Compositional Glass: A State with Inherent Chemical Disorder, Exemplified by Ti-rich Ni3(Al,Ti)1 D024 Phase

Abstract

A compositional glass is a state with an unavoidable disorder in chemical compositions on each site, characterized by frustration and freezing of the compositional degrees of freedom at low temperature. From this state a full atomic long-range order is unachievable by a reasonable thermodynamic treatment. There is a similarity between a spin glass (a magnetic state with disorder in spin orientations) and a compositional glass (with disorder in site occupations by chemical elements): both have frustrated ground states and a frozen disorder at low temperatures T < T_f (here T_f is called the freezing temperature). While it is possible to perform a ground-state search in a compositional glass, the resulting set of the fully ordered structures does not adequately represent the real solid with an inherent atomic disorder. Compositional glasses constitute a class of materials, which is insufficiently understood, but is of high industrial importance. Some of the phases in the precipitated alloys (including steels, high-entropy alloys, and superalloys) might be compositional glasses, and their better understanding would facilitate materials design. Due to their strength at high operating temperatures, superalloys are used in combustion engines and particularly in jet turbine engines. Precipitation strengthening of nickel superalloys is an area of active research. Local phase transformations inside Ni₃Al-based precipitates are of particular interest due to their impact on creep strength. In the Ni₃(Al_1−xTi_x)₁ ternary system, the competing phases are Ni₃Al-type L1₂ (γʹ) and Ni₃Ti-type D0₂₄ (η), while D0₁₉ (χ) is higher in energy. These three phases differ by the stacking of atomic layers: locally, the last two look like the internal and external stacking faults in L1₂. We compute enthalpies of disordered and ordered Ni₃(Al_1−xTi_x)₁ ternary structures, examine phase stability, investigate the ground states and competing structures, and predict that the Ti-rich Ni₃(Al_1−xTi_x)₁ D0₂₄ phase is a compositional glass with the atomic disorder on the Al/Ti sublattice. To resolve apparent contradictions among the previous experiments and to confirm our prediction, we perform X-ray diffraction and scanning electron microscopy analysis of the cast Ni₃(Ti_0.917Al_0.083)₁ sample. Our measurements appear to confirm the ab initio computed results. Our results elucidate properties of compositional glasses and provide a better understanding of precipitation strengthening mechanisms in Ni superalloys.

1. Introduction

The efficiency of heat engines increases with temperature T of a heater if an environment is used as a cooler. Thus, higher-efficiency engines require appropriate materials suitable for use at higher operating temperatures. Due to their strength at elevated T, superalloys are used for critical high-T applications in energy, automotive, and aerospace industries. In particular, nickel-based superalloys are used in jet engine turbine disks [1]. These alloys are strengthened by Ni₃Al γʹ precipitates [2], grain boundary strengtheners (e.g., carbon or boron) [3], and solid-solution strengtheners [4]: refractory elements (W, Re, Ta, Mo, Nb) [5] and rare earth elements (La, Ce, etc.) [6]. Local phase transformations inside Ni₃Al-based precipitates result in additional strengthening and improved creep of Ni superalloys [7,8,9,10,11].

Here we focus on the atomic ordering and phase stability in the Ni₃(Al_1−xTi_x)₁ ternary system. The competing phases in Ni₃(Al_1−xTi_x)₁ are L1₂ (γʹ)—the ground state (GS) of Ni₃Al, and D0₂₄ (η)—the ground state of Ni₃Ti. We find that D0₁₉ (χ) phase is higher in energy and less thermodynamically relevant. L1₂, D0₁₉, and D0₂₄ phases differ by the stacking of atomic layers, see Figure 1. The last two phases locally look like stacking faults in L1₂ [7].

In the Ni₃(Al_1−xTi_x)₁ ternary system the terminal binary L1₂ Ni₃Al and D0₂₄ Ni₃Ti structures differ. We investigate the phase stability of Ni₃(Al_1−xTi_x)₁ with a homogeneous atomic disorder on the (Al_1−xTi_x) sublattice, consider energy gain due to the atomic ordering, and construct the ground-state hull. We predict the existence of the fully ordered ground-state structures at intermediate compositions—in addition to the known binary L1₂ Ni₃Al and D0₂₄ Ni₃Ti structures. A subset of the predicted ground-state structures is shown in Figure 2. The computed energies are plotted in Figure 3.

