One of the key features of high entropy alloys is their severe local lattice distortions, which can lead to beneficial and unusual combinations of mechanical properties. Most reported high entropy alloys (HEAs) are made of size- and chemically similar elements, but if including a component with a distinct size difference was possible, further increase in distortions could be foreseen. However, such additions have typically been disregarded as such alloys are thought to be unmixable. Here, we predict the possibility of mixing such large elements into well-studied HEAs with the help of high or moderate pressure. Miscibility of three large elements in two alloy systems has been studied: Mg/Zr addition in CoCrFeMnNi alloy and Pb addition in MoNbTaTiV alloy. These new compositions are metastable at ambient conditions but can be stabilized with high pressures and probably quenched to ambient with suitable temperature–pressure protocols. We predict that some of the promising candidates can be thermodynamically stabilized at 15–20 GPa, or even lower pressures at elevated temperatures. If synthesized and quenched to ambient conditions, these metastable HEAs would possess ultra-large lattice distortions.

## I. INTRODUCTION

High entropy alloys (HEAs) or compositionally complex alloys (CCAs) is a research field that has grown rapidly during the last 20 years.^{1–4} The enormous chemical and compositional space available for property tuning has triggered theoreticians to enter the field attempting to provide guidance for experiments. A high value of mixing entropy (Δ*S*_{mix}) lowers the Gibbs free energy (Δ*G*_{mix}) of these alloys which, in turn, helps in the formation of stable single phase solid solutions. The most widely studied HEAs contain at least four of the nine following elements: Al, Co, Cr, Cu, Fe, Mn, Ni, Ti, and V.^{3–11} Five of these nine elements are the components of “Cantor alloy” (CoCrFeMnNi), which was first reported in 2004.^{3} Refractory group of HEAs/CCAs consists of alloys containing at least four of the nine refractory elements: Cr, Hf, Mo, Nb, Ta, Ti, V, W, and Zr.^{12–19} This group is studied less frequently than the 3d metal group of CCAs. They are often based on MoNbTaW, CrMoNbTa, HfNbTaZr, or CrNbVZr compositions and can contain nonrefractory elements such as Al or Si to decrease alloy density and improve their properties.^{20–22}

Along with phase stability investigations, features unique to HEAs have naturally been in focus for both theory and experiments. One such structural feature, the inherent large local lattice distortion (LLD) caused by many different local chemical environments, has been pointed out as a key phenomenon in HEAs.^{23} These distortions, where atoms displace from ideal crystal sites and bond lengths show a large spread in values, have been shown to influence dislocations in several ways allowing for unusual and beneficial combinations of mechanical properties.^{24} It is known that lattice distortions influence not only the activation of kink nucleation and motion of screw dislocation but also strengthen the local pinning for edge dislocations.^{25–27} Severe LLDs in HEAs can significantly strengthen the resistance to dislocation movements leading to the overall improved mechanical properties of alloys, such as increased yield strength and hardness.^{28–30} Also, the distortions are believed to reduce thermal conductivity and contribute to anomalous thermal expansion,^{31} providing a mechanism to reduce defect mobility and slow down corrosion.^{32}

From Hume–Rothery rules, it is known that large size mismatch between components hinders the formation of single-phase alloys. Thus, the most reported HEAs are made of size- and chemically similar elements,^{33,34} which reduces their possibility to show dramatically different behavior from less complex counterparts. However, we are here inspired by the work of Dubrovinskaia *et al.*^{35} which demonstrated that high pressures can be used as a route to beat the miscibility gap in binaries between Fe and Mg, which are the metals with vastly different atomic radii. They found that at pressures of 25 GPa and above, more than 10 at. % of Mg could be solved in Fe. This alloy was found to be metastable at ambient pressure after quenching, despite the complete lack of miscibility present in equilibrium. It was also reported that 9 at. % Mg could dissolve in Ni at 86 GPa and more than 12 at. % Mg dissolves in Co at 105 GPa.^{35}

The goal of this work is to find previously not-considered metastable HEAs with ultra-large LLDs displaying unusual and hopefully superior properties of mechanical, thermal, and transport type. We suggest that via high-pressure synthesis, large but compressible elements can be introduced into HEAs made of smaller atoms and that they can be trapped in the lattice at ambient pressure if first cooled and then de-pressurized, causing ultra-large LLDs. To identify promising chemistries, we use *ab initio* calculations to simulate HEAs where at least one element is of distinct size difference. We simulated three cases: (1) Mg and (2) Zr for inclusion in Cantor alloy which is 3d-HEA, and (3) Pb into MoNbTaTiV which is 4d/5d-RHEA.

