# Elasticity of Phases in Fe-Al-Ti Superalloys: Impact of Atomic Order and Anti-Phase Boundaries

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## Abstract

**:**

_{71}Al

_{22}Ti

_{7}, we use transmission electron microscopy (TEM) to detect their two-phase superalloy nano-structure (consisting of cuboids embedded into a matrix). The chemical composition of both phases, Fe

_{66.2}Al

_{23.3}Ti

_{10.5}for cuboids and Fe

_{81}Al

_{19}(with about 1% or less of Ti) for the matrix, was determined from an Energy-Dispersive X-ray Spectroscopy (EDS) analysis. The phase of cuboids is found to be a rather strongly off-stoichiometric (Fe-rich and Ti-poor) variant of Heusler Fe

_{2}TiAl intermetallic compound with the L2

_{1}structure. The phase of the matrix is a solid solution of Al atoms in a ferromagnetic body-centered cubic (bcc) Fe. Quantum-mechanical calculations were employed to obtain an insight into elastic properties of the two phases. Three distributions of chemical species were simulated for the phase of cuboids (A2, B2 and L2

_{1}) in order to determine a sublattice preference of the excess Fe atoms. The lowest formation energy was obtained when the excess Fe atoms form a solid solution with the Ti atoms at the Ti-sublattice within the Heusler L2

_{1}phase (L2

_{1}variant). Similarly, three configurations of Al atoms in the phase of the matrix with different level of order (A2, B2 and D0

_{3}) were simulated. The computed formation energy is the lowest when all the 1st and 2nd nearest-neighbor Al-Al pairs are eliminated (the D0

_{3}variant). Next, the elastic tensors of all phases were calculated. The maximum Young’s modulus is found to increase with increasing chemical order. Further we simulated an anti-phase boundary (APB) in the L2

_{1}phase of cuboids and observed an elastic softening (as another effect of the APB, we also predict a significant increase of the total magnetic moment by 140% when compared with the APB-free material). Finally, to validate these predicted trends, a nano-scale dynamical mechanical analysis (nanoDMA) was used to probe elasticity of phases. Consistent with the prediction, the cuboids were found stiffer.

## 1. Introduction

_{2}TiAl which crystallizes in the Heusler L2

_{1}-structure. Material properties of stoichiometric Fe

_{2}TiAl were intensively studied by quantum-mechanical calculations including thermodynamic and magnetic properties [22,23], electronic properties [24,25], and elastic properties under ambient conditions [26] or under hydrostatic pressures [23]. The theoretical predictions related to the level of spin-polarization were confirmed by experiments by Kourov and co-workers [27]. Interestingly, there is a long-lasting discrepancy related to the low-temperature magnetic moment of Fe

_{2}TiAl which was theoretically predicted nearly an order of magnitude higher than the experimental value. Density functional theory (DFT) calculations resulted in 0.9 μ

_{B}per 4-atom formula unit (abbreviated as f.u.) in Refs. [24,28] in contrast to experimental values, 0.1 μ

_{B}/f.u. reported in Ref. [25] or 0.11 μ

_{B}/f.u. in Ref. [29] for T = 4.2 K. Our recent study [30], which was focused on this discrepancy, indicates that off-stoichiometric Fe-rich and Ti-poor states of Fe

_{2}TiAl, which were not considered in previous theoretical studies, play a very important role. Our calculations of the off-stoichiometric Fe

_{2}TiAl predicted low magnetic moments comparable with the experimental values.

_{x}Al

_{1−x}(0.4 < x < 0.75) alloys around the equiatomic stoichiometry was computed in Ref. [34]. In particular, the disordered phases were studied by the coherent potential approximation (CPA) and the intermetallic compounds by the tight-binding linear muffin-tin orbital (TB-LMTO) method. Large local magnetic moments were found in the case of transition metal antisite defect in FeAl when studying magnetism in Fe-Al (in agreement with the experimental findings). In another theoretical study [35], a series of quantum-mechanical calculations for a large set of different Fe-Al compositions and local atomic arrangements was performed to explain the anomalous volume-composition dependence in Fe-Al alloys. It was found that the spin-polarized calculations of Fe-rich compounds resulted in an anomalous lattice-constant behavior in qualitative agreement with experiments. However, the nonmagnetic and fixed-spin-moment calculations of Fe-Al states provided only linear trends without any anomaly. These studies clearly identified the change in magnetism of iron atoms caused by an increasing number of Al atoms in the first coordination sphere of the Fe atoms as the decisive driving force for the anomalous behavior. In a further study, Fähnle et al. [36] applied a cluster-expansion method to predict the phase diagram for the system Ni-Fe-Al including binary Fe-Al states.