Interestingly, the energy decrease due to atomic ordering in the Ti-rich D0₂₄ phase is very small (see Figure 3); hence, we do not expect development of a long-range order (LRO) on the Al/Ti sublattice in experimental samples at these compositions, while a weak short-range order (SRO) is expected due to the repulsion between the solute Al atoms. This phase is an example of a compositional glass, generic properties of which we discuss.

In addition to the structural phase transition between L1₂ and D0₂₄ phases, there is a magnetic transition in the Ni₃(Al_1−xTi_x)₁ ternary system. Dependence of atomic magnetic moments on composition is given in Figure 4.

The kink in the energy versus composition dependence (see Figure 3), present in all three phases, is caused by a rapid change of the electronic density of states (DOS) at the Fermi level E_F, which is in the pseudo-gap in the Ti-rich structures, see Figure 5.

This article is organized as follows. Methods and materials are described in Section 2. Results in Section 3 clarify phase stability and atomic ordering in the Ni₃(Al_1−xTi_x)₁ ternary system, while our experiment confirms our theoretical predictions. Discussion and summary are in Section 4 and Section 5, respectively.

2. Methods and Materials

2.1. Theory

Physically different systems can be described by similar mathematical equations. Hamiltonian formalisms are equivalent for a magnetic system composed of interacting particles with spins (moments with a quantized projection on the chosen axis) and for an alloy with a variable composition on each site, occupied by one of the available chemical elements. On an occupied site an atom can be displaced from the ideal crystallographic position [12]; a vacancy can be treated as another atomic type with zero nuclear charge. A magnetic quantum system can be described in terms of the spin variables s_i. For spins s = ½ (here we omit the reduced Plank constant ℏ), on each site i there are only 2 possible values of $s_i=\pm \frac{1}{2}$, corresponding to either up or down spin orientations. The same spin-like variables s_i can describe occupation of each site i in a binary alloy, which is composed of only 2 chemical elements (for example, Al and Ti). One can extend this formalism to describe either spin or atomic disorder on a particular sublattice, if all the other sublattices are fully ordered. One can use the spin-like variables with a larger amplitude and more projections, for example, {−1, 0, 1} for a spin 1 magnetic system or for a ternary alloy. Next, there is a one-to-one correspondence between descriptions of an alloy in terms of the spin-like (s_i) and occupational variables x_i(X), where X is a particular chemical element (e.g., Ti). The values of x_i(X) are between zero and one: x_i(Ti) = 1 if the site i is occupied by Ti and 0 otherwise. For a binary alloy, composed by Al and Ti only, at each site i there is a constraint x_i(Al) + x_i(Ti) = 1, due to which one can exclude one variable $x_i\left(\mathrm{Al}\right)=\left(1-x_i\right)$ and keep only x_i(Ti) ≡ x_i. The equation $s_i=(x_i-\frac{1}{2})$ connects the occupational and spin variables, in which $s_i=-\frac{1}{2}$ for Al and $+\frac{1}{2}$ for Ti.

Historically, the Ising model [13] and its generalizations [14] described spin ordering in magnetic systems, while the cluster expansion [15,16] described atomic ordering in multicomponent systems. Hamiltonian formalisms for both are mathematically equivalent. In both cases, such systems can be frustrated. A spin glass [17] is an inherently disordered state of a frustrated magnetic system, and in the compositional space we expect the existence of a similar state, which we call a compositional glass. As a formality, below are two definitions.

A spin glass is a magnetic state, in which magnetic moments have orientational disorder due to frustration, caused by a balance between attractive and repulsive interactions. A spin glass [17] is characterized by randomness in orientations of spins and freezing of spin dynamics below the freezing temperature T_f.

A compositional glass is a state with an inherent chemical disorder, characterized by a randomness in the local chemical compositions on a set of sites (each site is occupied by one of the constituting chemical elements), and freezing of disorder below temperature T_f, at which atomic diffusion stops. A compositional glass is an analogue of a spin glass: both systems are described by similar equations, have comparable properties of mathematical solutions, but differ by a physical interpretation of mathematical symbols.