## II. METHODOLOGY

Table I lists the two base alloy systems studied in this work. In the new five-component HEAs, we kept four components from the original HEA made of smaller elements and added the element with a big atomic radius as the fifth component. We simulated only equimolar compositions. For Cantor-alloy related solid solutions, three crystal structures have been considered: face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal close-packed (hcp). At ambient conditions, Cantor alloy has the fcc lattice. However, at high pressures, it undergoes a phase transition from fcc to hcp, which begins at ∼15 GPa and is complete by 30 GPa.^{36} Alloying with components with small valence electrons concentration (VEC) can reduce the average VEC of solid solutions, which may cause stabilization of the bcc phase. It is known that a VEC of 8 suggests the stabilization of the fcc structure while a VEC of 6.8 stabilizes the bcc structure.^{37} Knowing that the addition of Mg or Zr reduces VEC in the Cantor alloy, it would be relevant to assume a stabilization of the bcc structure. For the second alloy system, MoNbTaTiV-like RHEAs with the addition of Pb, we considered only the bcc structure because we could not find any evidence of phase transition in MoNbTaTiV induced at high pressures. Also, the addition of Pb (with VEC = 4) reduces the average VEC of Pb-containing RHEAs, which should stabilize the bcc structure.

Cantor alloy . | MoNbTaTiV RHEA . |
---|---|

FeCoCrMn + Mg(Zr) | MoNbTaTi + Pb |

FeCoCrNi + Mg(Zr) | MoNbTaV + Pb |

CoCrMnNi + Mg(Zr) | MoNbTiV + Pb |

FeCoMnNi + Mg(Zr) | MoTaTiV + Pb |

FeCrMnNi + Mg(Zr) | NbTaTiV + Pb |

Cantor alloy . | MoNbTaTiV RHEA . |
---|---|

FeCoCrMn + Mg(Zr) | MoNbTaTi + Pb |

FeCoCrNi + Mg(Zr) | MoNbTaV + Pb |

CoCrMnNi + Mg(Zr) | MoNbTiV + Pb |

FeCoMnNi + Mg(Zr) | MoTaTiV + Pb |

FeCrMnNi + Mg(Zr) | NbTaTiV + Pb |

*P*on the thermodynamic stability of solid solutions can be determined from the behavior of mixing enthalpy $\Delta H m i x$. Our goal was to simulate the solubility of a bigger element (solute) in HEAs (solvent) made of smaller elements at different pressures. Therefore, $\Delta H m i x$ was calculated with respect to the enthalpies of a bigger element $ H B E$ and a four-component HEA $ H H E A$, as

There are a few reasons why we chose to calculate the mixing enthalpies in a way as shown in Eq. (1) and not with respect to only pure elements, or including competing ternary or binary alloy phases or ordered intermetallic compounds. Mainly, in this way, it shows the thermodynamic solubility of the bigger element in the four-component alloy made of much smaller elements which are themselves closely related to well-known solid solutions forming five-component alloys. Experimental synthesis of alloys is typically possible only at elevated temperatures when the diffusion and mixing processes are activated. At high temperatures, the configuration entropy of the four-component alloy would likely make it thermodynamically stable and prevent phase separation into clusters as well as the formation of intermetallic phases, and the inclusion of the fifth component would further increase a stabilizing effect of configurational entropy. Therefore, we did not consider pure elements or intermetallic phases as reference states for the calculations of $\Delta H m i x$.

