Delft University of Technology Multiple linear regression and thermodynamic fluctuations are equivalent for computing thermodynamic derivatives from molecular simulation

The reactivity of the surface of multicomponent metals such as High Entropy Alloys (HEAs) is rapidly gaining importance for corrosion and catalytic applications, but the mechanisms of surface segregation in these complex materials are poorly understood. Here we investigate with ab initio calculations the segregation thermodynamics in the magnetic Cr-Mn-Fe-Co-Ni HEA in vacuum, in the presence of O at the surface, and in the oxide phase. We predict a weak segregation of Ni for Cr-, Mn-, and Fe-rich alloys in vacuum and a very strong segregation of Cr and Mn upon exposure to O.

Despite the original focus was primarily on bulk, interest in the surface structure, composition, and reactivity of HEAs recently emerged, triggered by the investigation of the corrosion [17,19,20] and catalytic [21][22][23] behavior of these alloys. Owing to the vast compositional space of HEAs, the concentration of the elements can be tuned to obtain specific surface structures in order to design corrosion-resistant high performance HEAs, high-entropy coatings for conventional materials (e.g. steels), or highly active, stable, and selective high-entropy catalysts.
A major issue in HEAs is the precipitation of secondary detrimental phases, usually richer in composition of one or more constitutive elements [24][25][26]. Since element segregation is present in the bulk, even more so it should be observed on surfaces, where segregation energies are usually 10-100 times larger in magnitude. Recent studies however observed a seemingly uniform distribution of elements in high-entropy nanotubes [27], nanoparticles [28][29][30][31], and nanoporous structures [32], hinting that surface segregation may be negligible in the investigated HEAs at least in non-reactive conditions. This stands in stark contrast with common bimetallic alloys, such as PtPd, CoCr [33], Ag 3 Pd [34], where segregation is commonly reported.
A detailed understanding of the segregation mechanisms in HEAs is of paramount importance for the stability of HEA components [35]: a strong segregation of passivating elements could increase corrosion resistance on one hand, but could compromise the long-term stability of potential high-entropy catalysts on the other. Despite this pivotal role, the degree of surface segregation in HEAs has not been quantified exhaustively in the literature. Ab initio simulations of oxidation of a highentropy alloy were performed for Mo-W-Ta-Ti-Zr [36], but surface segregation was not considered.
In this article, we address the fundamentals of surface segregation for the prototypical Cr-Mn-Fe-Co-Ni HEA [2,37] with first principles calculations. Segregation mechanisms for this alloy are quite controversial: for CrMnFeCoNi, the formation of an external Mn 2 O 3 layer and an inner Cr 2 O 3 layer upon prolonged exposure to air has been observed [38][39][40], but very recently Wang et al. [41] found mainly Croxides and hydroxides and Fe-oxides in the native surface layer.
Here we calculate the segregation energies of the component elements of face centered cubic (fcc) Cr-Mn-Fe-Co-Ni as a function of the alloy composition, in vacuum and in the presence of 1 monolayer (ML) of atomic O on the surface. Additionally, we compute formation free energies of various oxides containing the component elements to evaluate the segregation also in the oxide phase. We consider the fcc (111) surface because it is the most close-packed. We show that Ni segregates to the surface in vacuum for Cr-, Mn-, and Fe-rich alloys, whereas Cr and Mn segregate to the surface when O is present for every alloy composition.
In Section 2 we provide the details of the calculation of segregation https://doi.org/10.1016/j.apsusc.2020.147471 Received 13 June 2020; Received in revised form 24 July 2020; Accepted 4 August 2020 and formation free energies with first-principles calculations and with standard canonical d-band models. In Section 3 and 4 we discuss the surface segregation in vacuum and with adsorbed O. In Section 5 we present the formation free energy of different oxides.

Calculation of segregation and formation free energies
2.1. Ab initio calculation of the segregation and formation free energies