An example of a compositional glass will be considered in Section 3.

2.2. Computations

Density functional theory (DFT) is used to address fully ordered and partially disordered periodic atomic structures [18]. Two DFT codes are used: the all-election Green’s function-based KKR-CPA [19,20] code MECCA [21] and the full-potential pseudo-potential VASP [22,23]. We use the same PBEsol exchange correlation functional [24] in both codes.

The Korringa–Kohn–Rostocker (KKR) approach [19,20] was combined with the coherent potential approximation (CPA) [25] to consider a homogeneous substitutional chemical disorder on the Al/Ti sublattice [21]. The truncated basis set with the maximal orbital angular momentum quantum number l_max = 3 included s, p, d, and f orbitals. We used the atomic spheres approximation (ASA) [26] with the periodic boundary corrections [27]. The ideal c/a ratio of the lattice constants was used for the close-packed hexagonal phases in KKR-CPA.

The Murnaghan equation of state (EoS) [28] was fitted to the E(V) data with at least 5 DFT points (with a step from 1% to 1.5% in the lattice constant, with an energy minimum near V₀) to find the equilibrium volume V₀ at each composition for the phases with a homogeneous disorder on the Al/Ti sublattice, see Figure 3. New DFT calculations were performed at the equilibrium volume to find atomic magnetic moments in Figure 4. We checked that the EoS energy at V₀ (shown in Figure 3) and the directly computed DFT energy E at V₀ agree within the DFT error bar at each composition.

The Thermodynamic Tool Kit (TTK) [29] was used to generate a set of periodic structures with various decorations of atoms on the Al/Ti sublattice in L1₂ and D0₂₄ phases. An iterative method described in Section 3.4 was used to add more ground state candidates into DFT consideration.

The VASP code [22,23] with PAW pseudopotentials [30] was used to compute energies of the fully ordered structures, which were fully relaxed at zero pressure. The plane-wave energy cutoff was increased to ENCUT = 650 eV. During relaxation we used the Gaussian smearing (ISMEAR = 0) with SIGMA = 0.043 eV, corresponding to k_BT at T = 500 K. For the fully relaxed structures we also used the tetrahedron method with Blöchl corrections (ISMEAR = −5). We used the Γ-centered Monkhorst–Pack k-mesh [31] with at least 50 points per inverse ångström. The spin-polarized calculations were performed for magnetic systems. For consistency, full-potential corrections (assumed to be linear versus composition) were applied to the KKR-CPA results using VASP energies at the terminal compositions.

2.3. Experimental Sample Preparation and Characterization

A 30 g ingot composed of 79.24 wt.% Ni, 19.75 wt.% Ti, and 1.01 wt.% Al was produced using vacuum induction melting (VIM) at the NASA Glenn Research Center Materials Processing Laboratory. The 30 g button was then cut, mounted in polyfast epoxy, and polished. The microstructural Ni₃(Ti,Al) sample was polished using SiC grit paper followed by 0.5 μm suspension and a final 0.05 μm colloidal silica finish. Scanning electron microscopy of the sample was performed using a Tescan MAIA3 at 15 kV of acceleration using a secondary electron detector. Chemical maps were performed using an Oxford Ultim Max Silicon Drift Detector and Aztec Software. Phase identification of the button sample was performed using both a Panalytical Empyrean X-ray diffractometer with a position sensitive detector (Galipix) in scanning 1D mode utilizing Mo-Kα radiation and Bruker D8 Discover (area detector) X-ray diffractometer utilizing Cu-Kα radiation. Ten separate SEM images were obtained, each containing a surface area of 0.02 mm [2]. The micrographs were then cropped and transformed into 8-bit images. They were then segmented using the adaptive threshold plug-in within the image processing software ImageJ [32]. The threshold plug-in segments the image using the gray-scale contrast between the darker γʹ precipitates and lighter D0₂₄ matrix. The software determines a gray-scale cut-off and a minimum size of a precipitate. Hence, any pixel that is determined “dark” enough to be considered a γʹ precipitate must also have a minimum number of connecting “dark” pixels for it to be considered a precipitate and included in the segmentation step. Otherwise, the plug-in will consider that pixel as noise and exclude it. After the segmentation, the segmented images were transformed into binary images and the segmented phases were analyzed to determine their area fractions.

