Reconsidering short-range order in complex concentrated alloys

The seemingly contradictory state of research on short-range order in many-component alloys is addressed through a critical review of the characterization of face-centered-cubic 3d systems. Despite the paucity of direct observations, the ordering of many widely studied alloys is argued to be primarily interesting for its potential ubiquity. To clarify this situation, future research directions are proposed with reference to historical results, including a review of the fundamental principles of ordering and clustering.


Introduction
Few topics in contemporary metallurgy have caused as much controversy as the prospect of chemical short-range order (SRO) in "high-entropy" or, more generally, "complex concentrated" alloys (CCAs). The phenomenon has, in recent years, become intensively investigated in face-centered-cubic (fcc) alloys containing several concentrated 3d elements, which are the focus of this article, although we will discuss many broadly applicable principles.
At first glance, the attention given to SRO in this class of materials may seem surprising; 3d CCAs are primarily interesting for their mechanical properties, but many studies have reported that the bulk deformation of one of the more strongly ordering alloys is practically unaffected by the development of SRO during annealing. [1][2][3][4] It has further become apparent that many purported measurements of SRO by transmission electron microscopy (TEM) were likely detecting something else. 4,5 These revelations could reasonably be interpreted to indicate that ordering in these systems is generally minimal and unimportant when present.
In this article, however, we will argue that the SRO of 3d CCAs remains of great scientific interest, albeit within a scope that is less well explored. Specifically, we discuss how the presence of an effectively ubiquitous degree of ordering in many conventionally processed alloys could reconcile a diverse range of theoretical and experimental results. In order to technically discuss this perspective and to guide future investigations, we first examine historical research on the topic, highlighting some important results that may not be widely appreciated today.

Binary alloys
Before considering SRO in many-component systems, it is helpful to review the physics of chemical rearrangement in binary solid solutions, of which there are two distinct forms. In the first, two elements simply segregate, or "cluster," forming separate phases in the long-range limit. Alternatively, attractive interactions between unlike atomic species can lead to the formation of various periodic lattice decorations, or orderings.
In either case, atomic distributions are usually described in terms of the frequencies of chemical pairs. The likelihood of finding elements A and B as nth-nearest neighbors is expressed by the Warren-Cowley (WC) parameter α (n) AB . In a random solution, the probability of finding B at a site r n from an A site is simply the fractional concentration of B, For the binary alloy depicted in Figure 1b, panel c shows α AB (k) for the (001) plane in reciprocal space, as determined from Equation 2.
While somewhat less immediately intuitive, the reciprocal-space view is attractive because binary α AB (k) is closely related to measurable diffuse-scattering intensity, as will be discussed later. Many structures can also be succinctly described by a small number of wave vectors. Clustering, for instance, is associated with k approaching [000], which corresponds to perfect phase separation.
Nonzero values of k represent ordering, which can be interpreted in terms of concentration waves expressing the average chemical environment of a crystal. The probability of finding either element at a lattice site at r is a superposition of concentration waves ∼ e 2πik·r with weights related to α AB (k). 8,9 While SRO necessarily involves many periodicities, specific wave vectors associated with LRO structures are generally dominant.

