Grain boundary segregation models for high-entropy alloys: Theoretical formulation and application to elucidate high-entropy grain boundaries

Grain boundary (GB) segregation models are derived for multi-principal element alloys (MPEAs) and high-entropy alloys (HEAs). Differing from classical models where one component is taken as a solvent and others are considered solutes, these models are referenced to the bulk composition to enable improved treatments of MPEAs and HEAs with no principal components. An ideal solution model is first formulated and solved to obtain analytical expressions that predict GB segregation and GB energy in MPEAs and HEAs. A regular solution model is further derived. The GB composition calculated using the simple analytical expression derived in this study and data from the Materials Project agrees well with a prior atomistic simulation for NbMoTaW. The simplicity of the derived analytical expressions makes them useful for not only conveniently predicting GB segregation trends in HEAs but also analyzing nascent interfacial phenomena in compositionally complex GBs. As an application example, the models are used to further derive a set of equations to elucidate an emergent concept of high-entropy grain boundaries


I. INTRODUCTION
Grain boundary (GB) segregation (aka adsorption) is one of the most classical and important phenomena in physical metallurgy and ceramics.Various statistical thermodynamic models for GB segregation have been developed and discussed by McLean, 1 Fowler and Guggenheim, 2 Hondros and Seah, 3 Lejček and Hofmann, 4 Wynblatt et al., [5][6][7] and many others (see, e.g., a book by Lejček 8 ).While many of the prior models focused on binary systems, statistical thermodynamic models for GB segregation in conventional multicomponent alloys with one principal element, most notably the classical Guttmann model for ternary alloys, 9 have also been derived.
1][12][13] Understanding GB segregation in HEAs and MPEAs is both scientifically interesting and technologically important.A limited number of atomistic simulations have been performed to model GB segregation in HEAs and MPEAs, 14,15 which generally represent complex and time-consuming tasks.GB segregation in HEAs has also been modeled by simplified density and CALPHAD-based approaches. 16,17Overall, GB segregation in HEAs and MPEAs has not yet been thoroughly investigated, in part, due to the compositional complexity that makes quantitative analyses challenging.
Recently, an emergent concept of high-entropy grain boundaries (HEGBs) has been proposed 18 and further elaborated 19 as the GB counterparts to HEAs.It was proposed that effective GB entropy can be positive and increase with an increasing number of components in saturated multicomponent alloys (where the chemical potentials are pinned by precipitated secondary phases). 18,19uch HEGBs can possess reduced GB energy and an increased total adsorption amount with increasing temperature, which can stabilize nanocrystalline alloys (nanoalloys) at high temperatures. 18,19rior statistical thermodynamic models for GB segregation assume one principal component as the solvent with one or more minor component(s) as the solute(s).Differing from classical models, here I formulate GB segregation models referencing the bulk composition to enable improved treatments of HEAs and MPEAs with no principal component.Such models can also be applied to a conventional multicomponent alloy with one principal component (e.g., to analyze the so-called type I HEGBs in Sec.V B).In this work, an ideal solution model is first formulated and solved to obtain analytical expressions to predict GB segregation and GB energy.Subsequently, a regular solution model is derived.It is shown that the GB composition calculated using a simple analytical expression derived in this work and data readily available from the Materials Project agree well with a prior sophisticated atomistic simulation for NbMoTaW. 20The models and derived analytical expressions are useful for not only conveniently predicting trends in GB segregation in HEAs but also analyzing nascent interfacial phenomena in compositionally complex GBs.As an application example, the derived models are used to further derive a set of useful equations for ideal and symmetric model systems to quantitatively elucidate the emergent concept of HEGBs in Sec.V.

II. THEORETICAL FORMULATION
Using the Gibbs approach, the GB energy of a multicomponent alloy can be expressed as the interfacial excess of the grand potential The first two terms are the contributions from GB excess of internal energy (u XS GB ) and GB excess of entropy (s XS GB ), respectively, and T is the temperature.In the third term, μ i is the chemical potential and Γ i is the corresponding GB adsorption amount (interfacial excess) of the ith component (i = 1, 2, 3, … , N).