Segregation free energy
We computed the segregation free energy of an element by comparing the surface free energy of a slab with disordered configurations on the surfaces with that of a slab where the surfaces are fully composed of that element only. Since the disordered and segregated slabs have different number of atoms of each species, we corrected our surface free energies with the chemical potentials of the elements in the Cr-Mn-Fe-Co-Ni alloy. The surface segregation free energy of an element i hence reads [42] where G slab (i) and G slab (ran) are the total free energies of the slabs with the surfaces occupied by element i and by a disordered configuration of the atoms, respectively; N s is the number of atoms on the two surfaces of the slabs; = n N j (i) s ij and n j (ran) are the number of atoms of type j on the surfaces with element i and with the disordered configuration, respectively; and µ j is the chemical potential of the element j.
In the evaluation of the free energies and chemical potentials we have neglected the vibrational, magnetic, and electronic contributions and retained only the configurational part using Stirling's approximation: where c j indicates the concentration of j in the alloy. The chemical potential of j is [42] where A denotes an alloy with a certain composition (CrMnFeCoNi, for instance). To decrease the computational effort, we did not compute E (A) explicitly, but estimated it via the average of the E (A j ) Eq.
(3) and (4) are exact in the limit c 0; in practice, we found that utilizing a finite c of about 1 2% introduces negligible errors.

Oxide formation free energies
We computed the formation free energy of an oxide with chemical formula M x O y as
For the PAW method, we employed the generalized gradient approximation in the PBE parametrization [52]. The 4s and 3d electrons were treated as valence for Cr, Mn, Fe, Co, and Ni, and the 2s and 2p electrons for O. The energy cutoff was fixed to 400 eV. The width of the Methfessel-Paxton function [53] for the electronic smearing was 0.1 eV. All the calculations were spin-polarized. The k-points sampled the Brillouin zone with -centered Monkhorst-Pack grids [54,55]. We modeled fcc(111) surfaces as × × 3 3 9 slabs with 81 atoms with 15 Å of vacuum to separate periodic images and used a k-mesh of × × 4 4 1. For the calculation of the bulk chemical potentials we considered × × 3 3 6 supercells with 54 atoms with the same orientation as the slabs (z-direction parallel to [111]) and used a k-mesh of × × 6 6 4. We relaxed the atomic positions until the interatomic forces decreased to less than 0.05 eV/Å. To minimize artificial ordering due to the finite size of the supercells, we considered 5 different supercell realizations and for each of them 120 different permutations of the positions of the atoms for the calculations of the surface free energies (Eq. (2)) and chemical potentials (Eqs. (3)), respectively. This allowed us to obtain statistical uncertainties of about 40 meV/at. for the segregation free energies.
For the EMTO method, we converged the ground-state electron density using the local density approximation and then calculated the total energy with the generalized gradient approximation from the obtained charge density. We considered the same valence electrons as in the plane-wave pseudopotentials for the component elements. We employed the Fermi smearing with a width of 0.1 eV and obtained our results at the corresponding finite electronic temperature. The k-points sampling was performed with the Monkhorst-Pack scheme. We treated the alloys within the coherent potential approximation (CPA) [56,57] with screened Coulomb interactions [58][59][60]. The screening parameters were obtained with the locally self-consistent Green's function method [61][62][63] with the same procedure as in Ref. [64]. They take the following values: Cr 0.767, Mn 0.794, Fe 0.316, Co 0.756, Ni 0.804 [65]. Within CPA, we modeled fcc(111) surfaces as 7-layer slabs with roughly 16 Å of vacuum and used a k-mesh of . For the calculation of the chemical potentials we considered fcc unit cells with a k-mesh of × × 16 16 16. We performed calculations for the non magnetic (NM), paramagnetic, and ferromagnetic (FM) states. We treated the paramagnetic state with the disordered local moment (DLM) approximation [66,67].

Canonical relations for the segregation
A qualitative description of the segregation in transition-metal alloys can be obtained also from canonical models exploiting the properties of the d-partial density of states (pDOS) and the d-band filling, such as the Friedel model and the Hammer-Nørskov model. These models are not quantitatively accurate, but are useful to rationalize important trends with respect to the composition.

Friedel model for the segregation in vacuum
The Friedel model [68] considers a d-valent alloy with a rectangular density of states of width w with a band filling f. In this model, the surface segregation energy of another metal, with band filling f , is related to the difference in the surface formation energies of the two metals [42]: where z s and z b are the numbers of nearest neighbors for the surface (9 for fcc(111)) and for the bulk (12 for fcc), respectively, and the squareroot dependence derives from a second-moment approximation to tightbinding [69]. The average d-band filling of Cr-Mn-Fe-Co-Ni is calculated from the number of d-electrons of each element. We used the following number of d-electrons, found by integrating the pDOS for each metal in the fcc structure up to the Fermi level: Cr 4.3, Mn 5.0, Fe 6.1, Co 7.0, Ni 8.1. For example, the band filling for the equiatomic CrMnFeCoNi is 0.61. Using the computed pDOS, we estimated the bandwidths w and parametrized them as a function of f (see the Appendix for details).