## 2. Materials and Methods

_{2}TiAl with the Heusler L2

_{1}structure. The stoichiometric Fe

_{2}TiAl is chosen as an example so as to show crystallographic relations without considering any disorder (which is important for the studied phases). All atomic positions and supercell shape and volume were relaxed. Atomic positions were relaxed so as to reduce the residual forces acting upon atoms under 0.001 eV/Å. All magnetic states were initially ferromagnetic but the magnetic degrees of freedom were also allowed to change when searching for minimum-energy states (but limited to either collinear or anti-collinear arrangement of local magnetic moments of atoms).

## 3. Results and Discussion

#### 3.1. Transmission Electron Microscopy of Fe-Al-Ti Phases

_{71}Al

_{22}Ti

_{7}revealed a two-phase superalloy microstructure (see Figure 2a). The subsequent EDS study (Figure 2b–d) and line-scans (Figure 2e,f) provided the chemical composition of both phases. The chemical composition agreed on average with the published thermodymanical assessment of the Fe-Al-Ti system [16]. In this assessment the composition Fe

_{71}Al

_{22}Ti

_{7}is located in a two-phase region within the ternary Fe-Al-Ti phase diagram. The phase Fe

_{66.2}Al

_{23.3}Ti

_{10.5}of cuboid precipitates is Fe-rich and the Ti-poor variant of the Fe

_{2}TiAl Heusler L2

_{1}-structure. The composition of the matrix was estimated to be Fe

_{80}Al

_{19}Ti

_{1}.

#### 3.2. Theoretical Calculations of Thermodynamic and Elastic Properties

_{62.5}Al

_{25}Ti

_{12.5}(20 atoms of Fe, 8 atoms of Al and 4 atoms of Ti) as an approximant of the above-discussed chemical composition (Fe

_{66.2}Al

_{23.3}Ti

_{10.5}). While we know the overall composition, which is different from the stoichiometric Fe

_{2}TiAl Heusler compound, we do not know where the additional Fe atoms are located. Therefore, we have considered three different supercells which differ in the distribution of atoms. As the first polymorph we consider a completely disordered solid solution (A2). It is modeled using the general special quasi-random structure (SQS) concept [47] and generated by the USPEX code [48,49,50] (see Figure 3a). The second variant consists of an ordered sublattice of regular Fe atoms in the Heusler L2

_{1}structure (see yellow dashed lines in Figure 3b). However, Al, Ti as well as excess Fe atoms form a solid-solution (modeled by another SQS) on the Al and Ti sublattices of the L2

_{1}structure (the B2 case). The third polymorph consists of regular Fe atoms and Al atoms that occupy their corresponding sublattices within the L2

_{1}structure (see yellow and blue dashed lines in Figure 3c). The excess Fe atoms and Ti atoms form a random solid solution (an SQS) at the Ti sublattice (the L2

_{1}variant). Let us remind that the three above-described variants (which we call A2, B2 and L2

_{1}) have the same composition Fe

_{62.5}Al

_{25}Ti

_{12.5}and, for example, the L2

_{1}case is thus an off-stoichiometric variant of the stoichiometric Heusler compound with the L2

_{1}structure.

_{x}Al

_{y}Ti

_{z}) = (E(Fe

_{x}Al

_{y}Ti

_{z}) −$x\xb7{\mu}^{\mathrm{Fe}}-y\xb7{\mu}^{\mathrm{Al}}-z\xb7{\mu}^{\mathrm{Ti}}$)/($x+y+z$) where $x,y,z$ are the numbers of Fe, Al and Ti atoms in the supercells and μ’s are their chemical potentials, i.e., energies of elemental bcc ferromagnetic Fe, non-magnetic fcc Al and non-magnetic hcp Ti. The formation energies are −0.183 eV/atom (Figure 3a, A2), −0.272 eV/atom (Figure 3b, B2) and −0.318 eV/atom (Figure 3c, L2