The Gibbs adsorption equation states These are the two fundamental interfacial thermodynamic equations to analyze GB segregation.Here, I shall note a typo in Eq. ( 2) in the prior perspective on HEGBs, 19 where it should have no dP term in the Gibbs adsorption equation because the GB excess volume is zero (v XS GB ¼ 0; albeit it did not affect the subsequent analysis) in the Gibbs convention.However, GB free volume (that can be defined as Δv free GB ; (@γ GB /@P) T ¼ À P i Γ i (@μ i /@P) T ¼ À Here, GB segregation is interfacial adsorption.I shall note that the "GB segregation" and "GB adsorption" are equivalent in thermodynamics, and they are used interchangeably in this article.

A. Derivation of an ideal solution model and its analytical solution
Let me consider a large-angle twist GB in a multicomponent ideal solution, as schematically shown in Fig. 1 (similar to a case of a twist GB with J max = 1 in the Wynblatt et al. model for a binary alloy 7,21 but being generalized for a multicomponent alloy here).The GB excess of internal energy for an ideal solution can be expressed as Here, n is the number of atoms in the atomic plane [so that the number of "GB sites" per unit area is Γ 0 ; 2n for this twist GB (Fig. 1), assuming the adsorption is limited within the two GB planes, which is true for ideal solutions but only an approximation for regular solutions]; X GB i and X Bulk i are the GB and bulk atomic fractions of the ith component, respectively; z is the total coordinates (number of bonds per atom); z v is the number of bonds per atom between two adjacent layers; e ii is the self-bonding energy of the ith component (referenced to free atoms so that e ii , 0); Q is the fraction of the broken bonds at the GB core between two GB planes at the twist GB (Q can often be set to 1/6 to represent an average large-angle general GB to match the empirical relation γ GB /γ Surface ¼ 1/3); and ΔE strain i (>0) is the strain energy of the ith component in the bulk, which is assumed to be relaxed at the GB planes.
FIG. 1. Schematic illustration of the lattice model for a twist grain boundary (GB), which is adapted from Wynblatt et al.'s model for a twist GB with J max = 1 in a binary alloy, 21 but generalized for a multicomponent alloy (including HEAs and MPEAs) here.Each atom has z bonds (z ¼ z l þ 2z v ), including z l lateral bonds within the atomic layer and z v bonds between two neighbor atomic layers parallel to the twist GB.It is assumed that Q fraction of the bonds is broken at the GB core between the two GB layers of the twist GB (Q is often set to 1/6 to represent a general large-angle GB to ensure γ GB /γ surface ¼ 1 3 to match typical experimental values).It is assumed, for simplicity, the adsorption is limited at the two GB layers (while acknowledging multilayer adsorption can take place in strong segregation regions in regular solutions 21,22 ).The differential bonding energies (due to broken bonds) and strain energy relaxation at the GB sites can drive GB adsorption (aka segregation).
For a HEA or MPEA without a principal component, ΔE i strain should be calculated using the classic Friedel model 23 but with respect to the weighted mean atomic radius and modulus of the bulk phase, where r i and K i are the atomic radius and bulk modulus of the ith component, respectively, and r and G are the weighted means of the atomic radius and shear modulus for the "matrix" phase It is noted that this model suggests that strain energy relaxation can take place for an HEA even without GB segregation.This is reasonable because the lattice distortion can cause significant atomic-scale strains inside a bulk HEA phase in a random solution.
Assuming a random solution, the GB excess configurational entropy can be expressed as where k is the Boltzmann constant.The adsorption amount (interfacial excess) of the ith component is given by The chemical potential of the ith component in an ideal solution is Combining the above equations, GB energy for a multicomponent ideal solution can be expressed as Here, je ii j ¼ Àe ii as e ii , 0. Using the jth component as a reference, the above equation can be rewritten as where γ (0) GB,j ( ¼ 1 2 Γ 0 Qz v je jj j) represents broken bonds contributed GB energy of the pure component j without any adsorption and relaxation.If we take j = 1 to represent the only principal component (the so-called "solvent") and ΔE strain 1 vanishes as X Bulk 1 ! 1, Eq. ( 10) reduces to an equation that is used for conventional multicomponent alloys (referenced to component 1, the "solvent"). 24etting (@γ GB /@X GB i )j X GB k ,k=i,j ¼ 0 (dX GB i ¼ ÀdX GB j and dX GB k ¼ 0 for all k = i or j), a Langmuir-McLean type GB adsorption equation 1,3,4 can be derived as Here, Δh (0) ads:(i!j) , where "0" in the subscript denotes the ideal mixing approximation, is defined to represent the enthalpic change associated with swapping one atom of the ith component from the bulk phase with one atom of the jth component at the GB, Here, a negative segregation enthalpy (Δh (0) ads:(i!j) , 0) implies the GB enrichment of the ith component (with respect to the jth component).