Hammer-Nørskov model for the segregation with oxygen
Within the Hammer-Nørskov model [70], the binding energy between a metal and an adsorbate species (and consequently also the segregation energy of that metal in the presence of adsorbates) decreases for increasing baricenter of the d-band of the metal with respect to its Fermi energy. An extension of the Hammer-Nørskov model to magnetic systems reads [71] E are the baricenters of the d-band, f are the d-band fillings and g d are the pDOS for each spin channel .

The equiatomic CrMnFeCoNi alloy
For the equiatomic CrMnFeCoNi (Cantor) alloy we fixed the lattice parameter to the experimental value of 3.60 Å [2, [72][73][74][75][76][77]. The segregation free energies of Cr, Mn, Fe, Co, and Ni at room temperature in this alloy are presented in Fig. 1. The different bars refer to the PAW and EMTO methods in different magnetic states. The NM segregation free energies are also compared to the Friedel model. The error bars for the PAW method are statistical and due to the finite size of the supercells (see Section 2.2).
In the NM state, the EMTO method underestimates slightly the segregation free energies of Co and Ni, even if relaxation is not taken into account in the PAW method. The NM segregation energies follow approximately the trend given by the Friedel model. Noteworthy, this simple approximation underestimates the segregation free energies by only roughly 0.15 eV/at. At fixed atomic positions, magnetism generally increases the segregation energy, with FM having a stronger effect than DLM, but exceptions are found for Fe and Co. The largest impact of magnetism is on Ni, where different magnetic states lead to segregation energies which differ by as much as 0.15 eV/at. Atomic relaxation has the biggest effect on the segregation energy of Mn, and is accompanied by a relatively large change of the magnetic moment of this element (more than 1 µ B ). This highlights a strong volume-dependence of the magnetic moment of Mn.
We find that the expected segregation of Ni, with the lowest surface formation energy among the five elements, is mitigated by magnetism and configurational entropy. We considered also surface configurations with Co and Ni segregating simultaneously at the surface but did not found substantial differences in segregation energies with respect to the two metals alone. Based on the PAW results in the FM state, we conclude that segregation in CrMnFeCoNi in vacuum at room temperature may be negligible and that the most realistic configuration of the surface at room temperature is likely a Cr-lean solid solution.
The left panel of Fig. 2 displays the magnetic moments of Cr, Mn, Fe, Co, and Ni at different positions in the slabs (on the surface layers, on the subsurface layers, and in bulk-like layers) computed in the FM state with the PAW and EMTO methods. The most notable difference between the two methods is the ferromagnetic alignment of Mn in the subsurface layer for the EMTO method; we observed this ferromagnetic alignment also in the PAW method with unrelaxed atomic positions, but it disappeared when atomic positions were relaxed; this is imputed to the already mentioned strong dependence of the magnetic moment of Mn on interatomic distances. The magnetic moments converge approximately to the bulk value already from the second layer below the surface for all component elements. The right panel of Fig. 2 shows the magnitude of the converged magnetic moments in the DLM state obtained with the EMTO method as a function of the position in the slab. In the bulk at the experimental volume and in the idealized DLM state, only for Mn and Fe finite local magnetic moments are observed in agreement with previous reports [37]. The inclusion of longitudinal spin fluctuations in other works stabilized the local magnetic moments in the DLM state [64,78] in the bulk. These details are however beyond the scope of the present work.

Off-equiatomic compositions
As Ni is the element most prone to segregation in vacuum, we calculated the segregation energy of this element for other compositions. We also tested Co, but found that its segregation energy is always higher than that of Ni.
For the PAW method we considered additional systems with stoichiometry A 40 B 15 C 15 D 15 E 15 . For the EMTO method we studied all quinary compositions with at least 10% of each element with increasing/decreasing concentration steps of 10% (e.g. A 60 B 10 C 10 D 10 E 10 , A 50 B 20 C 10 D 10 E 10 , A 40 B 20 C 20 D 10 E 10 , etc.). In addition, we considered quaternary, ternary, and binary Cr-and Ni-rich alloys with Cr and Ni concentrations up to 90% to span two corners of the compositional space. For each system we computed the equilibrium volume via a Birch-Murnaghan fit [79,80] of the equation of state.
The segregation energies of Ni at 0 K for the various Cr-Mn-Fe-Co-Ni alloys are reported in Fig. 3 as a function of the d-band filling for the PAW and EMTO methods and for the Friedel model. The three series of data highlight a clear band filling effect, with the segregation energy of Ni reaching a minimum for a band filling of 0.50-0.55, corresponding to Cr-and Mn-rich alloys, and approaching zero for Ni-rich alloys. The equiatomic CrMnFeCoNi alloy described in the previous Section is apparently an outlier for the PAW data points. This highlights a particularly stable configuration of the random surface and bulk (relatively low chemical potential of Ni) for the equiatomic composition.
We argue in general that segregation is negligible for > f 0.65 (Coand Ni-rich alloys) even at low temperature, whereas there is a weak (E segr ranges from −0.2 to −0.1 eV/at.) segregation of Ni (and Co) at lower band fillings (apart from the equiatomic composition).