_{1}). The L2

_{1}polymorph (Figure 3c) is thus thermodynamically the most stable one. The other two, A2 and B2, can be considered as models for non-equilibrium or high-temperature states. Considering our computational supercells as models for materials with different levels of disorder, we also evaluate the ideal configurational entropy ${S}^{\mathrm{conf}}$. As different polymorphs have different numbers of ordered and disordered sublattices (with different number of lattice sites), we use a generalized formula (see, e.g., Ref. [53]) derived for the sublattice model [54] ${S}^{\mathrm{conf}}=-R{\sum}_{\alpha}{a}_{\alpha}{\sum}_{i}{f}_{i}^{\alpha}ln{f}_{i}^{\alpha}$ where R is the universal gas constant, i runs over different chemical species, $\alpha $ over different sublattices, ${a}_{\alpha}$ is the ratio of lattice sites of a sublattice $\alpha $ with respect to the total number of all lattice sites and ${f}_{i}^{\alpha}$ is the concentration of a chemical species i on a sublattice $\alpha $. In our case all studied variants have only one disordered “sublattice” which contains either all atomic sites (the A2 variant), a half of them (the B2 variant) or one quarter of all lattice sites (the L2

_{1}variant). The ideal configurational entropy is therefore ${S}^{\mathrm{conf}}=-Ra{\sum}_{i}{f}_{i}ln{f}_{i}$ with the a coefficient equal to 1, 1/2 and 1/4 for the A2, B2 and L2

_{1}variant, respectively, and ${f}_{i}$ being the concentrations of chemical species on the disordered (sub-)lattice. The configurational entropy of the three studied variants is (in units of R) equal to 0.90026, 0.51986 and 0.17329 for the A2, B2 and L2

_{1}variant, respectively. The more ordered variants are predicted to have lower formation energies but they have lower configurational entropies. Consequently, depending on the actual values, order-to-disorder transitions are possible at elevated temperatures. In order to address them we approximately evaluate the free energies of formation ${F}_{\mathrm{f}}$ for each variant using the formula ${F}_{\mathrm{f}}={E}_{\mathrm{f}}-TS$, where we put the entropy term S equal to ${S}^{\mathrm{conf}}$ discussed above. A L2

_{1}-to-B2 transition is predicted at 1267 °C while the other transition temperatures (L2

_{1}-to-A2 and B2-to-A2) are much higher than the considered temperature range, and thus the ordered phase is certainly stable here. A direct comparison of the predicted and experimental L2

_{1}-to-B2 transition temperature is, unfortunately, not possible because (i) the Fe-Al-Ti phase diagram [16] for 1200 °C has no two-phase region for compositions close to the overall composition of our samples (Fe

_{71}Al

_{22}Ti

_{7}) and (ii) it is uncertain whether phases with off-stoichiometric L2

_{1}structure exist at these temperatures for compositions close to that which we simulate for the cuboids (the competing orderings are, in fact, the simulated ones, i.e., the B2 and A2, see Ref. [16]). Regarding magnetic and structural properties of the studied variants, their magnetic moments (in μ

_{B}per 32 atoms) are 37.35 (A2), 14.88 (B2) and 9.94 (L2

_{1}) and their volumes (in Å

^{3}per atom) are 12.59 (A2), 11.84 (B2) and 11.78 (L2

_{1}). The lower volumes roughly correspond to lower magnetic moments.

_{1}polymorph of the phase of cuboids (Figure 3c).

_{1}phase but the other atoms are randomly distributed over the remaining sublattices (see Figure 3b, B2). Finally, ${C}_{11}$ = 303 GPa, ${C}_{12}$ = 135 GPa and ${C}_{44}$ = 136 GPa for the SQS where the sublattices of regular Fe and Al atoms are ordered and defect-free while the excess Fe atoms and Ti atoms form a solid solution on the original Ti sublattice of the stoichiometric Heusler structure (see Figure 3c, the L2

_{1}case). The elastic properties were then visualized in the form of directional dependences of the Young’s modulus in the lower row of Figure 3. As can be seen, the overall stiffness grows with increasing chemical ordering (from Figure 3a to Figure 3c) while the elastic anisotropy (expressed by the Zener anistropy ratio 2${C}_{44}$/(${C}_{11}$–${C}_{12}$)) is 4.04 (the A2 case), 1.72 (B2) and 1.62 (L2

_{1}), respectively.