Alternatively, the bulk composition can be used as a reference, which can be a better approach to treating an HEA (or MPEA) without a principal component.Defining and Equation ( 9) can be rewritten as where Differentiation of Eq. ( 15) assuming dX GB i ¼ ÀdX GB j and dX GB k ¼ 0 for all k = i, j (i.e., swapping one atom of the ith component from the bulk with one atom of jth component at the GB) implies that the following equation holds for any pair of i and j (so that the value of the following expression is a constant independent of i or j): Here, g 0 and κ 0 are constants.Thus, X GB i can be solved as a function of κ 0 , The constant κ 0 can be determined by Thus, an analytical solution is obtained, Combining Eqs. ( 15) and ( 17) produces Using Eq. ( 19), an analytical solution for GB energy for a multicomponent ideal solution is obtained as follows:

B. Derivation of a regular solution model
The ideal solution model can be further extended to a regular solution model.For a regular solution, the pair-interaction parameter is defined as A regular solution reduces to an ideal solution when all ω ij ¼ 0. GB energy for a multicomponent regular solution can be expressed as where u XS(ω) GB is a regular solution (pair-interaction) excess term that should vanish when all ω ij ¼ 0.
For a regular solution, a Guttmann-like 9,25 adsorption equation can be derived (assuming fixed GB sites of identical adsorption enthalpy for simplicity) as where Here, Δh (0) ads:(i!j) ¼ Δh (γ) ads:(i!j) þ Δh (ε) ads:(i!j) , expressed in Eq. ( 12) for an ideal solution, includes the contributions from bond/surface energy Δh (γ) ads:(i!j) ¼ 1 2 Qz v (je ii j À je jj j) and strain energy (Δh (ε) ads:(i!j) ¼ À(ΔE i strain À ΔE j strain )), and it is independent of the composition.The third term, pair-interaction contribution Δh (ω) ads(i!j) , which will be derived next, depends on the bulk and GB compositions (X Bulk k and X GB k ) and pair-interaction parameters (ω ij ).Thus, Δh ads:(i!j) is not a constant for a regular solution (vs a constant Δh (0) ads:(i!j) in the Langmuir-McLean type model for an ideal solution), so Eq. ( 25) represents a Guttmann-type 25 adsorption equation (as shown subsequently).
For a regular solution, Δh (ω) ads(i!j) represents the pair-interaction excess segregation enthalpy (beyond the ideal-solution term) to swap one atom of the ith component in the bulk phase with one atom of jth component at the GB.To assess this term, let me first evaluate the pair-interaction excess internal energy to remove one atom of the ith component in a homogenous bulk phase, which can be expressed as Here, Δu (ωÀBulk) i is defined as pair-interaction excess bond energies to create a vacancy via removing one atom of the ith component (by breaking z bonds) in a regular solution [where e ij ¼ 1 2 (e ii þ e jj ) þ ω ij ], referenced to (subtracted by) the bond energies in an ideal solution [where e ij ¼ 1 2 (e ii þ e jj ) and ω ij ¼ 0].Thus, the pair-interaction excess term is ω ij per i-j bond (e ij ).With this definition, the well-known expression of the regular solution excess enthalpy (i.e., internal energy, assuming no change in volume) of mixing per molar can be derived as where N A is the Avogadro constant and Ω ik ; zω ik N A is the regularsolution parameter.Here, factor ½ is included because the total bond energies of a bulk phase are twice the sum of broken-bond energies associated with removing each atom (because every bond is shared by two atoms).