Many-component alloys
This method of binary pair analysis can be applied to manycomponent systems, at the expense of some elegance; α AB (k) is no longer directly measurable and the complete determination of ordering waves requires system-specific analysis. 9,10 Still, the theoretical SRO of M-component CCAs is often described in terms of M (M − 1)/2 correlations between unlike pairs, in real ( α (n) AB ) or reciprocal ( α AB (k) ) space. (d) Real-and (e) reciprocal-space representation of equilibrium short-range order/clustering in a model quaternary alloy. Element A (dark blue) exhibits ordering tendencies associated with the wave vector k = (1 1 2 0) , while element D (light blue) appears to cluster ( k → (000) ); elements B (gold) and C (red) are more randomly distributed. fcc, face-centered cubic; bcc, body-centered cubic; LRO, long-range order.
┼ The prefactor, which cancels for A = B , recovers a standard form of α AB (k) 6 for the definition of α (n) AB in Equation 1. The sum includes an n = 0 term that always equals 1.
In many systems, however, it can be simpler and more intuitive to first consider M (or M − 1 ) like-pair terms (i.e., α (n) AA or α AA (k) ). In contrast to unlike pairs, α AA (k) still represents ordering associated with an A-concentration wave with vector k . These values are also simpler to determine experimentally. 6 Despite greater compositional complexity, the ordering of many recently studied CCAs appears to primarily involve segregation or sublattice formation by a single element. A system could exhibit multiple such processes, including the formation of complex LRO from the sequential development of sublattices, but individual ordering reactions are often, though by no means always, associated with one chemical species. As a hypothetical example, a model CCA containing both ordering and clustering elements is depicted in Figure 1d, with corresponding reciprocal-space intensities shown in Figure 1e.
Due to the interdependence of pair correlations, the clustering or ordering of a single element is redundantly reflected in many separate pair parameters. For example, the formation of a Cr sublattice in CrFeCoNi is similarly apparent in the Cr-Fe, Cr-Co, and Cr-Ni pair correlations 11 , but could be approximately described by the distribution of Cr-Cr pairs. The theoretical SRO of nearest neighbors in CrCoNi appears similarly determined by the frequency of Cr-Cr bonds; Ni-Ni and Co-Co correlations remain small and remaining values largely conserve probability. 12,13 Of course, specific CCAs will likely display novel ordering phenomena that could require the full picture of concentration waves 9,10 or even many-body WC parameters. 14 The interpretability of all mean-field descriptions can also be compromised by the formation of local compositional heterogeneity in complex alloys. Nonetheless, the analysis of individual species in the binary formalism provides an intuitable starting point that invites the application of various historical results.

Phase transitions
At a sufficiently low temperature, T tr , the interactions driving ordering or clustering induce a phase transition, below which the equilibrium correlation distance becomes arbitrarily long-ranged. Below T tr , equilibrated clustering becomes phase separation and SRO gives way to LRO. The microstructure of real alloys, however, often corresponds not to an equilibrium configuration, but rather an arrested phase transition, such as partial phase separation into intermediatesized clusters. In many popular 3d systems, LRO develops very slowly and in some cases appears practically limited to nanoscale antiphase domains. 15 While this form of order remains fundamentally short-ranged, it could also be considered incomplete LRO to distinguish from the equilibrium SRO found above T tr .
Incomplete LRO, which is often controllable through heat treatments, has been the focus of most historical [16][17][18] and many recent 2,4,11 studies on SRO. While tuning order in this manner could in principle enable the tailoring of materials properties, most alloys are not processed in any way intended to promote ordering. Equilibrium short-range order or clustering (SRO/C) can nonetheless form as samples are cooled from high temperatures (e.g., in the region of the phase diagram highlighted in Figure 1a). The prevalence of this type of SRO/C, which can be further enhanced upon initial cooling below T tr , 19 is considered in the remainder of this article.
First, however, it should be noted that the structure approached by SRO above T tr can differ from the LRO found below T tr , even though the interactions driving chemical rearrangement are typically equivalent. As dictated by Landau's theory of phase transitions, crystallographic considerations require that many LRO structures form by first-order phase transitions, which generally involve nucleation and growth processes that impose initial energy barriers on ordering. Thus, the incomplete forms of these structures that could occur above T tr can be energetically unfavorable. 8,20 Certain other decorations can form without any such barriers by continuous (second-order) phase transitions. In the picture of concentration waves, the SRO forming above T tr can be associated with the wave vectors of these structures, even if their long-ranged forms are never realized. Of course, high-temperature SRO typically shares key features with the LRO ground state, such as the minimization of same-species nearest neighbors. The Cr-Ni system clearly demonstrates this phenomenon; CrNi 2 alloys can form MoPt 2 -type LRO below 863 K, but the SRO measured above T tr 21 resembles a concentration wave with k = (1 1 2 0) , which is associated with distinct structures such as the Al 3 Ti prototype.