3. Results

3.1. Relevant Phases

The L1₂ (cP4, Cu₃Au-type, cubic $Pm\overline{3}m$ space group no. 221), D0₁₉ (hP8, Ni₃Sn-type), and D0₂₄ (hP16, Ni₃Ti-type, hexagonal P6₃/mmc space group no. 194) atomic structures are shown in Figure 1. They differ by the stacking of 2-dimensional (2D) atomic layers along the cubic (111) direction in L1₂ or the hexagonal (0001) direction in D0₁₉ and D0₂₄ structures, as well as by c₁/a₁ ratio, where c₁ is the lattice constant along the stacking direction, divided by the number of layers, and a₁ is the interatomic distance between the nearest periodic Al/Ti sites within the 2D layers. The periodic stacking of the 2D layers is [ABC] in L1₂, [AB] in D0₁₉, and [ABAC] in D0₂₄. The Al/Ti sublattice is simple cubic (sc) in L1₂, hexagonal close-packed (hcp) in D0₁₉, and double hexagonal close-packed (dhcp) in D0₂₄, see Figure 1. In all three phases, the 3D Al/Ti sublattice is composed by the stacked 2D triangular lattices.

3.2. Ground-State Candidates

The known ground states at the terminal binary compositions are L1₂ Ni₃Al and D0₂₄ Ni₃Ti. The lowest-energy structures at three selected Ni₃(Al_1−xTi_x) ternary compositions are shown in Figure 2. The exhaustive set of L1₂-based ground states includes the three structures at x = 0, 0.25, and 0.5 (see Figure 3). In contrast, the D0₂₄-based ground states include many structures with well-spaced Al solute at 0.875 ≤ x ≤ 1; one of them at x = 0.875 is shown in Figure 2c. This Ni₃Al_1/8Ti_7/8 structure is hexagonal, with the maximally spaced Al solute; the Al sublattice forms a close-packed structure (hcp with AB stacking in Figure 2c); the competing structures differ by a stacking of the close-packed Al sublattice (other possibilities include [ABC] for fcc or [ABAC] for dhcp, see Figure 1). Due to the repulsion between Al atoms at a short distance, all D0₂₄-based ground states have a well-spaced Al solute and prefer mixing for the atomic ordering on the Al/Ti sublattice. However, at small Al concentrations at x(Ti)→1, there are many competing structures with well-spaced Al solute and similar energies; the energy differences between those are within the DFT error. In addition, the energy difference ΔE_d between the states with a homogeneous disorder on the Al/Ti sublattice and the ordered ground states with the well-spaced Al solute is small and constitutes only 2.37 meV/atom or 9.48 meV per Ni₃Al_1/8Ti_7/8 formula unit (f.u.) at x = 0.875; it reduces to 1.42 meV per Ni₃Al_1/12Ti_11/12 at x = 11/12; 2.85 meV per Ni₃Al_1/24Ti_23/24 at x = 23/24; and 0 meV/Ni₃Ti at x = 1. At x = 7/8, chemical entropy of the Al/Ti homogeneous atomic disorder is $S_d=k_B\left(x \ln \left(x\right)+\left(1-x\right)\ln \left(1-x\right)\right)=3.247·{10}^{-5}$ eV/K per Ni₃Al_1/8Ti_7/8 f.u. (here k_B is the Boltzmann constant) and is zero for the fully ordered structure; the expected ordering phase transition temperature is below ΔE_d/S_d = 292 K, while the experimental atomic diffusion is negligible below 500 K. Hence, due to the insufficient energy gain from forming the long-range order at 0.875 < x < 1 (see Figure 3), experimental Ti-rich D0₂₄-based Ni₃(Al_1−xTi_x) samples are expected to have only short-range order on the Al/Ti sublattice. This prediction makes it unnecessary to point at the fully ordered ground states, which differ from the actual state with an inherent atomic disorder at 0.875 < x < 1. We expect that Ni₃(Al_1−xTi_x) alloys at 0.875 < x < 1 will remain in D0₂₄ phase and have a disorder (no LRO) on the Al/Ti sublattice, while SRO [33] will be caused by a repulsion between Al solute atoms. An alloy with an inherent occupational disorder behaves as an analogue of a spin glass in occupational variables, with a corresponding similarity between magnetic (spin) [13] and occupational degrees of freedom. Such a state of matter is called a compositional glass.

