Phase stability, physical properties and strengthening mechanisms of concentrated solid solution alloys

Current Opinion in Solid State and Materials Science xxx (2017) xxx–xxx

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Current Opinion in Solid State and Materials Science journal homepage: www.elsevier.com/locate/cossms

Phase stability, physical properties and strengthening mechanisms of concentrated solid solution alloys q Z. Wu a, M.C. Troparevsky a, Y.F. Gao a,b, J.R. Morris a,b, G.M. Stocks a, H. Bei a,⇑ a b

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Materials Science and Engineering Department, University of Tennessee, Knoxville, TN 37996, USA

a r t i c l e

i n f o

Article history: Received 7 November 2016 Revised 4 May 2017 Accepted 23 July 2017 Available online xxxx Keywords: Concentrated solid solutions High entropy alloys Phase stability Physical/mechanical properties Strengthening mechanisms

a b s t r a c t We review recent research developments in a special class of multicomponent concentrated solid solution alloys (CSAs) – of which the recently discovered high entropy alloys (HEAs) are exemplars – that offer a new paradigm for the development of next generation structural materials. This review focuses on the role of inherent extreme chemical complexity on the phase stability, electronic, transport, and mechanical properties of this remarkable class of disordered solid solution alloys. Both experimental observations and theoretical models indicate that the phase stability of HEAs goes beyond the original conjecture that these alloys are stabilized by configurational/mixing entropy; rather, it results from competition between the homogeneously disordered phase and phase separation/intermetallic compound formation. Although the number of single-phase HEAs with equiatomic composition is limited, those that do exist often exhibit remarkable electronic, magnetic, transport, and mechanical properties. For the mechanical response, we discuss the solution strengthening mechanism which governs the strength and deformation behaviors of the CSAs, as well as the increasing evidence that low stacking fault energies (deformation twinning) plays an important role in the low temperature strength and ductility of CrMnFeCoNi related alloys. We also review the current understanding of the role of the number and type of alloy elements in determining the electronic, magnetic, and transport properties, in particular the dominant role of magnetic interactions in the properties of 3d-transition metal based alloys. Finally, we emphasize that, despite rapid progress in characterization and understanding of the phase stability and physical/mechanical responses of CSAs, there remain significant challenges to fully exploring the new paradigm that these alloys represent. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Pure metals are rarely used as such; alloying is the key to meet various performance requirements such as strength, ductility, corrosion and oxidation resistance. For example, the addition of 20–36 wt.% zinc into copper significantly increases the strength, ductility and corrosion resistance of brass; alloying 12.5 wt.%

q This manuscript has been authored by UT-Battelle, LLC under Contract No. DEAC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http:// energy.gov/downloads/doe-public-access-plan). ⇑ Corresponding author. E-mail address: [email protected] (H. Bei).

silicon into Al improves the forgeability and lowers the thermal expansion coefficient of these alloys, which is essential for the forged automotive pistons; suitable amounts of Cr, Mo, Ni and C can be added into Fe to make steels with better corrosion resistance and room/high temperature strength and toughness [1–4]. Various nickel-based superalloys, widely used in extreme environments subjected to heat and pressure, are alloyed with Cr, Fe, Mo, Mn, Al, and Si [5–7]. All of these alloying systems are based mainly on one principal element. During alloying development and design, ordered intermetallics attract much attention as high temperature structural materials because of their high temperature strength, excellent creep resistance and other mechanical properties. These intermetallic compounds are mainly derived from Ti-Al, Ni-Al and Fe-Al binary systems [7–10]. The development of these systems led to an enormous amount of knowledge about alloying strategies of intermetallic compounds based on two or more Principal elements. The development of new processing technologies such as rapid solidification and mechanical alloying has facilitated

http://dx.doi.org/10.1016/j.cossms.2017.07.001 1359-0286/Ó 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Z. Wu et al., Phase stability, physical properties and strengthening mechanisms of concentrated solid solution alloys, Curr. Opin. Solid State Mater. Sci. (2017), http://dx.doi.org/10.1016/j.cossms.2017.07.001

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the exploration of other alloys, such as bulk metallic glasses (BMG). BMGs are typically Pd-, La-, Zr-, Fe-, and Mg-based multicomponent bulk amorphous alloys that exhibit high strength but low tensile ductility [11–15]. In this new century, the alloying design concept is being extended to alloys with multi-principal elements in equi- or near-equiatomic ratios [16–40]. The alloy compositions move from the edge or corner of the phase diagrams towards the center; therefore, all elements in the alloys are concentrated. From a general physical metallurgy point of view, increasing the number of alloying elements with varying crystal structures will simultaneously increase the probability of forming additional phases and/ or intermetallic compounds. However, recent experimental studies have shown that alloys with multiple principal elements can crystallize as solid solutions with simple micro/crystal structures [16,17]. The alloys composed of 5 or more elements at or near equiatomic ratio, which tend to form a random solid solution, are referred to as high entropy alloys (HEAs) [16]. One typical example is the Cantor’s alloy [17], which is composed of 5 elements with different room-temperature crystal structure, namely bodycentered cubic (bcc) Fe and Cr, hexagonal close-packed (hcp) Co, A12-structured Mn and face-centered cubic (fcc) Ni. The CrMnFeCoNi alloy can solidify from melt as single-phase solid solution alloy with a fcc crystal structure. The term HEA was adopted by Yeh et al. [16], based on the hypothesis that the high mixing entropy stabilizes these alloys. As shown by the expression for the ideal solution, the entropy of P mixing DSmix ¼ kB Ni¼1 xi lnðxi Þ, for a NAcomponent alloy, the configurational entropy is maximal at equiatomic composition (xi ¼ 1=N; i ¼ 1; N) and increases with N as DSmix ¼ kB lnðNÞ. Besides their micro/crystal structure, overall superior properties are usually observed for these alloys, including high strength and ductility, outstanding corrosion, wear and oxidation resistance, and good phase stability [41–50]. For instance, Gludovatz et al. showed that single-phase CoCrFeNiMn [18] and CoCrNi [19] had exceptional damage tolerance. The toughness of these alloys was larger than that of virtually all pure metals and most traditional alloys [51–58]. Hundreds of multi-component alloys have been prepared using 30 different elements. These alloys are mostly prepared by melting high-purity elemental materials such as Co, Cr, Fe, Ni, Al, Cu and Mn. The phase stability of HEAs has been thoroughly studied by both experimental characterization [16–50,59–63] using different kinds of techniques, such as X-ray diffraction, scanning electronic microscope (SEM), and transmission electronic microscope (TEM). Predicting and understanding the phase stability of these alloys has been the subject of numerous theoretical studies [64–70]. Semi-empirical approaches have been widely used during the last decade in order to predict the stability of HEAs. These approaches employ properties such as atomic-size mismatch, electronegativity and enthalpy of mixing of the alloys [67–70]. In spite of some successes of those predictions, more reliable models are needed to guide experiments in the search of new HEAs compositions. Although the number of high entropy alloys with equiatomic compositions (as truly stable single-phase solid solution) is limited, those that do exist often exhibit interesting physical and mechanical properties. Extensive research has been conducted to understand the unusual mechanical properties of numerous equiatomic alloys under different environment, stress states, temperature and strain rates, especially those concentrated alloys with fcc crystal structures [17–19,30–40]. Here, we review the recent development and understanding of some important aspects: phase stability, physical properties, and strengthening mechanisms of this new class of materials.