For each system in Eq. (1), the Birch–Murnaghan equation of states (BM-EOS)^{38} was used to describe the dependence of the total energy on volume, and the theoretical values of pressure were derived from the fit. Total energies were calculated for 12 volume points in a range from 0.7*V*_{0} up to 1.1*V*_{0}, where *V*_{0} is a ground state volume at zero pressure. The BM-EOS gives data points of *P* and corresponding *H*; then we used fifth-order polynomial regression to obtain the continuous values of *H* as a function of *P*.

In this work, we simulated only equimolar compositions; therefore, the configuration entropy is $ S mix=\u2212R( ln0.2\u22120.8$ $\xd7 ln0.25)\u22480.5R$, which is much smaller than in the case when the reference states are pure elements. However, at high temperatures around ∼1500–2000 K, one can expect a sufficient stabilization effect in $\u2018\u2018HEA+BE"$ solid solution. We should note that this configurational entropy is a first approximation of the full mixing entropy, and we did not consider other contributions such as magnetic or vibrational that, however, tend to be similar and cancel to a large degree in Eq. (2).^{39–41}

For the electronic structure and total energy simulations, we used two first-principles methods: (1) the exact muffin-tin orbital methods combined with coherent potential approximation (EMTO-CPA)^{42,43} and (2) the projector augmented wave method (PAW)^{44} implemented in VASP software.^{45,46} In PAW calculations, the chemical disorder of HEAs was simulated using the special quasirandom structure (SQS) method.^{47,48} Generalized gradient approximation (GGA)^{49} was used to describe the exchange and correlation effects. The cutoff energy for plane waves was set to 500 eV. Integration over the irreducible part of the Brillouin zone was carried out using the Monkhorst–Pack method^{50} on a grid of $3\xd73\xd73$ *k*-points. The convergence criterion for the electronic subsystem was set to 10^{–4} eV for subsequent iterations. The relaxation of atomic positions was carried out by calculating Hellman–Feynman forces^{51} and the stress tensor and using them in the conjugated gradient method. Relaxation was completed when forces on the ions became of the order of 10^{–2} eV/Å. For Cantor alloy related solid solutions, we considered the fcc, bcc, and hcp structures. We generated $6\xd75\xd74$ supercells for fcc and bcc alloys, and $4\xd75\xd73$ supercells for hcp alloys. All SQS supercells, for four- and five-component alloys, contained 120 atoms with equimolar composition. For systems based on the MoNbTaTiV RHEA, we considered only the bcc structure, and generated $6\xd75\xd74$ supercells. The structures of the generated SQSs and other input files of DFT calculations are available in Ref. 52.

In the EMTO-CPA method, electronic structure calculations are based on Green's function formalism. The use of the full charge density (FCD) method^{53} ensures the calculation accuracy of EMTO-CPA comparable with methods involving the full potential. The main advantage of the EMTO package is the ability to use the coherent potential approximation for the efficient simulation of disordered structures. In the EMTO-CPA method, full charge density was represented by a single-center expansion of the electron wave functions in terms of spherical harmonics with orbital moments $ l F C D m a x$ up to 8. Self-consistent electron densities were obtained within the local density approximation (LDA),^{54} while the total energies were calculated using GGA. Integration in the reciprocal space was performed over a grid of 36 × 36 × 36 *k*-points. Energy integration was carried out in the complex plane using a semielliptic contour comprising 24 energy points. Calculations were performed for a basis set including valence *spdf*-orbitals.

For Cantor-alloy-like HEAs, we considered a paramagnetic state as the magnetic structure. Magnetic properties were accounted for within the collinear picture. The magnitude and orientations of the collinear local moments were calculated self-consistently. The paramagnetic state of solid solutions was described within the disordered local moment (DLM) approximation.^{55,56} The DLM-SQS is constructed to model an equal composition of spin-up and spin-down atoms for each chemically non-equivalent alloy component as well as a disordered configuration of up and down moments.

For the promising HEAs with large lattice distortions, we carried out the phonon calculations using Phonopy software.^{57} Phonons were simulated for $1\xd71\xd71$ supercells which have the same size as generated SQSs (120 atoms in supercell). To calculate the force constants, the value of atomic displacements was set to 0.01 Å.