3.3. Relative Energies of Disordered Structures

The computed energies E_dis(x) of the relevant phases of Ni₃(Al_1−xTi_x)₁ with a homogeneous atomic disorder on the Al/Ti sublattice are compared with the energies of the fully ordered structures in Figure 3. For the Al/Ti disorder, we find a mixing tendency (indicated by a positive curvature of the convex E_dis(x) curve) in L1₂ phase at all compositions x, while a mixing tendency in both D0₁₉ and D0₂₄ phases is present only at large Ti concentrations. Negative mixing energies (relative to the terminal Ni₃Al L1₂ and Ni₃Ti D0₂₄) indicate that Al and Ti atoms prefer to mix (and not to segregate) on the Al/Ti sublattice. If mixing with a homogeneous disorder is already energetically favorable, then further relaxation of compositional degrees of freedom with a formation of SRO or LRO on the Al/Ti sublattice is expected to lower energy. This leads to an expectation of existence of ordered structures (some of which are the ground states) with lower energies than those of disordered states. Indeed, Figure 3 shows the computed ground-state hull, which is below the E_dis(x) curves for both L1₂ and D0₂₄ phases. As discussed in Section 3.2 and Section 4, small differences ΔE_d indicate that the actual state is a compositional glass.

3.4. Energies of Fully Ordered Structures and the Ground-State Hull

Using the Thermodynamic Tool Kit (TTK) [34], for both D0₁₉ and D0₂₄ phases we generated a hierarchical set of ordered structures (smallest first) with atoms at the ideal lattice positions and used density functional theory (DFT) to compute energies of the fully relaxed structures. Next, energies and correlation functions for a set of structures [35] were used in the structural inversion to construct a cluster expansion (CE) [29]. The CE Hamiltonian was used to predict the CE energies of a very large number of structures and to find new ground state candidates [36]. Energies of those were computed directly in DFT. With energies and correlation functions of those additional structures, another CE was constructed and another search for additional ground states was performed. This iterative procedure was stopped (with 68 L1₂-based and 2650 D0₂₄-based structures) when no new lower-energy (within the DFT accuracy) ground state candidate structures were suggested by the CE and confirmed by DFT.

A CE Hamiltonian provides a polynomial energy versus composition dependence for the structures with a homogeneous disorder on the Al/Ti sublattice. A low-degree polynomial does not reproduce the kink near x = 0.915 on the E_dis(x) curve for D0₂₄ phase in Figure 3. Hence, any CE with a finite cutoff for the multi-body interactions (we considered m-body clusters with m ≤ 7) does not accurately reproduce the E_dis(x) curve for this phase, although a CE is useful during an iterative search for new GS candidate structures.

The suggested ground-state structures are shown in Figure 2. DFT energies of the fully ordered structures (including the ground states) are in Figure 3. Importantly, in D0₂₄ phase there are many GS candidates at 0.875 < x < 1, and at each composition there are many competing structures with similar energies (within the DFT accuracy), which differ by the longer-ranged atomic correlations at larger interatomic distances. In other words, at x→1 the D0₂₄-based ground states are the structures with a well-spaced Al solute, and each ground state is accompanied by multiple competing structures with similar energies, see Figure 3.

3.5. Magnetism

Ni₃Al and Al-rich Ni₃(Al_1−xTi_x) alloys are magnetic [37], with the ferromagnetic (FM) spin ordering of Ni magnetic moments M_Ni, while the induced atomic magnetic moments of Al and Ti are small (and oriented oppositely to Ni). Ni₃Ti is non-magnetic, with zero atomic magnetic moments. At larger x, atomic magnetic moments and the total magnetization in each phase of Ni₃(Al_1−xTi_x)₁ become smaller and disappear, see Figure 4. Although at small x D0₂₄ phase has higher moments M_Ni than L1₂, all D0₂₄-based ground states at x ≥ 0.875 are non-magnetic. Disappearance of M_Ni is a magnetic phase transition, which happens near x = 0.364; 0.447; and 0.67 in L1₂, D0₁₉, and D0₂₄ phases, respectively.