Next, the excess pair-interaction energy to remove one atom of the ith component in a region with a compositional gradient (on the mth plane perpendicular to the compositional gradient, without any broken bonds) is given by where z l ¼ z À 2z v is the number of bonds per atom in the lateral atomic plane.After rewriting, the first term in the above equation is Δu (ωÀBulk) i [Eq.(27)] and the second term is resulted from the compositional gradient.It is further assumed, as a simplification, that GB adsorption is still limited to two GB planes at the core of this twist GB for a regular solution.Taking k for a twist GB and considering the existing broken bonds (of fraction Q of z v bonds per atom) at the twist GB, the excess pair-interaction energy for removing one atom of the ith component at the GB can be deduced as Subsequently, the pair-interaction excess energy term in the GB segregation enthalpy for swapping one atom of the ith component in the bulk phase with one atom of the jth component at the GB can be expressed as the sum of four terms associated with (1) removing one atom of the ith component in the bulk phase (Δu (ωÀBulk) i ) to create a vacancy, (2) removing one atom of jth component at the GB (Δu (ωÀGB) j ) to create a vacancy, (3) inserting one atom of the ith component at the GB to fill the vacancy (ÀΔu (ωÀGB) i ), and (4) inserting one atom of jth component in the bulk phase to fill the vacancy (ÀΔu (ωÀBulk) j ), which can be expressed as Here, Δh (ω) ads(i) is defined as It represents the difference in the pair-interaction excess energy terms for removing one atom of ith component in the bulk phase vs removing one atom of ith component at the GB.Thus, the pairinteraction excess GB segregation enthalpy can be expressed as In Appendix A, I show that a simplified case of this derived expression for N = 2 is fully consistent with Wynblatt et al.'s model for binary alloys. 7,21Thus, the current model can be considered as a multicomponent generalization of Wynblatt et al.'s binary alloy model for a case of bilayer adsorption at a twist GB. 7,21 Using , the above equation can also be rewritten as Here, the first term in the above equation is equivalent to that in Eq. (5b) in Ref. 24 [by taking The above equation can be rewritten to a form similar to the Guttmann model 9,25 Δh However, α0 ikÀj = αik À (α ij þ αkj ) because of the extra brokenbond term [so that the GB is no longer a regular solution with the broken bonds to follow the regular-solution-based Guttmann model exactly; noting that α0 For a regular solution, GB segregation (aka adsorption) enthalpy for component i can be defined as so that Δh ads:(i!j) ; Δh ads:(i) À Δh ads:(j) .Similar to Eqs. ( 17) and ( 20), the following two equations can also be derived: and For a regular solution, the GB composition (X GB i ) needs to be solved numerically using the above equation because Δh (ω) ads(i) also depends on X GB i .If the pair-interaction term Δh (ω) ads(i) is small, the above equation may be solved iteratively.Otherwise, one should search for the initial X GB i values that lead to convergence of iterating Eq. (38) or use a better algorithm to solve Eq. ( 38) numerically.
Practically, using the regular solution model needs to quantify N(N À 1)/2 pair-interaction parameters and solving Eq. ( 38) numerically, which may not be trivial for a HEA.Thus, the analytical expression shown in Eqs. ( 20) and ( 22) for an ideal solution, which can be readily quantified and produce results agree well with the atomistic simulations for a HEA (as shown in Sec.III), can often be taken as the first-order approximation to predict useful trends.
However, the following equation for GB energy, which has been rigorously proved for ideal solutions, may only be used as an approximation when Δg ads:(i!j) is not a constant for a non-ideal solution: Again, I emphasize that Eq. ( 41) is rigorously held for an ideal solution [i.e., Eqs. ( 15) and ( 22)].For non-ideal solutions, a more rigorous and accurate expression of GB energy can be obtained by quantifying u XS(ω) GB for a regular solution or, more generally, the non-ideal u XS(nonÀideal) GB and s XS(nonÀideal) GB contributions beyond ideal solutions.

III. A CALCULATION EXAMPLE
Using NbMoTaW as an example, GB segregation was computed using the analytical expressions derived for an ideal solution [Eqs.(20) and (22)] and the density functional theory (DFT) data of the lattice parameters, bond energies, bulk, and shear moduli taken from the Materials Project. 29The data and calculated relevant thermodynamic quantities are shown in Table I for an equimolar NbMoTaW. Figure 2 shows the computed GB composition and GB energy as functions of temperature for this equimolar NbMoTaW.Strong segregation of Nb and depletion of W and Ta, particularly at low temperatures, were predicted for the equimolar NbMoTaW by this model (Fig. 2).For the fixed GB composition, the lattice model predicted that GB energy increases with increasing temperature due to temperature-induced desorption [Fig.2(b)].
In general, the adoption of readily available data from the Materials Project allows consistent and convenient parameterization of the model.Along with the analytical expressions derived in this study, segregation trends can be forecasted for HEAs and MPEAs with ease, which are useful.
