Spin-driven ordering of Cr in an equiatomic fcc NiFeCrCo high entropy alloy (HEA) was predicted by first-principles calculations. Ordering of Cr is driven by the reduction in energy realized by surrounding anti-ferromagnetic Cr with ferromagnetic Ni, Fe, and Co in an alloyed L12 structure. The fully Cr-ordered alloyed L12 phase was predicted to have a magnetic moment that is 36% of that for the magnetically frustrated random solid solution. Three samples were synthesized by milling or casting/annealing. The cast/annealed sample was found to have a low temperature magnetic moment that is 44% of the moment in the milled sample, which is consistent with theoretical predictions for ordering. Scanning transmission electron microscopy measurements were performed and the presence of ordered nano-domains in cast/annealed samples throughout the equiatomic NiFeCrCo HEA was identified.
High entropy alloys (HEAs) are an exciting new class of materials and several systems have exhibited desirable combinations of properties not commonly exhibited by conventional alloys. HEAs are composed of at least four to five principal elements that are present in high concentrations of 5–35 at. %.1,2 Alloys with equiatomic ratios have the highest configurational entropy contribution to the free energy. Equiatomic NiFeCrCoMn, which is a fcc random solid solution (RSS), has a good combination of mechanical properties in hardness,3,4 tensile strength,4–6 ductility,4,6 and fracture resistance at cryogenic temperatures.5 Its balanced performance in strength and ductility was attributed to a low stacking fault energy.3 Non-equiatomic NiFeCrCoMn HEAs were also reported to remain a fcc RSS, which provides promising opportunities in the future for compositional engineering of the mechanical property combinations in this system.3,7,8
Structurally, HEAs form solid-solutions on simple lattices (e.g., fcc, bcc, and hcp). This is in contrast to other multicomponent alloys, such as bulk metallic glasses (BMG), which are structurally amorphous or conventional alloys with high solute concentrations, which tend to segregate into secondary or multiple phases. Whether an alloy forms as a BMG vs. HEA has been attributed to a balance between entropy, enthalpy, and relative atomic size.9 Many HEAs tend to have a near-zero enthalpy and are also composed of atomic species of similar metallic radius. The small enthalpy is believed to allow the entropy to stabilize these as a RSS at relatively low temperatures (i.e., room temperature).
Recent work by Otto et al. further demonstrated the delicate balance between enthalpy and entropy in stabilizing a single phase RSS.10 Starting with the established single phase HEA, equiatomic NiFeCrCoMn, the authors prepared five alloys that each substituted one host element at a time with a replacement element following the Hume-Rothery criteria. It was found that all five alloys became multiphase. The authors attributed this to the increased enthalpies between the substitutional and host elements and their results clearly demonstrated that the entropy will not always dominate the free energy when enthalpic contributions become more significant.
What remains unclear is the local or long range atomic distributions in HEAs. NiFeCrCoMn is one of the few systems that remained a single fcc phase during the investigation of Otto et al., but the elements do not share a common bulk crystal or magnetic structure. Ni, Fe, and Co are ferromagnetic (FM), while Cr and Mn are anti-ferromagnetic (A-FM). Klaver et al. investigated the relative energetics of mixing an A-FM solute (Cr) into a FM solvent (Fe) and found that Cr solutes had an energetic preference to spatially separate in the FM host due to a process called magnetic frustration (significant scatter in the local moment of Cr that leads to a rise in total energy).11 This was found to be most significant for Cr atoms sitting in their own first nearest-neighbor (1NN) shell and the effect diminished gradually as the distance between Cr atoms increased. Brif et al. recently performed x-ray spectroscopy measurements using scanning electron microscopy and demonstrated that Cr was uniformly distributed throughout the NiFeCrCo alloy.12 At the resolution of the technique, however, atomic level ordering would not be resolved. Lucas et al. searched for long range order in NiFeCrCo, which contains FM and A-FM elements, using x-ray and neutron diffraction and found none.13 The authors noted, however, that this did not completely rule out any chance for local ordering.
Here, we implement two complementary first principles approaches to predict the relative energetics of chemical ordering within the NiFeCrCo system. NiFeCrCo is selected because it is the base alloy for many other 5+ element HEAs. Similar findings are anticipated for NiFeCoMn but they are not explored here. We demonstrate that Cr has a strong energy preference for not sitting within its own 1NN shell. In this equiatomic alloy, Cr can only escape to the second nearest neighbor shell, which creates an alloyed L12 structure. This alloyed L12 is predicted to have a smaller total magnetic moment as compared to the RSS. To compare with predictions, we synthesized equiatomic NiFeCrCo under equilibrium and non-equilibrium approaches. The temperature dependence of the magnetic properties of all samples are measured. Cast alloys have a significantly smaller low temperature magnetic moment than ball milled samples. Cold working cast alloys is found to increase the measured low temperature moment. Finally, we apply advanced electron microscopy to identify ordered nanodomains in this alloy.