3.6. Electronic Structure

Considering the electronic density of states (DOS) n(E), we find that the L1₂-based GS at x=0 (Ni₃Al), 0.25, and 0.5 have a pseudo-gap above the Fermi energy E_F. In contrast, D0₂₄-based GS at x ≥ 0.875 (including Ni₃Ti at x = 1) have the pseudo-gap at or below E_F, see Figure 5. Considering n(E_F) versus composition, we find an electronic transition from the states with higher n(E_F) at x(Ti) < 0.875 to the states with low n(E_F) and the Fermi energy in the pseudo-gap at x ≥ 0.875. In practice, this electronic transition coincides with the structural transition between L1₂ (at smaller x) and D0₂₄ phases (at x ≥ 0.875).

The valence electron charge density in real space in the selected ground states is shown in Figure 6. Locally, the neighborhoods of each Al and Ti atom in Ni₄₈Al₂Ti₁₄ (Figure 6b) resemble those in L1₂ Ni₃Al (Figure 6a) and D0₂₄ Ni₃Ti (Figure 6c), respectively. This is also true for the other D0₂₄-based ground states at x ≥ 0.875, which differ from D0₂₄ Ni₃Ti by a substitution of a fraction of Ti atoms by Al solute, with Al-Al spacing distance not shorter than 7.1 Å in Ni₄₈Al₂Ti₁₄ (Figure 2c).

3.7. Comparison to the Previous Experiments

Fully ordered Ni₃(Al_1−xTi_x)₁ at T = 0 K should segregate between the L1₂-based Ni₃(Al_0.5Ti_0.5) and D0₂₄-based Ni₃(Al_0.125Ti_0.875) ground states at 0.5 ≤ x ≤ 0.875. However, from a small energy difference between ordered and disordered states, we expect that experimental Ni₃(Al_1−xTi_x)₁ samples will not form LRO on the (Al_1−xTi_x) sublattice. Ni₃(Al_1−xTi_x)₁ samples with disorder (only SRO) on the Al/Ti sublattice will be in D0₂₄ phase at x ≥ 0.915 and predominantly in L1₂ phase at x < 0.8, with a phase-segregated region between. Currently, there is no consensus among the previous experimental assessments of the phase boundaries of D0₂₄ and L1₂ phases in Ni₃(Al_1−xTi_x)₁ alloys, see Table 1. Due to disagreement among the previous experimental data [38,39,40,41,42,43,44,45], a new measurement is desirable.

3.8. Experiment

3.8.1. X-ray Diffraction

XRD analysis using the Cu-kα data of the cast Ni₃(Ti_0.917Al_0.083)₁ sample is revealed in Figure 7. The graph shows that all the peaks, except for 36.5° 2θ, match the expected peak positions for the D0₂₄ crystal structure associated with Ni₃Ti. The 36.5° 2θ is most likely associated with the Ni₃Al L1₂ phase. Quantification of the phase weight percent was not performed due to significant peak intensity discrepancies between the Cu-kα and Mo-kα scans. It was determined that this discrepancy was due to poor sampling statistics associated with the large grains found in the sample. Still, from the XRD analysis it is clear that this sample formed primarily the D0₂₄ crystal structure, while the scanning electron microscopy (SEM) analysis was performed to validate the 79.24 wt.% Ni, 19.75 wt.% Ti, and 1.01 wt.% Al composition, corresponding to 0.7500 at.% Ni, 0.2292 at.% Ti, and 0.0208 at.% Al in Ni₃(Ti_0.9168Al_0.0832)₁ at x = 0.9168.