2. Phase stability in multi-component concentrated solid solution alloys (CSAs) 2.1. Experimental observation Yeh et al. [16] synthesized a ten-component equiatomic alloy, CuCoNiCrAlFeMoTiVZr, by means of arc-melting the constituent elements. According to the Gibbs’ phase rule, F = C  P + 2, at constant pressure eleven different phases could co-exist in the microstructure of this alloy. Surprisingly, the alloy obtained by Yeh et al. showed a relatively simple as-cast microstructure where only three main solid-solution phases were found, (two bcc phases and an amorphous phase). A similar result was obtained in AlCoCuCrFeNi alloy, where three phases, fcc, bcc and B2, were found. In addition, Cantor et al. [17] fabricated two equiatomic alloys with 20 and 16 elements each. Although both alloys have multi-phase microstructures, a simple fcc primary phase is present, containing various elements but particularly rich in Cr, Mn, Fe, Co and Ni. Indeed, CoCrFeMnNi was the first single-phase alloy to be thoroughly studied. By adding an additional element (Nb, Ge, Cu, Ti or V) into CoCrFeMnNi, the resulting equiatomic alloys all solidify dendritically to form a fcc primary phases, with interdendritic segregation and some additional phases. The common feature is that the total number of phases is always far below the maximum number allowed by the Gibbs’ phase rule. Over 100 equiatomic alloys have been made which contain mainly Al, Co, Cr, Fe, Mn, Ni, Ti, Nb, Cu, V, Ta, and Mo. These alloys have been microstructurally characterized to identify their phase stabilities. Some equiatomic alloys that have been fabricated and phase-identified are listed in Table S1 of the supplemental materials. In addition to the HEAs, numerous sub-alloys, namely binary, ternary and quaternary alloys, which were called ‘‘mediumentropy alloys” by some researchers, are also investigated. It is obvious that the simple microstructures (e.g., single phase solid solution phases) of equiatomic alloys cannot be obtained by increasing the number of constituents. A majority of the materials consist of additional phase(s) (e.g., other solid solution phase and/ or intermetallic compounds). A key alloy, which served as a starting point for many other alloys, is CrFeCoNi. About 30 CrFeCoNi-based equiatomic alloys with one or more additional elements have been studied and their compositions are listed in Table S1 of supplemental materials. By adding a single metallic element [41-45], such as Cu, V, Ti, Nb, and Mo, into CrFeCoNi, most alloys exhibit complicated microstructures with extra phases. The addition of Cu into CrFeCoNi results in a microstructure composed of two fcc phases (with compositions of 21Ni-23Co-10Cu-23Cr-23Fe and 7Ni-2Co-87Cu2Cr-2Fe respectively) with similar lattice parameter; the addition of bcc Nb and Ti leads to the formation of Laves phases; the addition of V causes the formation of r phases. There are also a few elements that when added to CrFeCoNi result in a single-phase solid solution, such as CrFeCoNiMn (fcc) [30–40] and CrFeCoNiPd (fcc) [71]. There are a few other base systems that are obtained by removing one element from CrFeCoNi, resulting in FeCoNi-based, CrFeNi-based, CrCoNi-based and CrCoFe-based alloys, as shown in Table S1 of supplemental materials. Another important alloying system is composed mainly of refractory metals, such as Nb, Ta, Mo, and W. These refractory alloys crystallize as a single-phase solid solution with bcc crystal structure [41]. 2.2. Factors that affect the phase stability Above observations are contradictory to the initial concept of the HEA approach, i.e., increasing the number of constituents increases the possibility of forming single-phase solid solution

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alloys, but rather suggest that the type of additive elements is an important factor determining the phase stability of multicomponent equiatomic alloys. Two representative sets of experimental work are as follows. By adding Al, Co, Cr, Fe, and Si sequentially to the fcc-structured solid solution CuNi, the resulting alloys are all multi-phase-structured [72]. Similarly, starting from the quinary equiatomic alloy CrMnFeCoNi which was shown to be single-phase fcc-structured solid solution, Wu et al. [32] degenerated it down to various subsystems, namely of five quaternary, ten ternary, and ten binary alloys that can be formed from the combinations of constituent elements of the quinary alloy. Upon casting, near-melting-temperature homogenization, rolling and annealing, three quaternaries, CrFeCoNi, MnFeCoNi, and CrMnCoNi, four ternaries, FeCoNi, MnFeNi, CrCoNi, and MnCoNi, and two binaries, FeNi and CoNi, were found to be single-phase fcc solid solutions as shown in Fig. 1. Concentration of the elements also strongly affects the phase stability of multi-component equiatomic alloys, which not only modifies the relative volume fraction of each phase, but also can change the crystal structure if a critical value is reached. The effects of fcc Al on the microstructures of the base alloy CrFeCoNi were extensively investigated by Chou et al. [73]. The CrFeCoNiAlx alloys with x in the range of 0–2 were cast and homogenized and their crystal structures were characterized. Results show that alloys with x < 0.5 tend to solidify as fcc single phase solid solutions. With 0.5 < x < 1, two phases, fcc and bcc, were observed and the amount of bcc phases increases with increasing Al content. When x is higher than 1, only bcc phases were observed. This indicates that Al acts as a bcc stabilizer in this alloying system. Similar

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Al-effects were also reported in CrFeCoNiMn, CrFeCoNiCu, CrFeCoNiTi, and CrFeCoNiTiCuMnV systems where the addition of Al tends to promote the formation of the bcc phase. The bcc structure generally has a lower atomic packing density than the fcc. Thus it can more easily accommodate larger solute atoms like Al. Addition of Al is anticipated to induce the transition from a close-packed fcc to a loose-packed bcc structure to relax the lattice distortion energy. This reasoning is also based on the interactions between Al and the transition metals with hybrid pd-orbitals and enhancements of the directional bonding for bcc structure. Another bccstabilizer was Cr. Rather than bcc stabilizer, Nb and Ti facilitate the formation of laves phases [44,74], and Mo and V tend to stabilize the r phase in CrFeCoNi, CrFeCoNiAl, CrFeCoNiCu and CrFeCoNiAlCu alloys [42,75]. There are also a few fcc-stabilizers, including Ni, Cu and Co. Co has less capability of promoting fcc phase formation since its fcc crystal structure is only stable at higher temperatures. In contrast to Ni, which tends to promote the formation of single-phase fcc-structured solid solution, the addition of Cu, in spite of the fact that it can also stabilize fcc structure, will normally cause the formation of an additional fcc-structured phase due to the tendency of phase separation because of its positive enthalpy of mixing with many common elements. This effect has already been observed in a few alloy systems, including CrFeCoNiCu [41], CoCrFeMnCu [30]. In addition to the compositional effects, the processing condition was also an important factor that could affect the phase stabilities. Singh et al. [25] found that the microstructures of CoCrFeNiAlCu alloy are strongly cooling-rate dependent. High cooling rates favor the formation of nano-scaled polycrystalline

Fig. 1. (a) All the possible quaternary, ternary, binary, and pure metal subsets of the quinary CrMnFeCoNi high-entropy alloy. After casting, homogenization, rolling and annealing, those that are single-phase fcc are identified in red, while those that are multi-phase or have a different crystal structure are identified in black; Back-scattered electron images showing multiple phases microstructure of CrFeNi (b) and single phase microstructure of CoNi (c) equiatomic alloys after rolling and annealing. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Adapted from [32].

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phases while low cooling rates lead to the formation of typical dendritic microstructures due to elemental segregation. Grain refinement and suppression of elemental segregation were also observed in CoCrFeNiAl alloy by increasing the cooling rate [26]. The effects of annealing on the as-cast microstructures of CoCrFeNiCuAl were investigated by Zhang et al. [28] and Lu et al. [29]. Both results show that the fcc phase becomes the dominant phase as the annealing temperature increases. Other than affecting the relative volume fractions of each existing phase, formation of new phases can also be initiated during annealing. Two typical examples are CoCrFeNiTi and CoCrFeNiTiAl [22], in which annealing at 1000 °C for 2 h leads to the formation of FeTi phase. In CoFeNiCuAl alloy [23], which solidifies as bcc + fcc structure under as-cast condition, the phase component is not changed if the annealing temperature is below 600 °C. However, Cu-rich precipitates start to form in the dendrite region when annealed at 700 °C. Many needle and spherical precipitations with increasing size homogeneously distribute all over the dendrite region at annealing temperature higher than 700 °C. The microstructural evolution during thermomechanical processing was extensively investigated for the CoCrFeNiMn alloy and its fcc-structured equiatomic subsets [32]. The fcc single phase solid solutions are not changed by nearmelting temperature homogenization after casting. The good phase stability of the CoCrFeNiMn was further confirmed by Otto et al. [24], which showed that the single-phase nature remained after annealing at 900 °C for 500 days. This alloy is unstable and forms second-phase precipitates at lower temperature annealing. Crrich-phase (mostly in grain boundaries) forms at 700 °C, and three different phases (L10-NiMn, B2-FeCo and a Cr-rich bcc phase) precipitate at 500 °C. 2.3. Theoretical studies of the phase stability of CSAs Solid solutions generally have small heats of formation. However, it may be the case that the multicomponent disordered alloys are themselves unusually stable. This stability can be displayed by the values of their enthalpies of formation (DHf). Here we show the values of the enthalpies of formation for several 5 and 4component alloys in the fcc, hcp and bcc crystal structures in order to gain more insight into the remarkable stability of these alloys. The calculated heats of formation shown in Table 1 include the fcc alloys studied by Otto et al. [30], i.e., MnCrFeCoNi, CrMnFeTiNi, MoMnFeCoNi, VMnFeCoNi, CrMnVCoNi, and CrMnFeCoCu. Although the composition of the alloys were carefully chosen to follow Hume-Rothery rules, only the base alloy, CrMnFeCoNi, formed a single-phase solid solution, while the rest presented precipitation of intermetallic phases. It is worth noting that the entropy of mixing remains unchanged for all these alloys. There are a few important features to point out from the values in Table 1. Firstly, according to the calculation, there is only a small energy