Theoretical overview of 3d CCAs
The Cr-Ni system in fact appears to provide a template for the ordering of many fcc CCAs containing concentrated Cr, namely the derivatives of CrMnFeCoNi. These alloys consistently exhibit a ∼ 900 K phase transition that calorimetrically resembles the onset of MoPt 2 -type CrNi 2 in Cr-Ni alloys. 22 Compositionally comparable CrCoNi is predicted to form a similar phase with an effectively random Co-Ni solution in place of Ni, 13,23 while systems such as CrFeCoNi may prefer an AlNi 3 -type (L1 2 ) Cr sublattice. 11 Both structures minimize Cr nearest neighbors, as can be largely explained by electrostatic interactions arising from charge transfer among electronegatively distinct elements (e.g., Cr to Ni). [24][25][26] Some recent first-principles calculations have suggested that magnetic interactions can also affect chemical ordering, 11,12,27,28 although our understanding of the magnetism of Cr in concentrated fcc alloys appears incomplete; either standard theory is incorrect or the presence of significant antiferromagnetic order has been overlooked. 28 Nonmagnetic V displays similar but stronger ordering tendencies, with VCoNi appearing to form AlNi 3 -type LRO 29 that is consistent with theoretical predictions. 30 Manganese is also understood to promote ordering in some conditions, although its behavior is comparatively less well studied. A CuAu-type (L1 0 ) MnNi phase is found among the decomposition of CrMnFeCoNi 31 and has been further examined in FeMnNi; 32 more complex orderings have also been theorized, though not experimentally validated, for Cr-Mn-Ni and Cr-Mn-Fe-Ni alloys. 27 Copper is expected to cluster in FeCoNiCu alloys, 33 just as in Ni-Cu. 34 Chemically distinct elements, such as Al, can also more readily cluster or form LRO. 10 While some recently studied CCAs appear approximately pseudobinary, the underlying physics of chemical rearrangement can nonetheless differ from their binary counterparts. For instance, the ordering of CrCoNi closely resembles that of CrNi 2 with a 10% higher transition temperature, 22 but the magnitude of its theoretical ordering energy 13 is several times larger, implying significant entropic differences. Conversely, the calculations of Reference 11 suggest that the formation of a Cr sublattice in CrFeCoNi is associated with a relatively minor entropy reduction given the configurational freedom of the remaining elements; some more systematic study of these effects could prove insightful.
Of course, all previously referenced predictions were made under the assumption of thermodynamic equilibrium and comparatively fewer studies have tackled the formidable problem of CCA kinetics. Perhaps most notably, Du et al. 23 explored the evolution of SRO in CrCoNi via the discrete hopping of vacancies in kinetic Monte Carlo simulations parameterized using a carefully developed interatomic potential. Although necessarily involving significant approximation, their calculations suggest that SRO will at least initially form far faster than typical quenching rates; a time-temperature-SRO diagram calculated by this study is reproduced in Figure 2a. As an alternative approach, simpler mean-field kinetic models 35 could also prove useful for many-component systems, provided relevant thermodynamic and mobility data are available.
With this theoretical picture as a guide, we will explore the experimental characterization of SRO/C in 3d alloys, with particular attention given to findings that could indicate the presence of SRO/C in conventionally processed material. This work is, of course, incomplete and we will primarily consider systems for which multiple data are available, in particular Cr-Ni and CrCoNi.

Diffraction
Several decades ago, the SRO/C of single-crystal binary alloys was routinely characterized using the diffuse scattering of monochromatic x-rays or neutrons, that is the intensity in the Brillouin zone not due to the underlying lattice or multiplescattering events. After accounting for contributions from static lattice displacements, the elastic diffuse intensity scattered at wave vector k can be related to the α AB (k) described by Equation 2. 7,40 WC parameters α (n) AB , and hence α (n) BB , can be subsequently obtained. Table I summarizes a number of historical measurements of SRO in Cr-Ni and V-Ni alloys subject to various thermal histories in terms of the nearest-neighbor WC parameter, α (1) CrCr or α (1) VV . Many of the considered samples contained similar fractions of Cr or V as CCAs of current interest and, while entropic considerations could differ, the basic picture of ordering is expected to be similar. Long-term annealing below T tr increases the degree of order, but, for a given composition, the difference between the least and most ordered measurements is less than that between the least order and complete disorder ( α (n) AB = 0 ). Additionally, many of the least ordered values were obtained at elevated temperatures and SRO could further develop during cooling. These results, which are comparable to those for other binary systems, clearly indicate that perfect atomic-scale randomness should never be assumed, even in quenched samples.
Unfortunately, interpreting diffuse scattering becomes rapidly more challenging with increasing compositional complexity, as the intensity measured in many-component alloys simultaneously represents all interspecies correlations. Specific α AB (k) can nonetheless be extracted from multiple measurements obtained under distinct scattering conditions. For instance, Cenedese et al. 16 characterized SRO in a concentrated Cr-Fe-Ni alloy by analyzing diffuse neutron scattering from three samples with varying isotopic content, as included in Table I. This said, few comparable studies exist.
Distinct scattering measurements can also be obtained from "resonant" x-ray energies corresponding to the absorption edges of individual alloy elements. 7,41 In this manner, Schönfeld et al. 11 investigated the diffuse scattering of CrFeCoNi, providing perhaps the most complete characterization of SRO in a CCA since the popularization of high-entropy alloys. While not determining specific pair correlations, they used complementary electronic structure calculations to deduce the partial formation of an AlNi 3 -type (L1 2 ) Cr sublattice in a sample aged at approximately 0.8 T tr . Comparable data are not available for differently processed samples.
Many recent studies have instead favored TEM, which does not require a beamline facility or single-crystal samples. However, while various other phenomena can cause extra reflections that have been widely interpreted as ordering, the electron scattering contrast among 3d elements appears insufficient to detect SRO in many systems of interest, 4,5 at least using standard techniques-conventional x-ray diffraction appears similarly insensitive. 4 (Clustering, LRO, and even SRO involving distinct chemical species, such as Al or Pd in a 3d alloy, may be more readily visible in TEM.) Most remaining evidence for SRO in CCAs is thus indirect, as will be examined in the remainder of this article. Some spectroscopy also appears to support the presence of ordering, 42,43 although these measurements can vary subtly with SRO and require comparison with simulated configurations, the accuracy of which has been questioned. 1 As another approach, atom-probe tomography can qualitatively reveal real-space concentration waves along one sample dimension. 2