To further benchmark the model, the calculated results from the analytical solution derived in this work was compared with a prior hybrid Monte Carlo and molecular dynamics (MC/MD) simulation of a polycrystal with six randomly inserted GBs at 300 K conducted by Li et al. 20 in Table 3.In the hybrid MC/MD simulation of the polycrystal, the bulk (grain) composition deviated from the equimolar composition due to GB segregation and a small average grain diameter of ∼7.5 nm in a closed system (even if the overall composition is equimolar in the 11 × 11 × 11 nm 2 simulation supercell).For example, the predicted GB segregation of Nb, which was indeed observed in the hybrid MC/MD simulation, resulted in the depletion of Nb inside the grains in the hybrid MC/ MD simulation.To do a fair comparison, the analytical expression was used to calculate GB composition (X GB i ) for the non-equimolar bulk composition of Nb 0.155 Mo 0.246 Ta 0.280 W 0.319 , which matched the observed average grain composition in the MC/MD simulation.The (110) twist GB (z v ¼ 2 and Q = 1/6) was selected to represent a general large-angle GB with moderate segregation (z v /z ¼ 1/4).Note that the weighted average radius, modulus, and bond energy, as well as strain energies and segregation enthalpies calculated for this non-equimolar alloy (Table II) are slightly different from those calculated for the equimolar NbMoTaW (Table I).
As shown in Table III, the calculated GB composition (∼68% Nb, ∼24% Mo, ∼6% Ta, and ∼1% W) using a simple analytical expression derived in this study agrees well with the GB composition (∼58% Nb, ∼27% Mo, ∼14% Ta, and ∼2% W) obtained by the sophisticated hybrid MC/MD simulation. 20Not only the relative segregation trend is correct, but the calculated GB composition using the analytical expression [Eq.(20)], even if it is derived from a simplified ideal solution model, quantitatively agrees with the sophisticated hybrid MC/MD simulation (that needed first fitting a DFT supported machine learning potential) 20 with reasonable errors.
This success suggests that the derived analytical expressions, using the data readily available from the Materials Project, 29 can be robustly useful in predicting trends of GB segregation in HEAs and MPEAs.

IV. DISCUSSION OF THE DERIVED MODELS
GB segregation in HEAs has been so far rarely characterized or modeled because of the compositional complexity.The derived analytical expressions for the simplified ideal solution model, along with readily available high-quality data from the Materials Project, 29 present a facile method to quickly forecast GB segregation trends in HEAs and MPEAs.This model can be further extended to predict GB segregation in other types of GBs (beyond the twist GBs analyzed here), as well as anisotropic segregation (after deriving the relevant GB segregation enthalpy expressions for multicomponent systems in a follow-up study).The model for regular solutions has also been derived.However, obtaining consistent and reliable data and developing robust numerical algorithms to solve the regular solution model for HEAs are non-trivial, which should be investigated further in future studies.In this regard, the derived analytical expressions for the simplified ideal solution model, which can be readily parameterized by the high-quality data from the Materials Project, 29 is robustly useful.The predictability has been shown for NbMoTaW.
The current models also have a few limitations.The analytical solutions are limited to the case of fixed number of GB adsorption sites with an identical segregation enthalpy.It is known that multilayer adsorption, layering and prewetting transitions, and GB critical phenomena, can take place in strong segregation region in regular solutions. 6,21,22,30,31Such interesting interfacial phenomena can in principle be solved numerically with the lattice models (as shown for simpler dilute binary solutions with a principal component). 6,21,22,30,31However, it is still challenging to analyze such complex interfacial phenomena in HEAs and MPEAs due to their compositional complexity.Moreover, the lattice models do not consider interfacial structural transitions, [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] including GB TABLE I. Parameters and calculated results for the equimolar NbMoTaW.All input data were taken from the Materials Project. 29Bond energies were calculated from the E 0 ( ¼ À 1 2 zε ii ) fitted for the Mie-Grüneisen equation of state.Bulk and shear moduli using the Voigt-Reuss-Hill approximation were adopted.Note that the mean is identical to the weighted average value for an equimolar HEA.The calculations were performed for a large-angle (110) twist GB (z v = 2 and Q = 1/6 to represent a general GB).Using all data from density functional calculation (DFT) calculations in the Materials Project offers a consistent and accessible method for the parameterization of the model.disordering (particularly, the possible formation of liquid-like GBs in coupled premelting and prewetting regions), [34][35][36]44,[46][47][48][49][50][51] which can take place and affect GB adsorption and GB energy. In additon, the lattice models do not consider GB free volume (assumed to be zero in the lattice model), which can also affect GB segregation and other properties.Future studies should be conducted to consider and investigate these effects, together with the compositional complexity.Yet, the simplicity of the analytical expressions [particularly, Eqs. ( 20) and ( 22) for an ideal solution] derived in this study for simplified models makes them robustly useful.