Table 1 Enthalpy of formation of several multicomponent equiatomic alloys. Alloy

DHf (meV/atom)

Single-phase

CrFeCoNi [fcc] CrFeCoNi [hcp] CrMnFeCoNi [fcc] CrPdFeCoNi [fcc] CrMnFeTiNi [fcc] VMnFeCoNi [fcc] MoMnFeCoNi [fcc] CrMnVCoNi [fcc] CrMnFeCoCu [fcc] NbMoTaRe [bcc] VNbMoTaW [bcc] NbMoTaW [bcc]

76.678 72.926 91.119 159.735 87.789 4.276 215.824 48.694 190.024 90.836 22.694 67.375

Yes N/A Yes Yes Yes No No No No Yes Yes Yes

difference between the DHf values of the fcc and hcp phases of CrFeCoNi, with HCP being slightly favored. While this finding appears to contradict experiment, the precise values may well be dependent on the use of a single instantiation of a SQS supercell, without any attempt to average over different supercell sizes and different SQS realizations [76]. Secondly, some DHf values of alloys that are known to decompose into multiple phases are smaller than those of alloys that demonstrably form single-phase solid solutions. For instance, the base alloy, single-phase CrMnFeCoNi, has a larger (less favorable) DHf than CrMnVCoNi, which phase separates. Also, it is worth noting that alloys such as CrFeCoNiPd that have a larger size mismatch of its atomic species form a single phase, and have a larger DHf than alloys that have a smaller size mismatch, such as CrMnFeCoCu. Lastly, all the fcc alloys presented in Table 1 have positive values of DHf. These findings clearly indicate not only that HEA solutions are not unusually stable, but also the lack of predictability that the values of the enthalpies of formation and the Hume-Rothery rules provide for this type of alloys. Taken together, these results clearly indicate that the T = 0 K enthalpy of mixing of the solid solution phase is not, by itself, a useful predictor of HEA formation and that a proper treatment of competing phases and finite temperature effects is called for. In order to gain further understanding of the stabilization of these alloys, one can also look at the atomic displacements from the initial fcc lattice configuration upon relaxation. Fig. 2 helps visualize the effect of the internal coordinates relaxation by displaying the magnitude of the displacement of each atom from its original (perfect fcc lattice) position. The displacement of the atomic positions is given as a percentage of the nearest neighbor (NN) distance for each alloy. We can observe from the cases plotted in Fig. 2 that the atomic displacements for the fcc alloys are quite large reaching 6% of the NN distance for the 5-component alloys. However, such relaxation only affects the values of the energies of formation by less than 5 meV/atom. One remarkable feature of the Cr-containing alloys is that the Cr atoms present a much larger displacement than the other atomic species, which is present in both the fcc and the hcp structures. This trend could be attributed to the larger size of the Cr atoms compared to other atomic species. However, in the CrFeCoNiPd alloy, the Cr atoms are the ones that display the largest displacements despite the Pd atoms being larger. The fact that the largest displacements do not stem from the largest atoms indicates that factors other than size play an important role on the stability of these alloys. This interesting trend of the Cr displacements may be linked to the different magnetic alignments of the atoms in the different alloys and it may show the key role of the magnetic interactions on the stability and structural properties of HEAs. Cr atoms tend to be antiferromagnetic to neighbor Cr atoms while also being antiferromagnetically aligned to neighbor Fe atoms. This can cause frustration that is likely responsible for the large displacement of the Cr atoms. On the other hand, Pd atoms, although larger, present a very small magnetic moment. It is worth noting a few striking differences between the fcc and bcc alloys regarding the atomic displacements and enthalpies of formation. In contrast to the trends present in the fcc alloys, the heats of formation of the bcc alloys are negative. This is surprising if one takes into account the larger atomic size mismatch present in the refractory alloys. In addition, the bcc alloys present larger atomic displacements than the fcc ones as displayed in Fig. 2. The larger magnitude of the atomic displacements is translated into a larger change in the enthalpies of formation due to the internal coordinates relaxation. The bcc alloys present a change in the values of DHf due to atomic relaxation on the order of 20 meV/atom, while the change for the fcc alloys is below 5 meV. It is also important to point out that the magnitude of the atomic displacement is roughly the same for all species present in the

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Fig. 2. Atomic displacements due to relaxation expressed as the percentage of the nearest neighbor distances. (a) Displacements in the CrFeCoNi alloy in the fcc crystal structure. (b) Displacements in the CrMnFeCoNi alloy in the fcc crystal structure. (c) Displacements in the CrPdFeCoNi alloy in the fcc crystal structure. (d) Displacements in CrFeCoNi (fcc) and displacements in VNbMoTaW (bcc).

bcc alloys, contrasting the behavior of Cr in the fcc alloys. This is consistent with the non-magnetic nature of the refractory alloys. 2.4. Predictive models: semi-empirical and ab initio approaches Determining the stability or phase diagram of a multicomponent alloy is very challenging. From a thermodynamic point of view, in order to assess the stability of an alloy one should calculate the Gibbs free energy of every competing phase; as the number of components in the alloy increases the number of possible competing compounds also increases. For instance, for a 5component alloy one would need to consider all the binary, ternary and quaternary compounds that can result from combinations of those 5 elements and compute the Gibbs free energy for each compound in different compositions, lattices and decorations. Since these calculations at present represent an insurmountable task, semi-empirical methods have played an important role in providing guidance in the search of new HEAs. Several semi-empirical approaches have been developed in order to predict the stability of HEAs [65,67–70,77,78]. These approaches largely follow the classic Hume-Rothery rules for solid solution formation and are typically based on atomic-size differences (d), electronegativities, and enthalpy of mixing (DHmix) of the multi-component alloys [67–70]. The main idea behind these models is that small size differences and small average heats of mixing are required for the formation of solid solutions. Alloys with large, favorable heats of mixing and large d tend to form intermetallic compounds. However, while small average DHmix and d are requirements for solid solutions, they are not sufficient or pre-

dictive for solid solution formation. For instance in the work of Otto et al. [30], substitutions that largely leave the values of DHmix and d unchanged cause several HEAs to form second phases. Similarly, Hume-Rothery substitutions – replacing one element with another of a similar size, electronegativity and crystal structure – similarly fail to preserve the single phase. Although semi-empirical models have been helpful in predicting the stability of HEAs, new models with more reliable predictive capabilities are needed in order to guide the experimental search of new alloys. The ideal model would be completely ab initio, not requiring any experimental input and would be able to narrow down potential single-phase candidates. Recently, an approach based on first principles calculations has been proposed, which makes remarkable strides toward this goal [66]. This model can predict which combinations of elements are most likely to form a single-phase HEA by applying a simple criterion based on enthalpy considerations. This approach utilizes ‘‘high-throughput” DFT calculations [66] of the energies of formation of binary compounds and requires no experimental or empirically derived input. The model correctly accounts for the specific combinations of metallic elements that are known to form single-phase HEAs, while rejecting similar combinations that have been tried and shown not to be single-phase. This model predicts that a set of elements will form a singlephase alloy if the enthalpy of formation of all the possible binary compounds formed by combinations of these elements fall within a certain range. This range is such that the compounds are neither too stable, leading to precipitation of that phase, nor too unstable, indicating immiscibility of the constituent species. This range can