3.8.2. Scanning Electron Microscopy

To validate the XRD results, scanning electron microscopy of the cast ingot was performed. A SEM image of the sample is shown in Figure 8a and the corresponding chemical maps of the area are revealed in Figure 8b. The chemical maps agree with the XRD result in that the majority of the cast Ni₃(Ti, Al) utilizes the D0₂₄ crystal structure. There are small thin areas of material that appear to be composed of Ni₃Al-type L1₂ phase due to the increase in Al content and the corresponding L1₂ peaks found in the XRD analysis. The Al chemical map also reveals that the Al content is not uniform in the D0₂₄ phase in the as-cast state where areas that surround the L1₂ phase are richer in Al. Figure 8 also shows the presence of smaller dark phases which are most likely Al- and Ti- rich oxides. Multiple SEM micrographs were taken of the Ni₃(Ti, Al) sample in the as-cast state to quantify the area fractions for the two present phases—D0₂₄ and L1₂. Since the two phases appear in different contrast when imaged with a secondary detector, they can be separated, and their corresponding area fractions quantified. From this analysis, the L1₂ phase corresponded to only 0.86% ± 0.12% of the total area, while the rest was composed of the D0₂₄ phase and oxides. This result appears to confirm the calculations performed on this ternary system.

4. Discussion

Compositional glasses constitute a class of materials with inherent chemical disorder. If ordering enthalpy is small compared to diffusion activation energy, then atomic LRO on each sublattice is unachievable by a reasonable thermodynamic treatment. At low temperatures atomic diffusion becomes negligible and atomic order (or disorder) is frozen. Thus, a compositional glass does not transform to a fully ordered state [46] upon cooling.

Ground-state search algorithms [36] are applicable to both conventional crystals and compositional glasses. For thermodynamic predictions it is necessary to consider not only the ground states, but also competing structures [35], which might be ordered or partially disordered. Ordering enthalpy ΔE_d is defined as the enthalpy difference between the state with a homogeneous disorder on a particular sublattice and the ground-state hull at the same composition. Diffusion activation energy E_A can be either computed from the first principles [47] or assessed from experimental data. According to the Arrhenius equation [48], diffusion rate D is proportional to exp(−E_A/k_BT); it becomes negligible below the freezing temperature T_f if the activation energy E_A is much higher than k_BT_f, where k_B is the Boltzmann constant. In Ni superalloys T_f > 500 K and k_BT_f > 43 meV, while ΔE_d < 10 meV in the Ti-rich Ni₃(Al_1−xTi_x)₁ D0₂₄ phase: ΔE_d(x) changes from 9.48 meV per Ni₃Al_1/8Ti_7/8 at x = 0.875 to zero at x = 1. Thus, ΔE_d < k_BT_f ≪ E_A at any x ≥ 0.875, and this phase is a compositional glass.

This example illustrates that in multi-phase materials (such as composites or precipitated alloys, superalloys, and steels) some of the phases can be compositional glasses; a better understanding of their fundamental properties is beneficial for materials design.

5. Summary

Compositional glass is a solid state with an inherent atomic disorder. An example of a compositional glass is Ti-rich Ni₃(Al_1−xTi_x)₁ D0₂₄ phase, which is relevant to precipitation strengthening in Ni superalloys—multi-phase materials of high industrial importance.

We analyzed the phase stability and atomic order in Ni₃(Al_1−xTi_x)₁ ternary system and identified the ground states and competing structures. We found multiple D0₂₄-based ground state candidates with a spaced Al solute at x ≥ 0.875 and confirmed 3 L1₂-based GS at x = 0, 0.25, and 0.5. We considered electronic, magnetic, and structural phase transitions. From the small enthalpy difference between the states with order and homogeneous disorder on the Al/Ti sublattice we anticipated that experimental samples at x ≥ 0.875 would not develop LRO on this sublattice. We predicted that Ti-rich Ni₃(Al_1−xTi_x)₁ alloy at x > 0.915 should be a compositional glass in D0₂₄ phase. Considering the previous experiments, we found no consensus in the literature regarding the boundaries of L1₂ and D0₂₄ phases. To resolve this contradiction, we conducted a new experiment at x = 0.9168, which appears to confirm our first-principles predictions.

Author Contributions

N.A.Z. provided the TTK code and advising to E.N.B., performed all DFT calculations, and wrote the manuscript. E.N.B. used the TTK code, performed data processing, and iteratively generated two sets of structures for L1₂ and D0₂₄ phases. T.M.S. performed experiment. J.W.L. participated in funding acquisition. All the authors participated in discussion and editing. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data is available from the authors upon a reasonable request.