Nb
In addition to predicting useful trends for GB segregation in HEAs and MPEAs via analytical expressions, the derived models and expressions can also be used to analyze nuisance interfacial phenomena in compositionally complex GBs.As an application example, the derived analytical expressions are used to further formulate a set of simplified equations to elucidate the emergent concept of HEGBs in Sec.V.

V. APPLICATION TO ELUCIDATE HEGBS A. General concepts
The rapid development of HEAs simulated a scientific interest in exploring the existence and character of their interfacial (GB) counterparts-"high-entropy grain boundaries (HEGBs)."The concept of HEGBs was originally proposed in 2016, 18 with a goal to utilize such HEGBs to stabilize nanoalloys against grain growth at high temperatures [Fig.3(a)].The relevant concepts were further elaborated in a short perspective article in 2023. 19However, rigorous full derivations have not been given in detail, in part, due to the lack of a useful GB segregation model for analyzing multicomponent alloys and HEAs.Here, such a model has been derived in this study.Thus, the analytical expressions derived here are used to further formulate a set of approximated equations to elucidate the emergent concept of HEGBs, with rigorous full derivations.
First, what is GB entropy?This may not be a trivial question.There are two general ways to define GB entropy.
In the Gibbs approach, GB excess of entropy (s XS GB ) is a welldefined quantity independent of the selection of the Gibbs dividing plane (differing from surfaces and other heterointerfaces with two different abutting phases).GB excess of entropy (s XS GB ) appears in the Gibbs adsorption equation [Eq.( 2)], a fundamental interfacial thermodynamic relation.
In an alternative and perhaps more useful definition analogous to the bulk thermodynamic relation dG ¼ ÀSdT þ VdP (so that S ¼ À(@G/@T) P ), one can define effective GB entropy to characterize the temperature dependence of GB energy, s eff: GB ; À @γ GB @T P, fixed X Bulk i or saturated

:
(42) The effective GB entropy for a unary GB can be derived from Eq. (2) as 19 where S V (¼ S/V, where S is the molar entropy and V is the molar volume) is the volumetric entropy and Δv free GB is the GB free volume (Δv free GB ¼ ÀΓ Á V).For a unary GB, s XS GB is from GB excess disorder (with structural, vibration, electronic, and other physical contributions).This equation implies that the effective GB entropy is not equal to GB excess of entropy (s eff: GB = s XS GB ) even for a simple unary GB, as there is an extra contribution from GB free volume (S V Á Δv free GB ).For a unary system, the effective GB entropy is usually positive due to the interfacial disordering and GB free volume, as shown in the above equation.
For a multicomponent GB, the effective GB entropy includes chemical, structural, and physical contributions (written below as FIG. 2. Computed (a) GB composition X GB i and (b) GB energy γ GB vs temperature curves for a high-angle (110) twist GB (z v = 2 and Q = 1/6 to represent a general GB with moderate segregation tendency) for an equimolar NbMoTaW, using the analytical expressions derived in this study.All input data were from DFT calculations in the Materials Project 29 to offer consistency and convenience for parameterization (while noting that bond energies estimated from DFT calculations are typically higher than those calculated from atomization enthalpies in prior studies, so the calculated GB energies are higher).For a fixed bulk composition, GB energy decreases with increasing temperature due to temperature-induced desorption.The current model does not consider interfacial disordering and GB free volume, which can reduce the GB energy with increasing temperature.
separate terms for simplicity, despite that they may not be independent terms), Here, s eff:(structural) GB represents structural contributions from GB free volume and interfacial structural disordering (akin to that in the unary systems discussed above but can be more complex and coupled with adsorption terms in multicomponent systems).In addition, s eff:(adsorption) GB represents chemical contributions from GB adsorption (segregation).Moreover, s eff:(phyical) GB represents the vibration, electronic, and other physical (e.g., magnetic) contributions to GB excess entropy.Although these terms are written, for simplicity, as separate terms in the above equation, they may not be independent.Various coupling effects can generally exist (and may not be easy to be expressed in any simple analytical forms).For example, coupling between GB adsorption and GB structural disordering can often be significant (particularly with the possible formation of liquid-like GBs at high temperatures [34][35][36]44,[46][47][48][49][50][51], which can also affect the GB free volume. I should note that the lattice models have no GB free volume ( P i Γ i ¼ 0 with identical V i for all i so that Δv free GB ¼ À , and various coupling effects should be considered and evaluated separately (and, if possible, added to the lattice models).