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be set by the ideal entropy of mixing and the annealing temperature of the alloy. The optimal range for the values of the heats of formation is rationalized as follows. The minimum value of the range is determined by the ideal entropy of mixing as Tann  DSmix, where Tann is the annealing temperature used in the experimental setting. The upper limit is set by the largest value of DHf for which the alloy does not phase-separate due to the immiscibility of any pair of elements. The upper limit of the enthalpy range (37 meV) was chosen to include all known singe-phase alloys. Moreover, Tann can be replaced by a critical temperature (Tcrit) below which diffusion is sufficiently slow that the enthalpic driving force is insufficient to result in phase separation. Utilizing different limits for the range of DHf can lead to different allowed combinations of elements that will form a single-phase alloy. It is important to note that the values of DHf should be compatible with the annealing temperature required to homogenize the alloy. For instance, the critical temperature can be a fraction of the average melting temperature, TM, of the constituent metallic species. Such a criterion is already consistent with the annealing temperature used experimentally. For example, the annealing temperature used by Otto et al. (1000 K) for CoCrFeMnNi [24] corresponds to a Tcrit = 0.55  TM, while the same Tann for CoCrFeNiPd [71] corresponds to Tcrit = 0.54  TM. For the bcc alloys, MoNbTaVW and MoNbTaW, Tann = 1673 K [42], which corresponds to a Tcrit of 0.56  TM and 0.53  TM, respectively. The model correctly accounts for the known (single-phase) HEAs, as well those that have been tried and shown not to be single-phase and it provides predictions of the element combinations that are most likely to form HEAs beyond those already known. Although this model only considers the formation of binary compounds it accurately predicts the formation of single-phase alloys, and it identifies closely related compositions that form multiple phases. In order to assess the formation of single-phase alloys, a 30x30 enthalpy matrix was constructed [66]. This matrix contains the lowest enthalpies of formation of all binary combinations of the elements: Mg, Al; all the 3d, 4d, and 5d transition metals, except Tc and Lu; only La was included from the Lanthanides. A 19x19 subset is shown in Fig. 3. Most of the values of the enthalpies of formation of binary compounds included in the enthalpy matrix were obtained from the binary alloy library of Curtarolo et al.[79] and the alloy database of Widom et al. [80]. Entries in the enthalpy matrix represent the DHf of the lowest energy structure of each binary compound relative to phase separation into pure elements. It is worth noting that each entry involves the

results of hundreds of DFT calculations for binary compounds, considering different compositions, lattices, and elemental decorations within each lattice type. Therefore, the full enthalpy matrix represents the distillation of tens of thousands of such calculations.

3. Electronic, magnetic and transport properties While phase stability is very important for alloy development and processing, understanding their physical properties are vital next steps in developing CSAs for specific applications. Moreover, the physical properties themselves are interesting from a scientific viewpoint. In particular, they provide insights into how control of chemical complexity – through the number and types of alloying elements – can be used to manipulate the underlying electronic structure. Furthermore, they provide insight and verification of theoretical modeling to understand the underlying bonding mechanisms, which are ultimately responsible for the exceptional properties of many CSAs: e.g. combined strength and ductility at cryogenic temperatures [18,19] and resistance to defect creation under irradiation [81]. A review of some recent progress in providing an ab initio understanding of the electrical, magnetic and thermal transport properties of CSAs and the theoretical methods that are required to accomplish this are the subject of this section.

3.1. Disorder effects in complex solid solutions The electronic, magnetic and vibrational properties of random solid solutions are fundamentally different from pure metals and ordered intermetallic compounds. The lack of long-range order means that Bloch functions, magnons and phonons are no longer eigenstates of the electronic, magnetic and vibrational subsystems. Even for the idealization of a random solid solution (Fig. 4a) where one imagines that the alloying components are randomly distributed on the sites of some idealized underlying periodic lattice, the chemical randomness results in the complete loss of periodicity. Beyond this, the fact that every site is in a different local chemical environment results in additional local lattice distortions (displacement fluctuations Fig. 4b), possible magnetic moment fluctuations that can either be collinear (Fig. 4c) or noncollinear (Fig. 4d) or all of the above (Fig. 4e). All these effects, together with possible departures from ideal randomness [concentration fluctuations or short ranged order (SRO)] (Fig. 4f), further

Fig. 3. (a) Enthalpy matrix displaying calculated enthalpies of formation of the lowest energy structures of binary compounds relative to phase separation into pure elements. (b), Prediction of multiple and single-phase alloys. Adapted from [66].

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Fig. 4. Illustration of the effects of disorder on solid-solution alloys. (a) Idealization of a 5-component equiatomic ABCDE solid-solution alloy; (b) displacement fluctuations; (c) magnetic moment fluctuations – collinear; (d) moment fluctuations - noncollinear; (e) a + b + c + d; (f) additional concentration fluctuations - e.g., short-ranged order.

modify the local symmetry and enhance the effects of disorder on the fundamental electron, magnetic and vibrational excitations. In general, experimental probes of the underlying excitations in disordered alloys probe configurationally averaged observables, i.e. properties averaged over all possible configurations fni g of the

atomic species consistent with their concentrations as implied by the angle brackets Ofni g illustrated in Fig. 5a. Theoretically, addressing the need to account for all the effects of disorder on configurational averaged observables that are illustrated in Fig. 4 poses a major challenge to ab initio DFT-based electronic structure meth-

Fig. 5. Illustrations of (a) the concept if configurational averaging i.e. averaging over all possible configurations consistent with the concentrations of the alloying species, and (b) the CPA single-site condition for self-consistently determining the scattering properties of the optimal (CPA) effective medium, for the case of an equiatomic 5-component alloy.

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ods. Ab initio methods for treating disordered alloys fall into two classes: firstly, those based on the coherent potential approximation (CPA) which allows the direct calculation of the configurationally averaged single site properties within the context of DFT – albeit within a single site or mean field approximation; and secondly, those based on supercell models in which the alloying elements are randomly distributed [82] on some underlying unit cell that is then periodically reproduced. This artificial reinstatement of translational invariants then makes it possible to used standard DFT electronic structure codes – provided that the number of atoms in the supercell is not too large (ideally < 100 and rarely more than 300). In terms of the formal development of the theory of electronic states of disordered systems, the CPA [83,84] can be viewed as sitting atop a hierarchy of effective medium theories that include the often used rigid band model (RBM) [85,86] and virtual crystal approximation (VCA) [87,88]. In the case of substitutional disorder, the notion that underlies such theories is to replace the calculation of the configurationally averaged properties of the real system (Fig. 5a) with the calculation of the properties of some equivalent ordered system [Fig. 5a (left)] that is chosen to give the best representation of the configurationally averaged properties of a real disordered alloy. For the RBM this system is chosen as one of the endpoint elemental metals and the effects of alloying are confined to the addition (subtraction) of electrons accordingly, as the electron-to-atom ratio of additional elements is larger (smaller) than that of the host. For the VCA, the effective medium is chosen as the average of the electron-ion potentials P of the pure elements V VCA ¼ a ca V a . For DFT, V VCA is then nothing more than the average ionic potential (or pseudopotential). In the CPA (Fig. 5b) the effective medium is chosen self-consistently. Although CPA was first conceived within a single-level tightbinding model [83,84], its implementation within multiple scattering theory methods for solving the Kohn-Sham equations of DFT results in a fully first principles theory of the electronic structure of solid solution alloys [88–91] and DFT [92,93]. Because of the close connection to the MST based Korringa-Kohn-Rostoker (KKR) band structure method for ordered systems, it is conventional to refer to the corresponding theory of disordered alloys as the KKR-CPA.

3.2. Ab initio KKR-CPA electronic structure methods Within MST based KKR-CPA methods, the CPA effective medium is represented as an infinite array of scattering centers each characterized by some unknown scattering t-matrix tCPA [Fig. 5b (right)]   that is determined from the condition snn t nCPA ; ft iCPA g; e ¼ P a;nn ðt n ; ft i g; eÞ where snn is the scattering path matrix for a ca s CPA CPA the site n. Just as the t-matrix can be thought of as converting an incoming wave and outgoing wave at a single site, the scattering path matrix snm ðeÞ for a collection of scatterers describes the conversion of an incoming wave at some site n into outgoing wave at some other site m, taking into account all possible scattering paths throughout the systems that begin at site n and end at site m. In terms of the scattering path matrix, the CPA condition that replaces the effective scatterer at the n-th site (t CPA ) by a real scatterer (t a ) produces no additional scattering at that site when averaged of all possible occupancies (species) [Fig. 5b]. Because of the close connection to the KKR band structure methods for solving the Kohn-Sham equations for pure metals and ordered compounds, it is conventional to MST implementation as KKR-CPA methods. Indeed, in the limit of a pure metal and ordered compounds, the equations of KKR-CPA reduce to those of KKR.

Once t CPA is has been determined, the basic electronic proper!

 a ð r Þ,  a ðeÞ, charge, n ties, single site electronic densities of states, n !

 a ð r Þ, are given by: and magnetization, m

 a ðeÞ ¼ 1=pI n

Z

Xn

!

 a ð r Þ ¼ 1=pI n !

 a ð r Þ ¼ 1=pI n

Z

! !  3 Ga r n ; r n ; e d r

ð1Þ

! !  f ðe  lÞGa r n ; r n ; e de

ð2Þ

Z

!