There are a number of challenges in the prediction of properties of RSS alloys, so two theoretical approaches using the generalized gradient approximation of the exchange-correlation functional as determined by Perdew, Burke, and Ernzerhof (GGA-PBE)14,15 were implemented here. The first was the well established Vienna ab initio simulation package (VASP)16,17 with atomic centers approximated with projector-augmented wave (PAW) pseudo-potentials.18,19 A k-point mesh was selected based on the convergence of the enthalpy to 1 meV/atom. Special quasi random structures (SQS), which best represent the targeted alloy within the confines of the periodic unit cell,20 were generated through a Monte Carlo algorithm.21 For an equiatomic alloy, the identity of the atom at each lattice site is not uniquely determined during SQS generation. Therefore, 24 permutations exist for the four component RSS SQS. We tested the convergence of enthalpy for 24, 64, and 120 atom SQSs with all possible permutations. We find that the average enthalpy of the formation for these RSSs are 0.309 ± 0.023, 0.302 ± 0.009, and 0.302 ± 0.007 eV per formula unit, respectively. Through averaging the permutations, even the small 24-atom SQS leads to a relatively converged enthalpy, albeit with more scatter. The second method implemented was the Exact Muffin-Tin Orbital combined with Coherent Potential Approximation (EMTO-CPA).22,23 The screening parameter for these calculations was set to 0.9 and a k-point mesh was selected based on the convergence of the enthalpy to 1 meV/atom. The CPA technique averages the Green's function to attain the randomness of atoms in a RSS. This one-site approximation neglects lattice distortions and also loses information on local chemical structure.
The local magnetic moments of each element as a function of the average magnetic moment, , of 1NN atoms for all permutations in the 24- and 120-atom SQS structures are plotted in Figures 1(a) and 1(b), respectively. FM Fe, Ni, and Co generally have positive moments, while the A-FM Cr atoms have a distribution of both positive and negative moments. Notably, Cr scatters significantly and is most negative when its neighbors have the largest positive moment, corresponding to when Cr is surrounded by a majority of FM neighbors. The magnetic moment of Cr becomes less negative as the average moment of the neighbors becomes smaller, which is a product, in part, of having more Cr in the 1NN shell. This strong dependence of the local moment is a result of magnetic frustration of the central Cr atom; similar to the trends identified in Ref. 11 for Fe-Cr alloys.
In binary Fe-Cr, there is a substantial repulsion when Cr resides in its own 1NN shell. This repulsion weakens as the distance between Cr atoms increases. For the equiatomic NiFeCrCo HEA, each element makes up 25 at. % of the system and, assuming no preferential segregation, Cr can only avoid sitting in its own 1NN shell by forming an alloyed L12 structure. In the alloyed L12 structure, one element occupies the cubic corner (CC) sites, while the face centered (FC) sites contain a RSS of the other three elements. Any displacements of the atom occupying the CC site will move it back into its own 1NN shell. Formation of the alloyed L12 structure also results in a reduction in configurational entropy as compared to the RSS and this penalty needs to be accounted for when comparing the relative energy of the RSS to the alloyed L12.
The relative energies of ordering one element (Ni, Fe, Co, or Cr) from the RSS to the CC site of the alloyed L12 are plotted in Figure 2(a). The EMTO-CPA allows for a continuous change in composition of each lattice site in the transition between the RSS and L12 (Figure 2(a, left)), while the SQS structures are only evaluated at the end points (Figure 2(a, right)). VASP and EMTO results are all spin-polarized. Cr is predicted as non-magnetic by EMTO, which contributes to differences in predicted enthalpies between the two methods. For comparison of trends, we align the enthalpies of the EMTO to those of VASP for the RSS only. No other constraints are applied other than this alignment of the RSS. Differences between the two electronic structure methods is found to be small. In addition to the enthalpy of formation, the plotted values also include configurational entropy as determined for an ideal solution at a temperature of 300 K. Configurational entropy is given by , where ci is the ratio of the number of atoms of a disordered component over the total number of disordered atoms. In the case of a partially ordered L12 phase, a convenient way of calculating configurational entropy is . Full estimates of the free energy requires contributions from the vibrational, magnetic, and electronic entropy, which are not included here. From the data presented in Figure 2, the most favorable ordering occurs through moving Cr to the CC site.
Figure 2(b) presents the magnetic moment for each element as a function of the average magnetic moment, , of the 12 1NN atoms in the six possible permutations of the alloyed L12 structure. In comparing these results to the similar data in Figures 1(a) and 1(b), the magnetic frustration is avoided as evidenced by the small scatter in the moments of each element. Fe, Co, and Ni all have aligned and positive magnetic moments, while Cr has a magnetic moment opposite of all its FM neighbors. This results in a reduction of the bulk magnetic moments when moving from the RSS () to the alloyed Cr L12 (). The uniform magnetic alignment also leads to an extremely small scatter on the order of 2 meV per formula unit in the predicted enthalpy of the alloyed L12 structure.