_{81.25}Al

_{18.75}(26 atoms of Fe and 6 atoms of Al) as approximants for the above-discussed chemical composition (Fe

_{80}Al

_{19}Ti

_{1}, the minor amount, 1%, of Ti is neglected). The first polymorph corresponds to a distribution where all atoms form a complete disordered solid solution (A2) modeled by a general special quasi-random structure, SQS (Figure 5a). The second variant is a B2-type SQS with the Al atoms on one of the two sublattices only, i.e., there are no 1st Al-Al NN pairs, Figure 5b, B2). The third polymorph is a D0

_{3}-type SQS with Al atoms on only one of the three sublattices, i.e., there are no 1st and 2nd Al-Al nearest-neighbor pairs, Figure 5c, D0

_{3}). We note that all the three variants of the phase of the matrix have the same concentration of Al (18.75 at.%) and the B2 and D0

_{3}cases are thus off-stoichiometric.

_{3}). The polymorph with the least disordered distribution (Figure 5c) and Al atoms further apart is thus thermodynamically the most stable in line with the discussed Al-Al repulsion as well as with experimental data obtained for the Fe-Al binary system (this tendency is behind the formation of the Fe

_{3}Al intermetallics with the D0

_{3}structure). The other two structures (A2 and B2) can be considered as models for non-equilibrium or high-temperature states. Similarly as in the case of the cuboid phase, we can evaluate the ideal configurational entropy for the three variants of the matrix phase. Their configurational entropies (in units of R) are equal to 0.48258, 0.33078 and 0.14058 for the A2, B2 and D0

_{3}variant, respectively.

_{3}variant (which has the lowest formation energy) at non-zero temperatures. The lowest formation energy obtained for the D0

_{3}variant is understandable as it reflects the nature of the Fe-Al system where the Fe

_{3}Al intermetallics with the D0

_{3}structure exists for 25 at.% of Al. The reasons for this behavior include the Al-Al repulsion (see above). Regarding magnetic and structural properties of the studied variants, their magnetic moments (in μ

_{B}per 32 atoms) are 56.57 (A2), 56.31 (B2) and 55.49 (D0

_{3}) and their volumes (in Å

^{3}per atom) are 11.84 (A2), 11.57 (B2) and 11.76 (D0

_{3}). The changes due to different atomic configurations are smaller than those in the variants of the phase of cuboids.

_{3}SQS without the 1st and 2nd Al-Al NN pairs (see Figure 5c). These results agree quite well with (close to T = 0 K) experimental data [62]: ${C}_{11}$ = 188 GPa, ${C}_{12}$ = 126 GPa and ${C}_{44}$ = 130 GPa. The elastic properties were then again visualized by the directional dependences of the Young’s modulus in the lower row of Figure 5. The maximum value of the Young’s modulus (in the 〈111〉 family of directions) grows with increasing ordering (from the A2, Figure 5a, to D0

_{3}, Figure 5c) while the elastic anisotropy (the Zener anistropy ratio) increases from 3.49 to 5.06 and eventually to 5.65.

#### 3.3. Nano-Scale Dynamical Mechanical Analysis of Fe-Al-Ti Phases

_{71}Al

_{22}Ti

_{7}solid solution partly orders upon cooling into the B2 phase (900–950 °C). It may then (below 800 °C) further decompose into a two-phase A2 + L2

_{1}composite material. Unfortunately, our experimental sample characterization techniques do not allow to determine the actual level of order in the two co-existing phases. Our quantum-mechanical calculations, which simulated different atomic arrangements with various levels of ordering in each of the constituting phase, were, in fact, performed to compensate (at least partly) for this lack of experimental data.

_{3}Al with the D0

_{3}lattice. This choice is motivated by the fact that the D0

_{3}lattice is a binary variant of the ternary L2

_{1}lattice, i.e., it has nearly identical atomic configurations as that shown in Figure 3c except for the off-stoichiometry.