!

f ðe  lÞGa ð r n ; r n ; eÞde

ð3Þ

where Xn is the volume of the Voronoi polyhedron about the n-th !

!

site and Ga ð r n ; r 0n ; eÞ is the corresponding CPA single site averaged green’s function

! !  Xh ! i a ! a ! a;nn a ! LL Ga r n ; r 0n ; e ¼ Z aL ð r n Þs 0 Z L0 ð r n Þ  Z L ð r > ÞJ L ð r < ÞdLL0 :

ð4Þ

LL0

!

!

where, Z La;n ð r n Þ and JLa;n ð r n Þ are the regular and irregular solutions of the single particle Schrödinger equation for (DFT) effective electron ion potential within the Voronoi polyhedron at the n-th site. A further quantity of interest is the Bloch spectral function (BSF), !

! !

!

e e; k ; k Þ. Formally, AB ðe; k Þ is the density of states AB ðe; k Þ ¼ 1=pI Gð per k-point. In the limit of an ordered system, the BSF reduces to a !

set of delta-functions at the band energies, em ðk ), namely: ! ! ! P ABOrd ðe; k Þ ¼ m dðe  em ðk ÞÞ. Thus plots of AB ðe; k Þ can be thought of as a generalization of the band structure of ordered systems to disordered alloys. A key feature of the KKR-CPA follows from the observation that the t-matrix of the effective medium corresponds to complex energy dependent potentials with the result that the effective medium is adsorptive giving rise to the finite electron mean free path or electronic lifetimes which manifest themselves in a smearing of the band structure in energy and wave vector. 3.3. Electronic and magnetic structure of fcc 3d-transition metal CSAs The properties of fcc 3d-transition metal based CSAs, including HEAs, are greatly affected by the tendency of the mid 3d-transition metal elements, Cr, Mn, Fe, Co and Ni to develop magnetic moments. The situation is very different in the bcc HEAs, such as NbMoTaW and VNbMoTaW that involve only early 3d, 4d, and 5d transition metal elements where magnetism is generally absent. Results of KKR-CPA calculations of the configurationally averaged local (single-site) magnetic moments for fcc equiatomic CrFeCoNi and CrMnFeCoNi alloys are shown in Fig. 6. The calculations were performed by using the local density approximation (LDA) and experimentally determined lattice spacing. The LDA potential was taken to be of muffin-tin form. For both systems, the local moments are very lattice spacing dependent and involve competing ferromagnetic (positive moments) and antiferromagnetic (negative moments) interactions. Furthermore, for CrMnFeCoNi it is possible to find multiple solutions where Cr and Mn sites have either positive or negative local moments. Clearly, at the equilibrium lattice spacing, both systems are magnetic. However, it also turns out the equilibrium lattice spacing, and therefore the local moments at that lattice spacing, depend strongly on the DFT exchange correlation function used in the calculations. As an example, for CrFeCoNi the LDA equilibrium lattice spacing is essentially coincident with the disappearance of the local moments, whilst for the Generalized Gradient Approximation and the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional, the DFT lattice spacing is close to the experimental value – an essentially identical conclusion found using supercell methods.

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Fig. 6. Calculated KKR-CPA lattice parameter dependence of the local, total and species resolve magnetic moments in equiatomic (a) CrFeCoNi and (b) CrMNFeCoNi high entropy alloys. The vertical lines denote the measured equilibrium lattice parameters and the solid dots denote the averaged moments based on a 2048 atom supercell model of Fig. 7.

The total and species dependent local moments of 11 equiatomic binary, ternary, quaternary and quinary alloys comprising various combinations of 3d-transition metal elements Cr, Mn, Fe, Co and Ni plus the 4d-tranition metal element Pd are shown in Table 2. From this table, it is clear that these alloy systems can be divided into two classes: 1) those with all of the species resolved moments being positive, i.e., the coupling is all ferromagnetic (FM), and 2) those with a mixture of positive and negative moments, i.e., they involve a mixture FM and antiferromagnetic (AFM) couplings. Overall, Fe, Co and Ni tend to couple to one another FM whilst Cr and Mn tend to couple AFM with the result that alloys containing only former elements have large net moments (generally falling on the well-known Slater Pauling curve) while those involving a mixture have much smaller net moments. Experimentally, it is known that many of the alloys that involve mixed FM/AFM coupling (e.g. CrCoNi and CrMnCoFeNi) do not order magnetically at any temperature [94]. These differences in the magnetic structure have a profound effect on the underlying electronic structure. This is illustrated in plots of the calculated KKR-CPA densities of states (DOS) and BSF along the major symmetry direction in the Brillouin zone (BZ) shown in Fig. 7. To illustrate the point that the BSF reduces to the usual band structure for the case of an ordered solid, the BSF of pure Ni is shown in the upper panel which is obtained by adding a small imaginary part (1 mRy) to the electronic energy which has

the effect of broadening the ideal d-function band structure peaks into Lorentzians. For the equiatomic alloys FeNi, CoFeNi, CrFeNi and CrMnCoFeNi, the effects of disorder giving rise to substantial disorder broadening in both the DOS and BSF are clearly evident. As noted previously [94–96] in which all elements couple FM, only the minority spin channel suffers significant disorder broaden, whist in alloys with mixed FM/AFM coupling both spin channels suffer significant disorder broadening both in energy and wavevector. The reason for this is easily understood. Upon alloying Ni with either Fe or Co (or both) the majority spin d-bands line up in energy resulting in very little disorder scattering as majority spin electron moves from a Ni to a Fe site. For the minority spin electrons the situation is very different in that the minority bands are split relative to the majority bands by an amount, 1 eV/lB, that is proportional to the size of the local moment for that element. Given that the local moment on Ni and Fe sites differs by more than 1.8 lB, this implies a scattering potential between Ni and Fe sites on the order of 1.8 eV which is a substantial fraction of the d-band width itself resulting in a large disorder scattering in the spin down channel. From this point of view, the addition of a FM coupling element will not significantly affect this situation, as is clearly the case for the DOS and BSF of NiFeCo shown in Fig. 7. For alloys with FM/AFM coupling again the situation is very different because, relative to the FM element, the spin-up and spin-

Table 2 Calculated total and species resolved magnetic moments of fcc CSA. The magnetic moments are given Bohr magnetons (lB). Alloy

Lattice Spacing (Å)

~ Total m

~ Ni m

NiFe NiCo NiPd NiCoCr NiCoMn NiFeCo NiFeMn NiFeCoMn NiFeCoCr NiFeCoCrMn NiFeCoCrPd

3.5826 3.5345 3.6954 3.5590 3.5977 3.5690 3.6160 3.5919 3.5715 3.5991 3.6730

1.611 1.141 0.536 0.269 0.223 1.602 0.184 0.470 0.661 0.272 0.627

0.682 0.645 0.831 0.147 0.237 0.671 0.157 0.291 0.283 0.137 0.384

~ Cr m

~ Mn m

~ Fe m

~ Co m

~ Pd m

2.540 1.636 0.241 0.162 0.809 2.497 1.736 1.807 0.654 0.139 1.265

1.298

2.167 1.930 1.840 2.455

0.825 1.243 1.637 2.132 1.228 1.084 0.821 1.464

0.096

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Fig. 7. Density of states (DOS) and Bloch spectral functions (BSF) along high symmetry directions for equiatoimic fcc NiFeCo, NiFeCr and NiFeCoCrMn alloys. The two left (right) panels show resolved DOS and BSF for spin up (down) electron respectively. Adapted from Refs. [81,94].

down bands are reversed with the spin-down band lying lower in energy than the spin-up band. As a result, both spin-up and spindown bands experience strong disorder scattering as they move from a FM to an AFM site. The result of this site-to-site moment reversal is a strong disorder scattering in both spin channels as is seen in Fig. 4 for CrCoNi and CrMnFeCoNi. Hence the stark contrast in electronic structure with the simple replacement of Fe in FeCoNi by Cr in CrCoNi.

and give rise to a low residual resistivity. On the contrary, in the mixed FM/AFM systems, the disorder smearing in both spin channels is a substantial fraction of the BZ dimension in both spin channels giving rise to a substantially higher residual resistivity. While this disorder smearing analysis is sufficient to give a qualitative understanding, a more complete calculation will require direct calculations with formal transport theories. 4. Mechanical properties and strengthening mechanisms