Acknowledgments

We thank Smirnov, A.V. and Johnson, D.D. for the KKR-CPA code, Noebe, R. for the experimental sample creation, Leary, A. and Rogers, R. for XRD help, Mendelev, M.I. and Scalettar, R.T. for discussion.

Conflicts of Interest

The authors declare no conflict of interest.

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Figures and Table

Figure 1. Atomic structures of L12 (γʹ), D019 (χ), and D024 (η) phases, shown for the fully relaxed Ni3Ti. The upper panel shows cubic (111) and hexagonal (0001) projections. The lower panel compares stacking of the 2D layers. Ti (Al) atoms are dark blue; Ni atoms are small yellow.

Figure 2. The ground-state candidates: L12-based orthorhombic structures at (a) x = 1/4 and (b) x = 1/2; (c) D024-based hexagonal structure at x=7/8. Ti is light blue; Al is red; Ni is small yellow. There are multiple structures with well-spaced Al solute on the GS hull at x > 0.875, see text.

Figure 3. Equilibrium volumes V0 [Å3/atom] and formation energies E [eV/atom] of Ni3(Al1−xTix)1 fully ordered structures (points) and phases with a homogeneous disorder on the Al/Ti sublattice (lines), based on L12 (red), D024 (black), and D019 (green) lattices. The ground-state (GS) hull is shown by the blue line, with marked L12-based (red squares) and terminal D024-based (black circles at x = 0.875 and 1) ground states. There are multiple GS candidates with spaced Al solute at 0.875 < x < 1.

Figure 4. Ni atomic magnetic moment (Bohr magnetons, μB/atom) versus Ti concentration x in three phases of Ni3(Al1−xTix) alloys with a homogeneous atomic disorder on the Al/Ti sublattice. Lines are 6th degree x(MNi) polynomials with even powers, fitted to DFT data points. Inset: 0.0005 e−/Å3 iso-surfaces of electronic spin density in cubic L12 Ni3Al unit cell with Ni (grey) and Al (black) atoms.

Figure 5. Electronic density of states (states/eV∙spin∙atom) of magnetic Ni3Al (a) and non-magnetic Ni3Ti (b) in L12 (red) and D024 (black) phases is compared to that of the L12-based GS at x = 1/4 and 1/2 (c) and D024-based GS at x = 7/8, 11/12, and 23/24 (d).

Figure 6. 0.05 e−/Å3 iso-surfaces of valence electron density in (a) cubic L12 Ni3Al at x=0, (b) hexagonal Ni48Al2Ti14 (see Figure 2c) at x = 7/8, and (c) D024 Ni3Ti at x=1, with Ni (grey), Al (red), and Ti (blue) atoms. The upper panel shows cubic 111 (a) and hexagonal 0001 (b,c) projections.

Figure 7. X-ray diffraction peaks of the of 79.24 wt.% Ni, 19.75 wt.% Ti, and 1.01 wt.% Al ingot suggesting the formation of the D024 Ni3Ti phase. No evidence of LRO on the Al/Ti sublattice.

Figure 8. (a) SEM micrograph of the cast 79.24 wt.% Ni, 19.75 wt.% Ti, and 1.01 wt.% Al ingot and associated (b) chemical maps.

Table 1

Phase boundaries of L1₂ and D0₂₄ phases in Ni₃(Al_1−xTi_x)₁ from our theory, our experiment, and literature. Here x (at.%Ti) is titanium atomic percent on the Al/Ti sublattice.

Temperature		xmax(L12)	xmin(D024)	Year	Ref.
Order	0 K	50	87.5	This	Theory
Disorder	∞	≈80	91.5	This	Theory
RT	300 K		≤91.68	This	Expt.
900 °C	1173 K	70	82	2016	[39]
900 °C	1173 K	67	83	2001	[40]
800 °C	1073 K	73	88	1999	[41]
1200 °C	1473 K	65	83	1999	[41]
850 °C	1123 K	64	96	1993	[42]
1250 °C	1523 K	62	90	1993	[42]
800 °C	1073 K	67	100	2005	[43]
800 °C	1073 K	63	100	1991	[44]
800 °C	1073 K	47	95	1965	[45]

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