Resistometry
Before SRO was ever directly measured-or even fully conceptualized-its presence was connected to resistivity anomalies, specifically what is still known as the "K-state." 15 As a general principle, the formation of order reduces the resistivity of a random solid solution, but its initial development can first lead to a noticeable increase. 15,44,45 Although only providing a relative indication of ordering, resistometry remains a relevant technique in the contemporary study of CCAs. For example, Schönfeld et al. 11 judged the practical completion of ordering in CrFeCoNi from the plateauing of resistivity during annealing. Two studies have more recently characterized the development of order in CrCoNi by measuring resistivity, 4,36 contrasting modest increases to the invariance of mechanical properties.
While most investigations have focused on the development of order during long-term annealing below T tr , Teramoto et al. 36 also considered the impact of SRO on the resistivity of quenched CrCoNi, concluding that conventionally achievable cooling rates produce a state comparable to equilibrium at only about 1.05 T tr . The comparable resistivities of samples annealed at various temperatures are reproduced in Figure 2b. As the theoretical value of α (1) CrCr at 1.05 T tr is approximately 0.25, 13 this result suggests the existence of a higher baseline degree of order than envisioned by many studies. Possibly supporting this picture, rethermalizing ultra-rapidly quenched CrMnFeCoNi 46 increases resistivity several times more than the annealing of conventionally processed CrCoNi. 4,36 It is, however, worth noting that resistivity could also be sensitive to nonordering phenomena such as the presence of impurities, potentially complicating interpretations.

Phase transformation and twinning
The presence of SRO in experimental samples can also be more speculatively inferred from a range of other properties. One simple observation motivating interest in the phenomenon is the remarkable stability of the fcc phase of many Co-rich CCAs that theoretically favor hexagonal close-packed (hcp) lattices. 11,12,47,48 Vibrational and magnetic entropy is expected to stabilize the fcc phase at the temperatures reached during conventional processing, but the structure of these CCAs remains largely unchanged at arbitrary low temperatures, even after mechanical deformation. 49 This behavior is contrasted by certain nonordering Co-Ni alloys, which similarly retain a metastable fcc phase after quenching, but exhibit both spontaneous and deformation-induced martensitic phase transformations. 50 Local structural transformations, including nanotwins and atomically thin hcp lathes, are still commonly observed in these CCAs, particularly in regions where local order would be disrupted by prior faulting, 47 which is consistent with simulations of alloys containing SRO. 51 Liu et al. 49 further speculated how the remarkable low-temperature work hardening of CrCoNi and CrMnFeCoNi could in part originate from the ordering-limited formation of planar defects, which were observed at a scale that could possibly impede dislocation motion, but not one that would meaningfully reduce ductility. The seemingly invariant nature of this deformation could indicate the presence of comparably impactful SRO in samples subject to various heat treatments. 2,4 Further supporting this hypothesis, the hcp phase of CrCoNi can be recrystallized after severe plastic deformation, 37 which would significantly reduce SRO by dislocation slip; a micrograph of an extended hcp structure obtained in this manner is reproduced in Figure 2c.