## III. RESULTS

### A. Solubility of Mg and Zr in Cantor alloy related HEAs

As we mentioned, for Cantor-alloy-like solid solutions, we considered three structures: hcp, fcc, and bcc. The total energies of all three structures are very close; therefore, in the main text of the manuscript, we show the results only for hcp alloys, and the data for fcc and bcc cases are given in the supplementary material.

The mixing enthalpies $\Delta H m i x$ of Mg-containing HEAs are shown in Fig. 1. The mixing enthalpy was calculated according to Eq. (1), and the reference states are bcc Mg and hcp four-component HEA. The bcc structure for pure Mg was chosen because it is a high-pressure phase, which becomes stable at pressures above 50 GPa.^{58} The calculations suggest that Mg-containing alloys can be stabilized at high pressures since their $\Delta H m i x$ becomes lower with increased pressure. Two methods, CPA and SQS, demonstrate good qualitative agreement. $\Delta H m i x$ in CPA is higher than in the SQS method. This is most likely due to lattice relaxation effects which are not accounted for in the CPA method. One can consider that the difference in $\Delta H m i x$ between CPA and SQS results at *P *= 0 GPa as a rough estimation of the lattice relaxation effect. According to Fig. 1, (FeCoNiMn)_{0.8}Mg_{0.2} is the easiest alloy to stabilize with pressure since its mixing enthalpy becomes negative at *P *≈ 40 GPa. The most difficult alloy to stabilize is (FeCrNiMn)_{0.8}Mg_{0.2}, its $\Delta H m i x$ becomes negative at *P *≈ 110 GPa according to SQS calculations. Figure 1 shows that if $\Delta H m i x$ of the alloy at zero-pressure is low, it becomes easier to stabilize the alloy with pressure.

The mixing enthalpies of the fcc and bcc alloys are shown in Figs. S1 and S2 in the supplementary material. Note that we did not consider the low-pressure state for pure Mg (hcp-Mg) as a reference phase. This was done because at high pressures, when the mixing enthalpy *H*_{mix} of most of the studied alloys becomes negative, Mg undergoes a phase transition from low- to high-pressure phases. In Sec. S2 in the supplementary material, we provide a detailed explanation for choosing high-pressure phases in the case of pure Mg, Zr, and Pb.

For Zr-containing Cantor-alloy-like solid solutions, we also focus on the hcp phase, but the information on mixing enthalpies for the fcc and bcc phases is available in the supplementary material (see Figs. S3 and S4). $\Delta H m i x$ was calculated with respect to the enthalpies of pure bcc Zr and hcp four-component HEA. The bcc structure for Zr was chosen because it is a high-pressure phase, which becomes stable at pressures above 30 GPa.^{59} For Zr-containing HEAs, we see a different behavior of $\Delta H m i x$ as a function of pressure (see Fig. 2). According to CPA calculations, the mixing enthalpies of all Zr-containing alloys become even more positive with pressure, meaning that none of these alloys can be stabilized with increased pressure. Due to the high computational cost of PAW-SQS simulations, we performed it only for one Zr-containing alloy, (FeCrNiMn)_{80}Zr_{20}, which is shown in Fig. 2(b). For this alloy, the PAW-SQS confirms the behavior of “ $\Delta H m i x$ vs *P*” obtained in EMTO-CPA, so it can be assumed that EMTO-CPA results are reliable. Since we are interested in alloys that can be stabilized with the help of external pressure, we did not perform PAW-SQS calculations for the other four Zr-containing alloys.