3.4. Disorder effects on the electrical resistivity Whilst there have been no published calculations of transport properties for CSA and HEA systems discussed here, measurements of the temperature dependence of the electrical resistivity of equiatomic CoNi, FeNi, FeCoNi, CrCoNi, CrFeCoNi, CrMnFeCoNi, and CrMnFeCoPd [94] is quite consistent with the picture of the effects of disorder on the transport properties in that they are sharply divided into classes, those with low resistivity and high resistivity. This is particularly the case for the zero temperature or residual resistivity, qR . For the FM coupled systems, CoNi, FeNi, FeCoNi residual resistivity values fall in the range 2 < qR < 10.4 lX cm while the values of the remaining alloys are in the range 77 < qR < 127 lX cm. Theoretically, in disordered alloys, the residual resistivity should be directly related to the disorder-induced electron mean path (MFP) or alternatively the electron lifetime since at zero temperature phonon scattering is frozen out. As such, the residual resistivity should be related to the degree of disorder smearing in the BSF at the Fermi energy, el

lMFP  1=Dk where Dk is a measure of the smearing averaged over the Fermi surface. Given that for the FM coupled alloys, smearing of the Fermi surface in the spin-up channel is small, a small fraction of the BZ dimension, these electrons can act as a short circuit

In addition to simple microstructures and interesting physical properties, a very exciting finding in the research community of high entropy alloys is the unusual mechanical properties, mostly observed in fcc single phase solid solution alloys such as the simultaneous increase of strength and ductility with decreasing temperature and the strong temperature-dependence of yield strength of CrMnFeCoNi alloy and its equiatomic subsystems [18,19,31, 36,38,40]. In the following sections, a few major points including solid solution strengthening, poly- and single-crystal deformation mechanisms, engineering applicability, and alloying design strategies originating from HEA concepts will be separately discussed. 4.1. Solid solution strengthening Solute atoms dissolved in solvent affect the mechanical behavior of metals to a varying extent depending on the amount and type of the solutes. Various elastic, electrical and chemical interactions could take place between the stress fields of the solute atoms and the dislocations in the matrix when two or more elements are combined such that a single-phase microstructure is retained and, of these, elastic interactions are believed to be most important. The dislocation mobility in the solvent lattice can be affected by the

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discrete foreign (solutes) atoms, depending on the energetics of, and the force resulting from, the elastic interaction of a dislocation with the strain field of a single solute atom. This is the conventional picture of solid solution strengthening [97–105]. Two widely-accepted significant contributions to this interaction are atomic size misfit and modulus mismatch between the solute and solvent. In reality, however, a dislocation moving through the solvent lattice could simultaneously interact with multiple solute atoms, and thus the net force exerted by all the solute atoms needs to be taken into account. For dilute solutions, the effect of a solute atom in the crystal structure of the solvent can be described unaffected by the other solutes. Taking into account the Friedel separation between strong obstacles encountered by a dislocation, Fleischer developed a description in which the critical shear stress to overcome obstacles varies as the square root of the solute concentration [106–108]. For concentrated solutions, instead, the solute atom-dislocation interactions are assumed to be intermediate- or far-field. Based on this, Labusch statistically treated the dislocation moving through an array of obstacles with a distribution of interaction strengths and obtained a critical shear stress that varies as the two-thirds power of solute concentration [109–111]. Compared to the decades-old theories of strengthening in binary solid solutions [112–120], the mechanical behavior of multicomponent solid solutions is relatively poorly understood. Gypen and Deruyttere [121] proposed a methodology that can be applied to calculate solid solution strengthening in multi-component alloys. A main assumption was that solute atoms possess no or only slight interaction with one another. Sagues and Gibala [122] did not take into account the different atomic size misfits produced by the continuously deformed crystal lattice when different solutes are present. The method is then applicable to multi-component alloys where a principal element is predominant. One of the specialties of CSAs is that the constituent elements are present in equal or near-equal atomic proportions. This could break down the picture of a dislocation moving through a solvent lattice and encountering discrete solute obstacles as the solute concentration and compositional complexity increase. Since there is no dominant constituent (i.e., no ‘‘solvent” or ‘‘solute”) in such equiatomic alloys, this new class of alloy is not a simple extension or extrapolation from the dilute solution limits but rather a distinct new state akin to a stoichiometric compound with fixed atomic ratios, albeit disordered. Rather than considering a dislocation moving through a solvent lattice and interacting with discrete solute atoms, it may be more appropriate to envisage the dislocation as moving through a mythical ‘‘average solvent” or ‘‘effective medium”. By adapting Gypen and Deruyttere’s approach and taking into account the continuous crystal lattice distortion and variable elastic distortion, recently, Toda-Caraballo et al. [123] presented a model that can be used to compute the solid solution hardening in HEAs. In this model, a matrix proposed by Moreen [124] to describe the mean value of the distribution of nearest neighbor distances, also called interatomic spacing distance, between atoms of different species was adopted. The fluctuations of each individual interaction distance were, depending on the combination of atoms and its coordinates in the lattice, was computed pair-wisely and captured in the matrix. And the frequency of the occurrence of each pair-wise interaction was considered. A packing factor for different crystal structures was also applied to account for the high packing density directions. Their model predicted strong strengthening effects of Cr in FeNiCo-based alloys and softening effects of Fe in CoCrNi-based alloy. There are also a few experimental studies that were conducted to understand the possible factors that could affect the strengthening effects in multi-component equiatomic alloys. Wu et al. [8] assessed the hardness differences at a given grain size, which pro-

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vide a measure of the strengthening effects of different solutes, of a family of alloys which are all possible equiatomic subsystems (namely, pure metal, binary, ternary, and quaternary alloys) of the quinary CrMnFeCoNi high-entropy alloy that can form singlephased solid solutions. Their results show that, among the five constituent elements, Cr possesses the strongest strengthening effects while Co causes only minor hardening. The yield strengths of this family of alloys were later compared at both ambient- and lowtemperatures and similar element-type effects were observed [31]. Size- and modulus- mismatch between the additive elements were assessed to identify possible mechanisms of the strengthening differences. In terms of the Seitz radii, the atomic sizes [125] of Fe, Ni, Co, Cr, and Mn are 1.411, 1.377, 1.385, 1.423, and 1.428 Å, respectively. The largest difference in these atomic sizes is only 3.7% suggesting that the difference in the hardness of the alloys is unlikely to be due to size misfits. In contrast, the Young’s moduli [126] of Fe, Ni, Co, Cr and Mn are 211, 200, 209, 279, and 198 GPa, respectively, with the largest difference yielding a misfit 40.9% between Cr and Mn. Therefore, although it is difficult to extrapolate the general principles governing strength beyond the binary alloys, the observed Cr-strengthening effects might be largely due to the modulus mismatch between Cr and the other elements. Thus the nature of the added element plays an important role in the hardening of equiatomic alloys. Wu et al. [127] also conducted a statistical study to investigate the alloying (type and number) effects on the mechanical properties of the same family of fcc equiatomic alloys. In their study, various physical and microstructural parameters (e.g., melting temperature, lattice parameter, density, Poisson’s ratio, shear modulus, maximum size and modulus mismatch, and annealing twin density) of CrMnFeCoNi and its sub-alloys were analyzed to establish statistically significant correlations with basic mechanical properties (e.g., yield and ultimate tensile strength, uniform elongation and strain hardening capability). Scatter plots were made for each pair of parameters to visualize the potential correlations followed by calculations of the correlation coefficient and level of significance using standard statistical procedures. Among the factors that were studied, yield strengths were found to correlate in most significant ways with the maximum modulus mismatch and only mildly with the number of alloying elements. 4.2. Deformation mechanisms Although both fcc-and bcc-structured single phase HEA solid solutions have been explored, the understanding of deformation mechanism of HEAs mostly relied on studies on the fccstructured single phase alloy, CrMnFeCoNi, and its sub-alloys. An initial assessment of the tensile properties of the quinary equiatomic CrMnFeCoNi by Gali and George [16] have shown that both the strength and ductility of this alloy increase with decreasing temperature, with the largest differences occur between 293 and 77 K. Wu et al. [31] later investigated all the fcc-structured single phase subsets which are shown in Fig. 1 of the CrMnFeCoNi alloy and found that their flow stresses are temperature dependent to varying degrees. Fig. 8 shows the representative stress-strain curves of the model CrMnFeCoNi and CrFeCoNi alloys at various temperatures. Further understanding of these temperaturedependent behaviors were provided by Otto et al. [128] in which microstructures of the CrMnFeCoNi alloy were characterized as a function of strain using TEM. During the initial stages of deformation (up to 2% strain), deformation in the CoCrFeNiMn alloy occurs by planar dislocation glide on the normal fcc {1 1 1}h1 1 0i slip system at all the temperatures (between 77 and 1073 K). The observation of both undissociated ½h1 1 0i dislocations and numerous stacking faults implies the dissociation of some perfect dislocations into 1/6h1 1 2i Shockley partials. At increasing tensile