To test the predictions made by first principles methods, samples were prepared in three ways for analysis by SQUID-VSM. One sample was prepared by ball milling pure elemental powders in high-purity argon in a stainless steel vial using a SPEX 8000 mixer mill for 24 h at room temperature. Two other samples were prepared by melting pure elemental metals in high purity argon atmosphere using an electric arc melter. Each arc melted sample was then cast into a water-cooled copper mold and annealed in Ar-2% H2 at 1273 K for 24 h. One sample was tested after annealing and the other was cold rolled to 80% rolling reduction before testing. SQUID-VSM measurements were conducted with a Quantum Design MPMS using an applied field of 198.9 kA⋅m−1 from 1.8 to 390 K at a rate of 0.08 K⋅s−1.
Magnetization as a function of temperature for all of the samples is presented in Figure 3. At 1.8 K, the highly disordered sample produced by ball milling exhibits the largest magnetization. Cast samples with low temperature moments are 44% of those measured for ball milled samples. A large difference in moments is expected from the results shown in Figures 1 and 2(b), as the contribution from Cr is much more negative in the ordered state, while the contributions from the other elements remain nearly constant. While there are slight compositional differences between the cast and milled samples, none were significant enough to explain the difference in magnetic properties. A smaller, yet still significant increase in magnetization is seen after cold working, which would disrupt order in the cast/annealed sample.
Local atomic and chemical structure were also explored through scanning transmission electron microscopy (STEM). Samples for electron microscopy were prepared from the cast HEA using a standard twinjet electropolisher. High-angle annular dark field (HAADF) STEM images were acquired using a probe-corrected FEI Titan G2 60–300 kV operated at 200 kV. Further details of the sample preparation and imaging conditions can be found in Ref. 25. The revolving STEM (RevSTEM) method was employed to remove drift distortion from the STEM images.26 Forty 1024 × 1024 pixel frames were acquired with a dwell time of 2 μs/pixel. A 90° rotation angle step was introduced between each successive frame.
Figure 4(a) shows a representative RevSTEM image of the cast HEA. The seemingly uniform intensity distribution results from small difference in the atomic number of constituent elements (ZCr = 24, ZFe = 26, ZCo = 27, and ZNi = 28), making it difficult to unambiguously distinguish ordering by visual inspection. Highlighting intensity variations in the image, we calculate the ratio of each atom column intensity to the average intensity of the four nearest neighbors through peak fitting and converting the STEM image to a matrix representation.27 The ratio map for the RevSTEM image is shown in Figure 4(b) where local regions of modulating large (red) and small (blue) ratios are evident. To further quantify this local ordering, we compare each block of 5 × 5 neighbors to a completely ordered pattern, which is then used to calculate a correlation coefficient (Figure 4(c)) for each block. A perfectly ordered structure with alternating intensity would result in a correlation coefficient of unity, while a disordered structure would have zero correlation. The correlation coefficient reaches a maximum of approximately 0.67, which, according to a p-value test generated through comparing these correlations to those of a simulated RSS, is inconsistent with the null hypothesis (p = 7 × 10−8). Therefore, the result strongly suggests the presence of ordered 2–3 nm domains in the bulk HEA.
We have demonstrated that magnetic frustration of Cr exists in NiFeCrCo random solid-solutions, which leads to significant scattering of the local magnetic moment. Ordering of Cr provides a way to eliminate the magnetic frustration and also leads to a reduction of free energy. Temperature dependent SQUID measurements indicate that this alloy has a finite magnetic moment at low temperatures, which is sensitive to synthesis approach. The equilibrium approach leads to a smaller magnetic moment, consistent with the theoretical predictions. RevSTEM measurements identified ordered nanodomains present throughout the sample. Although nanodomains were found here, future studies that explore a variety of annealing procedures may be warranted to ultimately achieve long range ordering.
C.N., A.J.Z., C.C.K., and D.L.I. acknowledge support for this work from the National Science Foundation from the Metals and Metallic Nanostructures program under Grant No. DMR-1104930. A.A., X.S., and J.M.L. acknowledge support from the Air Force Office of Scientific Research (Grant No. FA9550-12-1-0456) and the Analytical Instrumentation Facility (AIF) at North Carolina State University, which was supported by the State of North Carolina and the National Science Foundation. D.L.I. acknowledges support for J.W.H. from the NSF REU on Advanced Materials for Environmental Sustainability under Grant No. EEC-1156762. D.L.I. and C.N. would also like to acknowledge Levente Vitos for sharing his EMTO-CPA code, version 5.7, for work on this project. C.N. and D.L.I. thank M. C. Gao for sharing 64 atom SQS.