The following discussion focuses on the adsorption (a.k.a.GB segregation) effects on the effective GB entropy (s eff:(adsorption) GB ) using the lattice model derived.
Equation (2) suggests that GB adsorption reduces GB energy.5][56][57][58][59][60][61][62][63][64] However, temperature-induced GB desorption can counter this stabilization effect [Fig.3(b)] at high temperatures.Thus, it is beneficial to seek for conditions where GB energy reduces and GB segregation increases with increasing temperature in so-called HEGBs, which can offer both reduced thermodynamic driving pressures and increased kinetic solute-drag 65 pressures to stabilize nanoalloys [Fig.3(b)]. 19were used and the calculations were also performed for a large-angle (110) twist GB (z v = 2 and Q = 1/6 to represent a general large-angle GB).Individual lattice parameters: radii, bulk, and shear moduli and selfbonding energies for Nb, Mo, Ta, and W are listed in Table I.The weighted average radius, modulus, self-bonding energy, as well as the strain energies and segregation enthalpies, which depend on the bulk composition, are shown here.The computed GB composition at 300 K for this non-equimolar Nb 0.155 Mo 0.246 Ta 0.280 W 0.319 alloy is shown in Table III 20 Note that bulk (grain) composition deviated from the equimolar composition due to GB segregation and small grain size in a closed system in the MC/MD simulation.To do a fair comparison, the values of X GB i were calculated using the analytical expression based on the nonequimolar bulk composition (Nb 0.155 Mo 0.246 Ta 0.280 W 0.319 ) matching that observed in the MC/MD simulation.showing the stabilization of three nanoalloys with high-entropy grain boundaries (HEGBs) against grain growth at high temperatures, in comparison with nanocrystalline unary and binary nanoalloys. 18,52,53(b) A proposed mechanism 19 of stabilizing a nanoalloy against grain growth at high temperatures via HEGBs.To stabilize a nanoalloy, reduced thermodynamic driving pressure and increased critical kinetic solute drag pressure for grain growth have to achieve a balance below the solid solubility limit.Increasing temperature can destabilize the nanoalloy by inducing GB desorption, while HEGBs can counter this effect via increasing total adsorption with an increasing number of components. 19It was proposed 19 that HEGBs can simultaneously reduce the thermodynamic driving force and increase the kinetic solute drag, thereby increasing the high-temperature stability of nanoalloys.This figure is

FIG. 4. Schematic illustration of GB segregation (aka interfacial adsorption in thermodynamics
) in multicomponent alloys in three scenarios with different influences on effective GB entropies.(a) For a single-phase multicomponent alloy with a fixed bulk composition and a fixed number of GB adsorption sites, GB energy should decrease with increasing temperature due to temperature-induced desorption, so that the effective GB entropy is negative (without considering the effects of interfacial disordering and GB free volume, which can give rise to positive effective GB entropy).(b) Interfacial disordering can decrease GB energy with increasing temperature to produce a positive effective GB entropy in a unary system. 66In multicomponent systems, interfacial disordering (and widening), particularly the formation of liquid-like GBs, 34,36 can enhance GB adsorption.The coupling of interfacial disordering and adsorption may subsequently promote the formation of HEGBs, but this hypothesis has not been rigorously approved yet.(c) Type I HEGBs can form from GB adsorption effects in saturated alloys with (N À 1) precipitated secondary phases that pin the chemical potentials of the primary phase, where GB energy decreases with increasing temperature and effective GB entropy increases with the increasing number of components.A further analysis shows that the type II HEGBs of similar characters can also form in HEAs saturated with one or more precipitated secondary phase(s) that pins the chemical potential of the segregation element(s) in the (primary) high-entropy phase.The coupling of scenarios (c) and (b) may further increase the effective GB entropy, pointing to a future exploration direction.