Campari *et al.*^{60} synthesized experimental samples of Cantor alloy with the addition of 5 at. % Zr using the vacuum induction melting method. The samples show a dendritic solidification of the microstructure with a clear distinction between the dendritic and the inter-dendritic phases, and Zr is predominantly accumulated in the inter-dendritic secondary phase. The phase separation in the microstructure may suggest that the formation of a single-phase solid solution is not favorable.^{60}

The difference in the solubility of Mg and Zr in HEAs is interesting since both Mg and Zr are significantly softer than four-component HEAs. Therefore, one would expect that both Mg- and Zr-containing alloys would demonstrate similar behavior of the mixing enthalpies as the pressure increases. However, we see the opposite trend of $\Delta H m i x$ in these alloys. To understand the difference, one should look at how these alloys follow Vegard's law for atomic volumes. In supplementary material in Sec. S3, we provide detailed information on how the deviations from Vegard's law in these alloys were estimated. According to our estimation, all Mg-containing alloys demonstrate a negative deviation from Vegard's law, $\Delta V<0$, while all Zr-containing alloys demonstrate a positive deviation, $\Delta V>0$. According to Ref. 61, $\Delta V$ is equal to the first order derivative of mixing Gibbs free energy with respect to pressure, $\u2202\Delta G m i x/\u2202P$. In the case of Zr-containing alloys, $\u2202\Delta G m i x/\u2202P=\Delta V P = 0>0$, which explains the lack of miscibility of Zr in four-component HEAs at high pressures. On the other hand, all Mg-containing alloys demonstrate $\u2202\Delta G m i x/\u2202P=\Delta V P = 0<0$, which agrees with the trends of $\Delta H m i x$ shown in Fig. 1.

### B. Solubility of Pb in MoNbTaTiV-like RHEA

The second alloy system studied in this work is MoNbTaTiV RHEA with the addition of Pb. Only the bcc structure was considered for these alloys. The mixing enthalpies of Pb-containing RHEAs were determined with respect to the total energies of pure hcp Pb and bcc four-component HEA as shown in Eq. (1). The hcp structure for pure Pb was chosen because it is a high-pressure phase, which becomes stable at pressures above 14 GPa.^{62} According to Fig. 3, with increased pressure, $\Delta H m i x$ of alloys becomes lower, showing the stabilizing effect of high pressures. According to SQS calculations, the easiest alloy to stabilize is (NbTaTiV)_{0.8}Pb_{0.2}: its $\Delta H m i x$ becomes negative at *P* > 40 GPa, and the most difficult alloy to stabilize is (MoNbTaV)_{0.8}Pb_{0.2}: its $\Delta H m i x$ reaches negative values at *P* > 80 GPa. Similar to Mg-containing HEAs, one can see that the lower is the $\Delta H m i x$ of alloy at zero-pressure; the lower pressure is required to reach negative values of $\Delta H m i x$. Both CPA and SQS methods demonstrate good qualitative agreement. The SQS results at zero-pressure are lower, which is most likely because of the lattice relaxation effect that is not accounted for in CPA calculations.

### C. Atomic size difference and stabilizing effect of finite-temperatures

_{0.8}Mg

_{0.2}and (NbTaTiV)

_{0.8}Pb

_{0.2}solid solutions. Their mixing enthalpy becomes negative at

*P*> 40 GPa. It is important to note that the miscibility of elements with distinct size differences requires not only thermodynamic factors but also overcoming the barrier of atomic size difference. For HEAs, there is an empirical rule of atomic size mismatch $\delta $ which is used to predict whether it is possible or not to form single-phase solid solutions. However, this rule of

*δ*-parameter is not always accurate and often used as a recommendation. The usual Hume–Rothery law of atomic sizes works for binary systems, and it cannot be strictly applied for multicomponent systems. Therefore, we decided to make a rough estimation of the Hume–Rothery law, where the bigger element “BE” was considered as a solute and the four-component “HEA” as a solvent. We estimated the average atomic radius of the solvent from the average atomic volume of the four-component HEA. Then, the atomic size difference can be estimated as

According to the Hume–Rothery law, the single-phase solid solution is formed when *D *< 15%. In our case, compressibility of the solute is much higher than that of the solvent as it is clearly seen in Fig. 4. For both (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} alloys, high pressures allow to reduce the size difference between the solute and the solvent, which helps to form a single-phase solid solution. For (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} alloys, the requirement of *D *< 15% is achieved at pressures of 44 and 36 GPa, respectively. Interestingly, these pressures are very similar to the values when $\Delta H m i x$ of these alloys become negative in our direct calculations.

