaction), determined from an analytic second-variation of the DFT-based KKR-CPA grand potential with respect to concentrations fluctuations of at atomic site i and at atomic site j [28].
The chemical stability matrix (2) (k; T) reveals the unstable Fourier modes with ordering wavevector ko, or clustering if at ko=(000) at spinodal temperature (Tsp) [28]. Here, Tsp is the temperature where SRO diverges, i.e., −1 (ko;Tsp)=0, which signifies absolute instability in alloy and provides an estimate of order-disorder (ordering systems) or miscibility gap (in clustering systems).

Results and Discussion
Phase stability, structural, and magnetic property analysis for Cr-Ni-Al-Ti-Fe HEFSs, related to recent experimental work of Wolf-Goodrich et al. [29], are shown in Table 1.
We calculated the formation energy (Eform) of each HEFSs in BCC, FCC and HCP phases. The calculated formation energy in HCP phase for each alloy was a large positive number compared to BCC and FCC phases, therefore, not discussed. Our phase stability analysis The thermodynamic stability of multicomponent alloys is an important criterion to understand relative phase stability with respect to alloying element, which requires nontrivial sampling over infinitely large configurations in disorder phase [26,27]. Recently, Singh et al [34,37] extended the Hume-Rothery criteria [36,40]    . The expression suggests that change in lattice constant inversely related to change in volume or lattice constant as shown in Fig. 2a-c. We can see in Fig. 2a- [28,34,37], and the likely low-temperature long-range order [28], including its electronic origin.
In Fig. 3a-  and modes driving SRO that are manifested in SRO pairs in Fig. 3a Fig. 3a and Fig. 3c are the same, i.e., Ni-Al and Al-Ti pairs. This asymmetry atomic pairs in SRO and (2) pairs occur due to the conservation of the sum rule [28]. The Block-spectral function (BSF) and partial density of states (PDOS) for BCC Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 are shown in Fig. 4a,b. The BSF in Fig. 4a shows large disorder broadening (at)near Fermi energy (EFermi), where scale on right shows weak (black) to strong (red) disorder effect arising from mixing of different alloying species The possibility of secondary ordering phase arises due to presence of stronger ordering peak at H in Ni-Al pair and weaker ordering peak at P in Al-Ti pair. The SRO peaks at H+P are suggestive of L21 phase [25].
Indeed, the expectation of low-temperature ordering due to increased hybridization among alloying elements was also confirmed by the presence of strong SRO peaks at H-point (indicating B2-type mode) and H+P-point (indicating L21-type mode) for Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 HEFS in Fig. 4c. The maximal SRO peak at Hpoint in Fig. 4c shows Ni-Al dominated B2-type ordering. The strong Ni-Al peak at Hpoint is followed by Fe-Al and Ni-Ti SRO pairs. A fairly strong Fe-Al SRO can be attributed to the larger solubility of Fe than Cr at low-temperature in ordering phases.
Also, a weaker Cr-Ni peak in SRO at -point in Fig. 4c indicates the tendency of segregation, i.e., energetically Cr and Ni do not prefer same neighboring environment [28]. Note that a weak secondary Al-Ti peak at P-point as shown in inset of Fig. 4c is indicative of Ti enriched B2 phase. The presence of a weak ordering peak at P along with strong ordering peak at P is consistent with coexistant B2 and L21 phases as reported by Wolf-Goodrich et al. [29] in nearly same composition as Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 HEFS.
Unlike Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 (with strong B2 (H-point) ordering along with possible L21 ordering in Fig. 5a), the Cr0.20Ni0.10Al0.30Fe0.40 HEFS only shows possible B2 ordering with no-sign of L21 (in strong agreement with observations of Wolf-Goodrich et al. [29]). The direct energy calculation of low temperature ordering phases (B2 and L21) and comparing them with disorder phase (BCC) will allow us to establish the fact that predicted incipient long-range order phases may exist [28]. However, the determination sublattice occupation (on phase decomposition of disorder phase) is needed for energy calculation of ordering phases using DFT, which remains unknown. The concentration wave in Eq. (1) [42,44] Fig. 6a-b. The TDOS of Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 in Fig. 6a shows that both majority-spin and minority-spin channel have a pseudo-gap at the Fermi energy, indicative of increased energy stability [36]. The predicted Eform of -12.27 meVatom -1 (BCC), -47.49 meV-atom -1 (B2), and -65.99 meV-atom -1 (L21) in disorder and partially ordered phases also confirms our analysis. In Fig. 6b, the BCC phase shows pseudo-gap region both in up-spin and down-spin channel, which suggests strong stability. The low Eform of -21.42 meV-atom -1 also confirms our hypothesis. The TDOS of B2 and L21 phases are identical and show strong pseudo-gap region in up-spin channel, however, down-spin channel shows a peak structure just below the Fermilevel. Unlike TDOS of Cr0.20Ni0.10Al0.25Ti0.10Fe0.35 HEFS in Fig. 6a, the presence of large electronic density of states in Cr0.20Ni0.10Al0.30Fe0.40 HEFS in Fig. 6b at Fermi-level leads to weaker change in energy stability in B2 (-23.53 meV-atom -1 ) and L21 (-23.55 meV-atom -1 ) phases compared to disorder phase.

Conclusion
In summary, the density-functional theory based linear-response theory was used to directly calculate the short-range-order for all atomic pairs simultaneously relative to the homogeneously disordered BCC phase. We showed that the orderdisorder transformation, i.e., BCC-to-B2 and BCC-B2-L21, can be controlled by compositional tuning. The proposed hypothesis of SRO-controlled ordering transformation was exemplified in Cr-Ni-Al-Ti-Fe based ferritic-steels, and we show that the predicted ordering pathways are in good agreement with existing experiments. Our calculations also indicate the possibility of coexistence of ordering phases such as B2 and L21 below phase decomposition temperature. This study further emphasizes that SRO is important both from fundamental and application point of view as it is known to affect phase selection [16] and mechanical response [18]. Therefore, the tunability of SRO in multi-principal element alloys using purely chemistry provides unique insights for controlling phase transformation, which shows the usefulness of our theory guided design of next generation high-entropy ferritic steels.