Yield strength
The impact of SRO on the yield strength of alloys has also received some attention, although no conclusive effects have been experimentally demonstrated in bulk CrCoNi 1,2,4 or related CCAs. In solid solutions, the energy of a dislocation line segment generally depends on its local chemical environment, which varies significantly throughout a random crystal. Stationary dislocations adopt wavy paths through chemically favorable regions, balancing compositionally sensitive core energies with the elastic cost of curvature.
The formation of SRO tends to reduce chemical fluctuations, decreasing the relative stability of locally favorable configurations and thereby lowering the stress required to move dislocations. 38 Conversely, SRC further roughens the energetic landscape, increasing yield strength. 52 Both phenomena can also strengthen alloys due to the energetic cost of destroying local order during slip 53 (i.e., the creation of a diffuse antiphase boundary [DAPB]). Considering both energetic fluctuations and the destruction of order, SRC should only increase yield stress, but the impact of SRO can be more subtle due to the cancellation of these terms; illustrative calculations for a model system 38 are shown in Figure 2d. Consequently, yield strength should not be considered a reliable probe of SRO.
These principles can also explain why a number of computational studies have predicted that annealing will increase yield strength; many recent atomistic simulations of deformation in CCAs have relied on a description of CrCoNi 54 that predicts unphysical clustering among Ni atoms rather than the expected ordering of Cr. 12,13,23

Planar slip
After yielding, the initial passage of a dislocation destroys most local ordering within a slip plane, "softening" the plane for subsequent dislocations. 55 This slip-plane softening, combined with the effects of SRO/C on cross-slip, 56 is expected to promote planar slip behavior. DAPB formation energies can be estimated from planar dislocation arrays observed during in situ deformation experiments, 57 allowing for connection to theoretical ordering energies. However, while the observation of planar slip is typically considered to indicate SRO/C (or dispersed LRO precipitates 55 ), its dependence on cross-slip behavior means that the phenomenon is sensitive to crystal orientation, 39 as demonstrated in Figure 2e. As the latter effect has not been widely appreciated, future studies should exercise caution while interpreting the presence (absence) of planar slip as the presence (absence) of significant SRO/C.

Magnetometry
In alloys composed of magnetic 3d elements, bulk magnetization, which represents the sum of environmentally sensitive local atomic moments, can also reflect local ordering. 58 In the low-temperature limit, magnetization measurements are generally comparable to the predictions of electronic structure calculations, which quantitatively describe the magnetic ground states of many alloys such as FeCoNi, 59 which is not known to form significant SRO/C. However, the bulk magnetometry of systems containing concentrated Cr is often at odds with predictions for random solid solutions. 59, 60 This discrepancy could be explained by the presence of significant SRO; 12 Figure 2f displays how low-temperature magnetization measurements for Cr-Co-Ni alloys are far more consistent with calculations including ordering. ǂ Limited experimental evidence corroborates this picture; 61 for instance, the ferromagnetism predicted for random solid solutions has been observed in some thin films, which could be less prone to ordering at low deposition temperatures. 62 Severely deformed CrCoNi samples containing hcp martensite also exhibit some ferromagnetism, 37 although the precise origin of the small reported moment is unclear. The magnetic properties of CrM-nFeCoNi appear similarly sensitive to cold working and heat treatment, 63 which could be at least partially explained by the effects of these processes on SRO. Further investigation of this topic seems merited, although the study of alloys containing V in place of Cr could prove more immediately enlightening given the previously discussed uncertainties regarding the magnetism of Cr. 28

Conclusion
If SRO matters in 3d CCAs, it is likely because a significant degree is effectively ubiquitous in many systems, as has been suggested, though not proven, by a range of experimental results. Further advancement of the field will likely require new resonant x-ray diffuse-scattering experiments; even if the quantitative characterization of SRO is impractical, the state of order in conventionally processed samples could be relatively assessed and interpreted with the aid of theoretical models. Additionally, this analysis draws disproportionately from studies on CrCoNi and, while the physics of similar systems are not expected to radically differ, closer examination of more complex alloys seems required. Mark Asta is an executive associate dean of engineering and professor of materials science and engineering at the University of California (UC), Berkeley, and a faculty senior scientist at Lawrence Berkeley National Laboratory. He received his PhD degree from UC Berkeley in 1993 and has published more than 300 articles in the field of computational materials science. He has received ASM International's Materials Research Silver Medal and J. Willard Gibbbs Phase Equilibria Award, The Minerals, Metals & Materials Society (TMS) William Hume-Rothery Award, and is a Fellow of TMS and the American Physical Society. Asta can be reached by email at mdasta@berkeley.edu.