_{1}parameter is equal to 2.866 Å). Our calculations show that the B2-TiAl and B2-FeTi have lattice parameters that are way too large to form coherent particles, 3.181 and 2.944 Å, respectively. In contrast, B2-FeAl matches nearly perfectly the cuboid matrix with a lattice parameter theoretically predicted to be 2.872 Å. Moreover, elemental iron with computed lattice parameter of 2.829 Å would also structurally match the cuboid matrix very well. It should be noted that the equilibrium higher-temperature Fe-Al-Ti phase diagrams do not contain any of the additional coherently-coexisting phases which we considered in our analysis. However, the studied phases can be off their equilibrium and, therefore, the sub-nanometer internal phases can exist due to a sluggish diffusion as frozen-in metastable phases (or even unstable ones stabilized by surrounding material).

## 4. Conclusions

_{71}Al

_{22}Ti

_{7}. After confirming a two-phase sub-micron superalloy nano-structure (cuboids of one phase coherently embedded into a matrix of another phase) by TEM, we have determined the chemical composition of both phases by EDS data as well as those from a previous thermodynamic assessment (see, e.g., Ref. [16]). The phase of cuboids is observed to be a strongly off-stoichiometric (Fe-rich and Ti-poor) Fe

_{66.2}Al

_{23.3}Ti

_{10.5}variant of the Heusler Fe

_{2}TiAl intermetallic compound with the L2

_{1}structure. The phase of the matrix is a solid solution of Al atoms in a ferromagnetic body-centered cubic (bcc) Fe with composition Fe

_{81}Al

_{19}. Not having detailed information about the atomic distributions in the studied nano-phases from experiments, we have computed and compared properties of three different types of distributions of Al atoms in the phase of the matrix and three different distributions of atoms in the phase of cuboids employing the special quasi-random structure (SQS) concept. In particular, three different distributions of chemical species (A2, B2 and L2

_{1}) were simulated for the phase of cuboids in order to determine the preference of the excess Fe atoms for the various sublattices. Our calculations predict the lowest formation energy (and so the highest termodynamic stability) for a solid solution of excess Fe atoms and Ti atoms at the Ti-sublattice within the Heusler L2

_{1}phase (the L2

_{1}phase). Similarly, the studied polymorphs of the phase of the matrix include a general randomly distributed variant (A2) as well as B2 and D0

_{3}cases when the 1st or the 1st and 2nd nearest-neighbor (NN) Al-Al pairs are eliminated (the B2 and D0

_{3}variants, respectively). In agreement with the reported repulsion of Al atoms and the experimental data for the Fe-Al binary system, the SQS without the 1st and 2nd NN Al-Al pairs (the D0

_{3}variant) turned out to have the lowest formation energy. It should be noted that our assessment of the thermodynamic stability of different atomic configurations was based only on the evaluation of formation energies of static lattices and partly also of the configurational entropy (an analysis of other effects/terms, such as those related to phonons or magnons, we suggest as a topic for future studies).

_{1}phase is found to be elastically stiffer than the phase of the matrix. Notably, the Young’s modulus of 220 GPa is more than twice higher than that of 98 GPa in the case of the 〈001〉 family of directions. On the other hand, the stiffness of both phases is nearly equal along the 〈111〉 family of directions.

_{1}phase of cuboids. The studied APB is shown to soften the phase of cuboids. A quantitative estimate would, however, require more information related to the APB-affected material. As another impact of APBs, we predict a significant increase of the magnetic moment.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{1}phase of cuboids (Figure 3c). The ab initio calculated elastic constants are

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**Figure 1.**A schematics showing relations between a 16-atom cubic-shape supercell of stoichiometric Fe

_{2}TiAl with the Heusler L2

_{1}structure (

**a**) and a 32-atom supercell (

**b**) of the type which we used for our study. The dashed lines in the part (

**b**) show the 45°-rotated 16-atom cell exhibited in part (

**a**).

**Figure 2.**A transmission electron microscope (TEM) micrograph (

**a**) showing the superalloy nano-structure in our Fe

_{71}Al

_{22}Ti

_{7}sample. The corresponding chemical composition (

**b**–

**d**) as detected by the Energy-Dispersive X-ray (EDS) spectroscopy in the same spot of our TEM lamellae. The analysis is accompanied by one of our line-scans (

**e**) together with the corresponding chemical profile (

**f**).

**Figure 3.**(

**Upper row**) Schematic visualization of the employed 32-atom computational supercells with the chemical composition Fe

_{62.5}Al

_{25}Ti

_{12.5}(20 atoms of Fe, 8 atoms of Al and 4 atoms of Ti). These structures are used for modeling the phase of cuboids for which the equilibrium composition was determined to be Fe

_{66.2}Al

_{23.3}Ti

_{10.5}. The supercells differ in the distribution of atoms (see text). Yellow and blue dashed lines indicate ordered sublattices as in stoichiometric Heusler structure. (

**Lower row**) Visualization of the anisotropic elastic properties for each atomic distribution in the form of directional dependences of the Young’s modulus (using the SC-EMA software [51,52], scema.mpie.de).