In the case of Zr incorporated in Cantor-alloy related solid solutions, we also see that atomic size difference *D* decreases with pressure (see Fig. S10 in the supplementary material). For (FeCoNiMn)_{0.8}Zr_{0.2}, the requirement of *D *< 15% is achieved at *P* > 145 GPa, which is a much higher pressure than in the case of (FeCoNiMn)_{0.8}Mg_{0.2}. This is due to the higher stiffness of Zr compared to Mg: theoretical bulk moduli of pure Mg and Zr are 36 and 87 GPa, respectively. Since atomic size difference *D* in Zr-containing alloys decreases with pressure, it cannot explain why $\Delta H m i x$ of alloys increases with pressure (see Fig. 2). However, as we determined earlier, it is most likely due to a positive deviation from Vegard's law observed in all Zr-containing alloys.

Now, let us consider the effect of finite-temperatures on the thermodynamic stability of two promising candidates. Figure 5 shows the pressure dependence of mixing free energy $\Delta G m i x$ of (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} alloys. Here, $\Delta G m i x$ was calculated for temperatures up to 1500 K. As expected, the configurational entropy $\Delta S m i x$ helps to stabilize the alloys. At *T *= 1500 K, the mixing energy of both alloys becomes negative at pressures around 15 GPa. The estimation of atomic size difference shows that *D *> 15% at 15 GPa (see Fig. 4), which will probably prevent the formation of single-phase solid solutions. However, one should note that our estimation of *D* is carried out at *T *= 0 K and does not account for softening of both solute and solvent at high temperatures.

Both candidates, (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2}, are metastable if quenched to ambient pressure since they have positive mixing energies at zero pressure. However, to make a judgment of materials metastability, one needs to check if it is dynamically stable. In Fig. 6, we show the phonon densities of states (PDOS) of (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} calculated for zero pressure and zero temperature. Both alloys are dynamically stable since no imaginary frequencies were found for the whole set of wave vectors in alloys’ Brillion zones. In the case of (FeCoNiMn)_{0.8}Mg_{0.2}, the phonons were calculated for hcp, fcc, and bcc structures, and all three structures are dynamically stable and demonstrate a similar behavior of PDOS [see Fig. 6(a)]. So, it can be assumed that both (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} are metastable at ambient conditions.

### D. Local lattice distortions

*l*

_{i}from the average value of bond length

*l*

_{mean},

Figure 7 shows how the addition of Mg can influence the bond length distortions in FeCoNiMn alloy: at 50 GPa when the mixing enthalpy of (FeCoNiMn)_{0.8}Mg_{0.2} alloy becomes negative and after quenching to 0 GPa. At *P* = 50 GPa, the addition of Mg can drastically increase the range of bond length values. As a result, bond distortions in (FeCoNiMn)_{0.8}Mg_{0.2} alloy become significantly larger than in FeCoNiMn. After returning to ambient pressure, the bond distortions of (FeCoNiMn)_{0.8}Mg_{0.2} become even larger. This is because of the high compressibility of Mg-containing bonds: the most compressed bonds are Mg–Mg pairs. Severe lattice distortions in Mg-containing Cantor alloy can significantly strengthen the resistance to dislocations movement, leading to the overall improved mechanical properties of the alloy, such as increased yield strength and hardness.^{28–30}

Figure 8 shows the effect of Pb on bond length distortions in NbTaTiV alloy: (a) at 50 GPa when the mixing enthalpy of (NbTaTiV)_{0.8}Pb_{0.2} alloy is negative and (b) after quenching to 0 GPa. The influence of Pb is not as strong as in the case of Mg in Cantor alloy; however, one can see that Pb also increases the range of bond length distortions. After quenching to 0 GPa, the effect of Pb on local distortions becomes a little stronger. One should note that lattice distortions in four-component NbTaTiV alloy were already very large. It is expected since the loose-packed bcc structure has more free space to tolerate large lattice distortions compared to more close-packed fcc and hcp structures. The effect of Pb is to increase lattice distortions, but the effect is not very strong. However, the distortions in (NbTaTiV)_{0.8}Pb_{0.2} have almost the same range as those in (FeCoNiMn)_{0.8}Mg_{0.2}.