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Fig. 8. Engineering stress-engineering plastic strain curves of (a) CrMnFeCoNi and (b) CrFeCoNi alloys under quasi-static tension condition with an engineering strain rate of 103 s1. Adapted from [24,31].

strain to 20% at cryogenic temperatures, nanoscale deformation twins were observed. Secondary twinning systems were activated at even larger strain (38%). Thin nano-twins formed during deformation are able to serve as obstacles to dislocation motion and this strengthening effect is more pronounced as individual twins are arranged in bundle. Nano-twins can also interact with dislocations and cause the accumulation of a high density of sessile dislocations within the twin lamellae, leading to increased twin strength with deformation. These accumulated dislocations are potentially effective barriers to dislocation motion and can provide additional strengthening within the induced mechanical twins and increase the critical stress required to induce plastic deformation across the twins. Deformation twins can also create new interfaces and decrease the mean free path of dislocations during tensile testing (‘‘dynamic Hall–Petch”) [129–138] and thus produce a high degree of work hardening and a significant increase in the ultimate tensile strength. This increased work hardening prevents the early onset of necking instability and is a reason for the enhanced ductility observed at low temperatures (e.g., 77 K). Twinning in this alloy has also been observed after severe plastic deformation at room temperature by high-pressure torsion as well as after rolling [139]. The promotion of deformation twinning was also achieved by adding 0.5 at.% carbon to the CrMnFeCoNi alloy [38]. This twinning-induced strength and ductility was similar to other traditional materials, e.g., copper thin films [140–142] and the recently developed twinning-induced plasticity (TWIP) steels [143–146]. While twining can explain the good combination of tensile strength and ductility at cryogenic temperatures, it cannot explain the large increase in yield strength with decreasing temperatures in the CrMnFeCoNi high-entropy alloy since it was not observed in the early stages of plastic deformation. Yield strengths of fcc metals are normally temperature insensitive due to their negligible Peierls-Nabarro barriers [147–150]. This temperature sensitivity could be increased when solute atoms were added to form binary fcc alloys, such as Cu-Mn [151,152], Cu-Al [153–155], Cu-Ge [156– 158], Cu-Zn [159,160], Cu-Ni [161], Au-Ag and Al-Mg [162] alloys. The general trend is the increase of yield strength at lower temperatures and the shifting of both thermal and athermal portions of the yield strength vs. temperature curves to higher values as the solute concentration increases [163,164], suggesting an increase in the number of both short-range dislocation obstacles that can

be overcome by thermal activation and longer-range obstacles that cannot. As pointed out earlier, as a special class of alloys, CSAs can be treated neither as pure metals because of the presence of multiple alloying elements nor as a simple extension or extrapolation from the dilute solid solutions because of the highly concentrated compositions in all elements. An experiment-based mechanistic understanding of the temperature sensitivity of the yield strength was provided by Wu et al. [31] where the mechanical properties of all equiatomic binary, ternary, and quaternary alloys based on the elements Fe, Ni, Co, Cr, and Mn that were previously shown to be single-phase face-centered cubic solid solutions characterized as a function of temperature. As shown in Fig. 9, some alloys, such as CrMnCoNi, CrFeCoNi, CrCoNi and FeNi, show stronger temperature dependence of yield strength than other alloys, such as CoNi and pure Ni. By ruling out some possible contributions, such as grain boundary strengthening and precipitate strengthening, their analysis suggests that the temperature dependence of the yield strength

Fig. 9. The temperature dependence of the 0.2% offset yield stress of the equiatomic alloys and pure Ni. Adapted from [31].

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in the alloys may be determined by Peierls-barrier-dominated lattice friction, with the height of the Peierl’s barrier controlled by thermal influences on the width of the dislocation. Their results also show that during the initial stages of plastic deformation (5– 13% strain, depending on material), the temperature dependence of strain hardening is due almost entirely to the temperature dependence of the shear modulus, indicating the athermal nature of dislocation multiplication, accumulation and interaction during the early stages of deformation. The above Peierls-stress model in Fig. 9 appears to be limited to the concentrated alloys with the most noticeable temperature dependence and obviously does not work for those adjacent to pure Ni in this plot. Wu et al. [165] has conducted tests with varying both temperature and strain rates, so that the activation volume can be extracted. Referring back to Fig. 9, the alloys showing the weakest temperature dependence have activation volumes of several hundreds of b3 with b being the Burger vector. The concentrated alloys at the top of Fig. 9 have activation volumes of several tens of b3, which are still higher than the corresponding value in single-element bcc metals. The athermal strengthening of all these alloys is found to be consistent from predictions of Labusch-type models, based on the works of Gypen and Deruyttere [121] and Toda-Caraballo et al. [123]. Consequently, the corresponding thermal activation process should involve local bowing of gliding dislocations in the forest of solute atoms, as shown in Fig. 10. High temperature deformation mechanisms for the Cantor alloy were also reported. Stepanov et al. observed discontinuous dynamic recrystallization associated with the nucleation and growth of new grains via migration of the initial grain boundaries during uniaxial compression at 873–1273 K [33]. Through the studies on the evolution of lattice strains, peak width, and intensities of several h k l reflections using in-situ neutron diffraction, Woo et al. [34] suggested the dominant deformation modes are dislocation glide and diffusion-controlled dislocation creep at 800 and

13

1000 K, respectively. Lattice strain in the equiatomic CrMnFeCoNi alloy was also measured by Wu et al. [35] using in-situ neutron diffraction to elucidate the elastic and plastic deformation behavior. Their results indicated that the anisotropic behavior, including the lattice strain and texture, is similar to that of traditional fcc alloys. 4.3. Single crystal plasticity The complicated stress state of individual grains, different operating slip systems operating near a grain boundary from those elsewhere, and the acting of grain boundaries as obstacles to dislocation movement in a polycrystalline make it difficult to mechanistically understand the deformation mechanisms such as the slip behavior in these materials. Interesting questions include: is the strength different in tension and compression, and does the critical resolved shear stresses (CRSS) obey the Schmid law? Single crystals of some alloys, including quaternary CrFeCoNi [166] and quinary CrMnFeCoNi alloy [167] were successfully grown to reveal the fundamental mechanical behavior of the equiatomic alloy. Both results show that the multi-component equiatomic alloys deform like typical fcc materials. Dislocations tend to glide on the closepacked planes, {1 1 1}, along the close-packed direction, h1 1 0i, making {1 1 1}h1 1 0i primary slip systems and the CRSS obeys the Schmid law (Fig. 11). Tension-compression asymmetry is also not observed. Unlike pure fcc metal, CRSS in both the quaternary and quinary alloys are found to be strongly temperature dependent, which is believed to arise from the variation of intrinsic resistance to gliding dislocations with temperature. To compare the single crystal results in Fig. 11 to those in Fig. 9, it should be noted that the polycrystalline behavior contains the Hall-Petch effects. Upon testing many grain sizes and removing the Hall-Petch effects in Fig. 7, the tensile yield strength can convert to the CRSS by the Taylor factor. These results from Fig. 9 agree well with the single crystal measurements in Fig. 11.

Fig. 10. Schematic illustration of the thermally activated processes in equiatomic alloys. (a) Long-range bowing process corresponding to large activation volumes (e.g., NiCo). (b) Short-range effects for low activation volumes (e.g., NiCoCr). Adapted from [165].

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Fig. 11. (a) The 12 possible independent slip systems for fcc materials; (b) predicted and (c) observed slip trace pattern for (1 0 0), (0 1 1) and (1 1 1)-oriented single crystals under spherical indentations at 293 and 77 K, respectively; (d) summary of possibly activated slip systems, maximum Schmid factor, 0.2% offset yield stresses and critical  orientation tested in compression at room temperature. DIC strain resolved shear stress of equiatomic CrFeCoNi single crystals; (e) Stress–strain behavior of the ½591 measurements (insets 1 to 4) show the onset of slip at the axial stress of r = 140 MPa. Inset marked 5 shows the residual strain field upon unloading. Adapted from [166,167].