**Figure 4.**Results of quantum-mechanical calculations of interactions of Al atoms in a ferromagnetic body- centered-cubic (bcc) Fe. The calculated energies are shown as energy differences relative to the lowest obtained energy (when the Al atoms are located mutually as the 4th nearest neighbors (NN) pairs) which is set to be the energy zero (dash-dotted horizontal line).

**Figure 5.**Upper row: Schematic visualization of the 32-atom computational supercells with chemical composition Fe

_{82.25}Al

_{18.75}(26 atoms of Fe and 6 atoms of Al) used for modeling of the Fe-Al phase of the matrix. The supercells differ in the distribution of atoms (see the text for details). The left variant is a distribution when all atoms form a complete disordered solid solution modeled by a general special quasi-random structure, the A2 case (

**a**). The middle variant is a B2-type SQS with all the Al atoms on only one of the two sublattices, i.e., there are no 1st NN Al-Al pairs (

**b**). The right figure shows a D0

_{3}-type SQS with the Al atoms on only one of the three sublattices—there are no 1st and 2nd Al-Al NN pairs (

**c**). Lower row: Visualization of anisotropic elastic properties for each atomic distribution in the form of directional dependences of the Young’s modulus (visualized by the SC-EMA software [51,52]).

**Figure 6.**Nano-scale elastic properties of the studied Fe-Al-Ti superalloy detected by Hysitron nanoDMA (nano-scale dynamical mechanical analysis) detector.

**Figure 7.**Schematic visualization of the Fe-Al-Ti phase of cuboids modeled by a double 64-atom supercell without any anti-phase boundaries (

**a**) and with an anti-phase boundary (APB) (

**b**) together with the corresponding ab initio calculated local magnetic moments (

**c**,

**d**) and elastic properties (

**e**,

**f**) visualized in the form of a directional dependence of the Young’s modulus for both the APB-free and the APB-containing case, respectively. The magnitude of the local magnetic moments is visualized by the diameter of spheres representing the atoms (the scaling can be deduced from an example of one of the Fe atoms for which its local atomic magnetic moment of 2.5 μ

_{B}is listed). The change of the overall magnetic moment is very significant: the APB-containing state (see part (

**b**)) has the total magnetic moment higher by 140% than the APB-free one (see a similar effect in, e.g., Ref. [69]). Red dashed arrows in (

**a**,

**b**) indicate layering of Al atoms, which is altered due to the APB shift (full red arrow).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Friák, M.; Buršíková, V.; Pizúrová, N.; Pavlů, J.; Jirásková, Y.; Homola, V.; Miháliková, I.; Slávik, A.; Holec, D.; Všianská, M.;
et al. Elasticity of Phases in Fe-Al-Ti Superalloys: Impact of Atomic Order and Anti-Phase Boundaries. *Crystals* **2019**, *9*, 299.
https://doi.org/10.3390/cryst9060299

**AMA Style**

Friák M, Buršíková V, Pizúrová N, Pavlů J, Jirásková Y, Homola V, Miháliková I, Slávik A, Holec D, Všianská M,
et al. Elasticity of Phases in Fe-Al-Ti Superalloys: Impact of Atomic Order and Anti-Phase Boundaries. *Crystals*. 2019; 9(6):299.
https://doi.org/10.3390/cryst9060299

**Chicago/Turabian Style**

Friák, Martin, Vilma Buršíková, Naděžda Pizúrová, Jana Pavlů, Yvonna Jirásková, Vojtěch Homola, Ivana Miháliková, Anton Slávik, David Holec, Monika Všianská,
and et al. 2019. "Elasticity of Phases in Fe-Al-Ti Superalloys: Impact of Atomic Order and Anti-Phase Boundaries" *Crystals* 9, no. 6: 299.
https://doi.org/10.3390/cryst9060299