Here, $ c i$ and $ r i$ denote concentration and atomic radius of the *i*th component, respectively; and $ r i$ denotes the average atomic radius. Atomic radii of alloy components were obtained from Voronoi analysis (or Wigner–Seitz cells). Voronoi analysis was carried out for fully relaxed alloys at zero pressure. It gives volumes $ V i$ of polyhedra for each atom in the SQS supercell. Then, the atomic radius can be estimated as $ r i= ( 3 / 4 \pi ) V i 3$; and the average atomic radius is defined as $ r \xaf= \u2211 i c i \xd7 r$. One should note that within Voronoi analysis, the volumes/radii of atoms in supercells are influenced by their local chemical environment. We provide detailed information on the radius of each atom in simulated SQSs in the supplementary material (see Figs. S6 and S7).

Table II lists the atomic radii and the mismatch $\delta $ parameter for FeCoNiMn and (FeCoNiMn)_{0.8}Mg_{0.2} alloys. Atomic radii of elements become bigger after the addition of Mg. This is because Mg increases the lattice parameter of the (FeCoNiMn)_{0.8}Mg_{0.2} alloy; as a result, volumes $ V i$ and radii $ r i$ of atoms become larger. The inclusion of Mg in FeCoNiMn significantly increases the $\delta $ parameter, from 0.17% to 1.43%. For comparison, one may refer to the report by Tong *et al.,*^{63} where x-ray diffraction was used to measure lattice distortions in several refractory multi-principal element alloys (RMPEAs); and among those, HfNbTaTiZr, HfNbTiZr, and NbTiVZr alloys were reported to have severe local lattice distortions. Theoretically predicted $\delta $ parameters for these three alloys are 1.2%, 1.4%, and 1.8%, respectively.^{63} Our candidate, (FeCoNiMn)_{0.8}Mg_{0.2}, with *δ* = 1.43% may have distortions comparable to those reported for RMPEAs.^{63}

Alloy . | Atomic radius r_{i} (Å)
. | Average radius $ r \xaf$ (Å) . | δ (%)
. | ||||
---|---|---|---|---|---|---|---|

Fe . | Co . | Ni . | Mn . | Mg . | |||

FeCoNiMn | 1.370 | 1.372 | 1.376 | 1.371 | – | 1.372 | 0.17 |

(FeCoNiMn)_{0.8}Mg_{0.2} | 1.427 | 1.423 | 1.429 | 1.422 | 1.477 | 1.436 | 1.43 |

Alloy . | Atomic radius r_{i} (Å)
. | Average radius $ r \xaf$ (Å) . | δ (%)
. | ||||
---|---|---|---|---|---|---|---|

Fe . | Co . | Ni . | Mn . | Mg . | |||

FeCoNiMn | 1.370 | 1.372 | 1.376 | 1.371 | – | 1.372 | 0.17 |

(FeCoNiMn)_{0.8}Mg_{0.2} | 1.427 | 1.423 | 1.429 | 1.422 | 1.477 | 1.436 | 1.43 |

For the second promising candidate (NbTaTiV)_{0.8}Pb_{0.2}, in Table III, we also see that the addition of a bigger element, Pb, significantly increases the $\delta $ parameter. However, $\delta $ of the NbTaTiV alloy was already quite large, unlike that of FeCoNiMn. Nevertheless, both candidates, (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2}, demonstrate close values of an atomic size mismatch, meaning that one should expect similar LLDs from both alloys. This result is in qualitative agreement with the bond length distortions shown in Figs. 6(b) and 7(b), where both (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} alloys demonstrate similar ranges of bond length distortions at zero pressure.