4.4. Potential application of HEAs as structural materials When selecting a material for practical applications such as load bearing, one is normally faced with a strength-toughness dilemma. Materials engineers can apply many strengthening procedures, such as alloying, precipitating, and deforming, to enhance loadbearing capacity, but almost always to the detriment of the fracture toughness. It is also fairly easy to raise the fracture toughness level of structural materials, for example, through thermomechanical treatment at the expense of strength. Surprisingly, the high entropy CrMnFeCoNi alloy with high tensile strength level of 1 GPa and excellent ductility (60–70% tensile elongation) was found to exhibit exceptional damage tolerance, displaying remarkable fracture toughness which exceeds 200 MPam1/2 for crack initiation and rises to >300 MPam1/2 for stable crack growth at cryogenic temperatures down to 77 K (Fig. 12) [18,19,168]. These toughness values are comparable to the very best cryogenic steels, specifically certain austenitic stainless steels and high Ni-steels [51,53–55,57,169–171], which also have outstanding combinations of strength and ductility. To understand the combination of high strength and good toughness, in situ TEM was used to reveal the deformation mechanisms of the HEA [168], which revealed (1) the motion of the Schockley partial dislocations and the corresponding formation of stacking faults at the initial stage of deformation (Fig. 12a), (2) as applied stress increases, the movement of perfect dislocations (although with extreme difficulty) in localized bands containing arrays of many closely packed dislocations (Fig. 12b), and (3) the creation of nanoscale bridges that span the crack-tip region and

deform by nano-twinning (Fig. 12c). By removing Fe and Mn, the mechanical properties of the resultant CrCoNi ternary alloy were further significantly improved [19]. At cryogenic temperatures, strength, ductility and toughness of the CrCoNi alloy improve to strength level of 1.3 GPa, failure strain up to 90%, and KIc values of 275 MPam1/2. The remarkable fracture toughness combined with high strength and ductility puts this alloy among the most damage-tolerant materials in that temperature range, making these alloys ideal materials for low-temperature structural applications. To further examine the potential of the alloy as an engineering material for structural applications, Wu et al. investigate the behavior of the CrMnFeCoNi alloy [172] under welding which is a critical fabrication technology used in a wide variety of industries such as energy, aerospace, and ship building. The results show that CrMnFeCoNi has excellent weldability for potential structural applications. After welding, no solidification cracking was observed; moreover, the welded materials maintained the strength and ductility of the base metals (BMs) at both room and cryogenic temperatures. Similar to the base materials, the welded materials exhibit higher strength and ductility at lower temperatures owing to the enhanced twinning activity. Deformation twinning is more pronounced in the fusion zone (FZ) than in the BM region because of the coarse grain structure. 4.5. Alloying design deviating from equiatomic alloys The initially proposed alloying design concept was based on the alloying of five or more elements in equiatomic or near-equiatomic

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Fig. 12. (a) Bright-field TEM images that show the formation of SFs (the red arrows) at the crack tip (top left-hand corner) under loading of the CrMnFeCoNi high-entropy alloy. Beam direction is h1 1 0i; (b) TEM images represent the dynamic process of the planar slip of undissociated 1/2h1 1 0i type dislocations; (c) bright-field TEM image of a growing crack during in situ straining of the CrMnFeCoNi high-entropy alloy. The crack tip is located 500 nm away from the right-lower corner of this image; (d) Ashby map showing fracture toughness as a function of yield strength for high-entropy alloys in relation to a wide range of material systems. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Adapted from [19,168].

concentrations to produce materials with simple microstructures and promising properties. High entropy alloys were expected to benefit from phase stabilization through entropy maximization. More recent studies have relaxed the strict restrictions on high entropy alloy compositions by demonstrating the weakness of this connection. Through compositional variations based on the quinary CrMnFeCoNi alloy, Deng et al. [27] designed a lean twinninginduced plasticity single phase high entropy alloy (TWIP-HEA), Fe40Mn40Co10Cr10 with excellent room temperature mechanical properties that are comparable to those of advanced TWIP steels. The mechanical properties of the Fe40Mn40Co10Cr10alloy are comparable to those of the CrMnFeCoNi alloy, but with the occurrence of extensive deformation twinning at room temperature under tensile and also to those of some FeMnC and FeMnAlC TWIP steels. Recently, Li et al. [173] proposed an alloying design strategy deviating from the high entropy concept, that is a metastabilityengineering strategy (as shown in Fig. 13) in which the phase stability of materials is intentionally decreased to achieve two key benefits: interface hardening due to a dual-phase microstructure (resulting from reduced thermal stability of the hightemperature phase); and transformation induced hardening

(resulting from the reduced mechanical stability of the roomtemperature phase). This strategy has much similarity with the well-known transfor mation-induced-plasticity (TRIP) steels in which martensite, bainite and retained austenite are used to strengthen the ferrite matrix and deformation-induced austenite-to-martensite transformation to increase its strain hardening capacity. Major differences between TRIP steel and the metastability-engineered HEA strategy is the dramatic solid-solution strengthening of HEA through extensive alloying. This new alloy design strategy was demonstrated through a transformation-induced plasticity-assisted, dual-phase high-entropy alloy (TRIP-DP-HEA), Fe50Mn30Co10Cr10. In this dual-phase HEA, the two contributions (multi-phase hardening and solid-solution strengthening) led respectively to enhanced trans-grain and inter-grain slip resistance, and hence, increased strength. The increased strain hardening capacity that is enabled by dislocation hardening of the stable phase (fcc) and transformation-induced hardening of the metastable phase (hcp) produces increased ductility. The combined increase in strength and ductility distinguishes the TRIP-DP-HEA from other structural materials.

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Fe50Mn30Co10Cr10

Fig. 13. (a) The x-ray diffraction pattern, microstructure, and mechanical behavior of the model TRIP-high entropy Fe50Mn30Co10Cr10 alloy; (b) schematic sketches illustrating the deformation structures in the TRIP-HEA. Adapted from [173].

5. Summary and conclusions The recent development of a special class of concentrated solid solution alloys, including high entropy alloys, was reviewed with focusing on the phase stability, physical properties, and strengthening mechanisms of concentrated solid solution alloys. The major findings are summarized as follows. (1) Although the term ‘‘high entropy alloy” is still used to represent multi-component alloys containing five or more elements in equi- or near-equi-atomic concentrations, the initial concept related to the suppression of compound formation through the maximization of configurational entropy is being questioned. The microstructures of this class of alloys are controlled by the competition between phase separation and intermetallic formation. (2) Due to the inconsistency between initial concept and experimental findings, numerous theories and models, including those based on entropy, enthalpy, Hume-Rothery rules, and binary phase diagrams, were developed. However, as yet, none of them can perfectly predict the microstructure of HEAs. More accurate and universal microstructureprediction models are needed for new alloy design, development and application purpose. (3) In addition to the promising microstructures, CSAs provide a playground to systemically investigate the effects of number, species and concentration on the physical properties of the CSAs. Some CSAs exhibited interesting physical properties including magnetic, electrical, and thermal properties.

Phase stability and physical properties are critically dependent on the magnetic interactions of the alloying elements such as the contrast between Cr and Pd. However, knowledge of the physical properties of HEAs is still limited and more in-depth and comprehensive theoretical and experimental investigations are needed. (4) Although the number of ‘‘actual” HEAs with equiatomic compositions (as truly stable single-phased solid solutions) is limited, those that do exist such as the CrMnFeCoNi alloy, often exhibit remarkable mechanical properties, including high strength and ductility, especially at low temperatures. (5) Similar to the high-Mn TWIP steel, twinning plays an important role in the strength and ductility of low stacking fault energy CSAs, such as the CrMnFeCoNi and CrCoNi alloys, significantly increasing strain hardening and postponing necking at low temperatures (e.g., 77 K). (6) The remarkable toughness arising from the twinning effects combined with good weldability makes CrMnFeCoNi a good potential for engineering materials.

Acknowledgement This research was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. Work by GMS was performed as part of the Energy Dissipation to Defect Evolution (EDDE), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences.

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Please cite this article in press as: Z. Wu et al., Phase stability, physical properties and strengthening mechanisms of concentrated solid solution alloys, Curr. Opin. Solid State Mater. Sci. (2017), http://dx.doi.org/10.1016/j.cossms.2017.07.001