Segregation with an oxygen layer
To evaluate the impact of reactants on the surface, we analyzed the segregation in the presence of 1 ML of O adsorbed in the fcc hollow positions. For this we employed the PAW method for the equiatomic CrMnFeCoNi alloy as well as for prototypical off-stoichiometric compositions A 40  If the composition of the alloy is changed, the segregation energy of    Given that the strongest segregating element is Cr, we also studied the segregation in the subsurface layer if the surface layer is fully occupied by this element. The results are plotted as white bars in Fig. 4. It can be seen that Cr tends to segregate also to the subsurface layer, but much more weakly.
The magnetic moments of the most stable surface configuration with adsorbed O are displayed in Fig. 6. The O and Cr atoms on the surface have negligible magnetic moment. The moments of the atoms on the other layers do not differ substantially from those with a clean surface, apart from the lower magnetic moment of Mn in the subsurface layer.

Segregation in the oxide phase
To investigate the preference towards oxide formation for the various elements, which impacts the segregation in the presence of O, we computed the formation free energies of various metal-oxides. We shortlisted the structures by selecting all the stoichiometries within 40 meV/at. from the 0 K convex hull taken from the Materials Project [81,82]. For each oxide, we relaxed the atomic degrees of freedom, the shape, and the volume of the simulation cells. We found some minor discrepancies in comparing our formation energies with those from the Materials Project, where the Hubbard U parameter was introduced to correct the spurious self-interaction. These discrepancies regarded mainly the equiatomic oxides (MnO, FeO, CoO, and NiO) but have no impact on our conclusions.
The formation free energies of the candidate oxides at 300 K are shown in Fig. 7. To calculate the chemical potential of O, we fixed the partial pressure of this element to 0.2 atm. In these conditions, the formation of oxides is strongly exothermic for all the considered compounds. According to our predictions, the most stable oxide is Cr 2 O 3 and this points to the segregation of Cr also in the oxide phase. The trend of the oxide formation energies is the same as that of the segregation energy in the presence of oxygen: the elements with lower band filling, Cr and Mn, have a higher affinity to O than the other elements, especially Ni. The situation does not change substantially for offequiatomic compositions (A 40 B 15 C 15 D 15 E 15 ), with different chemical potentials.
A Cr-enrichment has been observed by Wang et al. [41] in the oxide phase of CrMnFeCoNi, in agreement with our findings. Other studies [38][39][40] reported the formation of an external Mn 2 O 3 layer and an internal Cr 2 O 3 layer. We impute this discrepancy to the very slow diffusion of Cr in the bulk [25] and oxide [39] phases, that hinders the formation of the thermodynamically most stable oxide (Cr 2 O 3 ) in favor of other metastable phases.

Conclusions
Surface segregation in HEAs is not conceptually different from conventional alloys and segregation patterns follow in general the canonical relations (Friedel model, Hammer-Nørskov model). However, magnetism and configurational entropy counteract chemical driving forces and in some cases segregation is prevented even at low temperature.
For Cr-Mn-Fe-Co-Ni in vacuum we found a weak segregation of Ni at low band filling ( < f 0.60) and negligible segregation at high band filling ( > f 0.65). In the presence of O, Cr and Mn segregate strongly on the surface and also in the oxide phase.
From a materials design perspective, it must therefore be considered that the composition of the surface of Cr-Mn-Fe-Co-Ni alloys may differ from that of the bulk for Cr-, Mn-, or Fe-rich alloys, or in the presence of reactants on the surface.
In general, surface segregation in HEAs is in most cases not negligible. This could turn out to be an issue or a desirable property, depending on the target application, for instance catalysis or corrosion. The utilized methodology and approach can be straightforwardly applied to other alloys, such as noble-element HEAs or refractory HEAs. For these alloy classes the canonical models may even work better due to the suppression of magnetic effects.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.