Alloy . | Atomic radius r_{i} (Å)
. | Average radius $ r \xaf$ (Å) . | δ (%)
. | ||||
---|---|---|---|---|---|---|---|

Nb . | Ta . | Ti . | V . | Pb . | |||

NbTaTiV | 1.599 | 1.602 | 1.601 | 1.568 | – | 1.592 | 0.902 |

(NbTaTiV)_{0.8}Pb_{0.2} | 1.634 | 1.634 | 1.634 | 1.616 | 1.682 | 1.640 | 1.349 |

Alloy . | Atomic radius r_{i} (Å)
. | Average radius $ r \xaf$ (Å) . | δ (%)
. | ||||
---|---|---|---|---|---|---|---|

Nb . | Ta . | Ti . | V . | Pb . | |||

NbTaTiV | 1.599 | 1.602 | 1.601 | 1.568 | – | 1.592 | 0.902 |

(NbTaTiV)_{0.8}Pb_{0.2} | 1.634 | 1.634 | 1.634 | 1.616 | 1.682 | 1.640 | 1.349 |

## IV. CONCLUSIONS

We studied the stability of novel high entropy alloys with additions of large and extremely size-mismatched elements into well-known HEAs that induce ultra-high local lattice distortions. We find that several of these alloys can be stabilized at high-pressure, explained by the higher compressibility of larger elements. The first-principles simulations were performed using PAW-SQS and EMTO-CPA methods. The miscibility of two alloy systems has been studied: (1) Mg and Zr added to Cantor-alloy-like solid solutions and (2) Pb added to MoNbTaTiV-derived refractory metal solid solutions. Overall, we investigated the stability of fifteen compositions of HEAs. The mixing enthalpies calculated within PAW-SQS and EMTO-CPA methods are in good qualitative agreement. We showed that all Mg-containing HEAs can be stabilized with high pressures, and the most promising candidate is (FeCoNiMn)_{0.8}Mg_{0.2} which can be stabilized at pressures about 40 GPa at zero K and as low as 15 GPa at 1500 K. It is plausible that a considerable amount of Mg, although less than the here studied 20 at. %, can be solved in Cantor-like alloys even at pressure ranges reachable by, e.g., large volume presses. Zirconium, on the other hand, cannot be dissolved in Cantor-like alloys since the mixing enthalpy of such alloys becomes even more positive at high pressures. Calculations for the third studied system, Pb incorporated in MoNbTaTiV-like solid solutions, show that they can be stabilized with high pressures. Another promising candidate is (NbTaTiV)_{0.8}Pb_{0.2}: its mixing enthalpy also becomes negative at pressures about 40 GPa at zero K and around 15 GPa at 1500 K. Both (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} candidates have very large LLDs. Mg can dramatically increase the lattice distortions in the Cantor alloy. Pb in (NbTaTiV)_{0.8}Pb_{0.2} also increases distortions, but its effect is not as big as in the Cantor alloy system with Mg. Both (FeCoNiMn)_{0.8}Mg_{0.2} and (NbTaTiV)_{0.8}Pb_{0.2} candidates demonstrate a similar range of LLDs. Phonon calculations suggest that both alloys are dynamically stable at ambient conditions. If synthesized, these alloys may have improved mechanical properties, such as increased yield strength and hardness. Moreover, the existence of such solid solutions would prove that it is possible to synthesize HEAs where components have big differences in atomic size. Based on our theoretical results, we predict that high-pressure synthesis can be a new route to obtain previously not-considered HEAs with ultra-large lattice distortions.

## SUPPLEMENTARY MATERIAL

See the supplementary material for detailed information on mixing enthalpies, deviation from Vegard's law, and atomic radii of elements in studied HEAs.

## ACKNOWLEDGMENTS

Computations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS) at NSC partially funded by the Swedish Research Council through Grant Agreement No. 2022-06725. B.A. acknowledges financial support from the Swedish Research Council (VR) through Grant No. 2019-05403, and 2023-05194 from the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFOMatLiU No. 2009-00971), from the Knut Alice Wallenberg Foundation (Wallenberg Scholar Grant No. KAW-2018.0194), as well as support from the Swedish Foundation for Strategic Research (SSF) through the Future Research Leaders 6 program, FFL 15-0290.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Boburjon Mukhamedov:** Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Writing – original draft (equal). **Björn Alling:** Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available in the Open Materials Database, Ref. 52. The data contain the input files of DFT simulations.

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