For a single-phase multicomponent alloy with a fixed bulk composition and a fixed number of GB adsorption sites, GB energy increases with increasing temperature due to temperature-induced desorption or de-segregation [Fig.4(a)].This effect is seen in Fig. 2(b) for NbMoTaW computed from the lattice model (albeit that effects of interfacial disordering and GB free volume, which are not considered in the lattice model, can reduce GB energy with increasing temperature).Thus, the effective GB entropy (from the adsorption contribution in the lattice model) is generally negative for a fixed bulk composition, 19 s eff: (adsorption, fixed HEGBs cannot exist from GB adsorption effects in this scenario.However, the formation of HEGBs can be promoted in two scenarios, as shown in Figs.4(b) and 4(c) and explained below.
On the other hand, it was proposed 18 and recently elaborated in a perspective article 19 that GB adsorption alone can lead to the formation of HEGBs in saturated alloys [Fig.4(c)] with chemical potential(s) pinned by precipitated secondary phase(s) to produce positive effective GB entropy, 19 s eff: (adsorption, saturated) GB ¼À @γ GB @T P, saturated (X Bulk i on the solvus) (. 0 possible): (46)   Such HEGBs may possess three characters, (1) GB energy reduces with increasing temperature, (2) the total GB adsorption amount increases with increasing temperature, and (3) the positive effective GB entropy increases with increasing number of components.
Two basic types of HEGBs are envisioned. 19Type I HEGBs can form in a saturated conventional multicomponent alloy with (N À 1) precipitated secondary phases that pin the chemical potentials of the primary phase [Fig.4(c)].Type II HEGBs can form in a HEA or MPEA with (N À 1) principal elements and one strongly segregating element, where one or more precipitated secondary phase(s) pins the bulk chemical potential(s).Here, I will also further propose and evaluate mixed type I and type II HEGBs in Sec.V C.
Subsequently, the GB segregation model derived in this study is used to further derive approximated equations to elucidate type I and type II HEGBs, as well as the general mixed cases, with full derivations that had not been provided previously. 19ere, I mostly adopt a simplification of ideal solutions where Δg ads:(i) ¼ Δh (0) ads:(i) , by using Eq. ( 22) that was rigorously derived for the ideal solution lattice model to elucidate the key characters of type I and type II HEGBs, as well as the general mixed cases.However, the key trends and characters revealed below in Sec.V can be generalized to non-ideal solutions with a more general Δg ads:(i) expression by assuming more approximations.
Considering a case of fixed bulk composition with (N À 1) segregating solute elements of identical X Bulk i and Δh (0) ads:(i!1) , the reduction of GB energy upon segregation depends on the total amount of solute atoms P N i¼2 X Bulk i , but not on the number of solute components (N À 1).Moreover, for negative Δh (0) ads:(i!1) , GB energy increase with increasing temperature (without consider interfacial disordering) due to the temperature-induced desorption.Thus, there is no HEGB effect for the GB adsorption contribution in this fixed bulk composition scenario.
However, HEGBs can exist in a fixed chemical potentials scenario, represented by a saturated multicomponent alloy. 19In this scenario of type I HEGBs, the chemical potentials are pinned by (N À 1) precipitated secondary phases in an N-component system (with no compositional degree of freedom according to the Gibbs phase rule), so that the bulk composition moves along the maximum solvus line with increasing temperature.
Assuming, for simplicity, a precipitate of a binary M x S y line compound (M = matrix component 1 and S = solute component i), the solvus for a dilute solution (X 1 ! 1) in the 1i binary system can be expressed (based on the derivations in Appendix B) as (for a dilute ideal solution), (48)   where Δg ppt: i(1) (%Δh ) is the free energy (enthalpy) of precipitation per atom of the ith component.Here, α i(1) ¼ 1 for an ideal solution.
Considering a dilute ideal solution (X 1 % 1) and approximating the solubility of the ith component in the multicomponent alloy with its solubility in the 1i binary system (X Bulk i % X

Binary Solvus 1Ài
% exp[Δh ppt: i(1) /(kT)]), plugging Eq. ( 48) into Eq.( 47) produces Δh ppt: i( 1) where Δh ads:Àppt: i ; Δh (0) ads:(i!1)À Δh ppt: i(1) (taking to be a positive value here) represents the free energy to swap one atom of the ith component at the GB with one atom of the 1st component in the precipitate.A more accurate (yet still approximated) solution for a non-ideal solution is where non-ideal Δg ads:Àppt: i (instead of Δh ) is adopted and the dimensionless coefficient f [→ 1 for a saturated dilute ideal solution with chemical potentials pinned by a set of (N À 1) binary line compounds)]is expressed as Here, (=1 for an ideal solution) is a ratio representing the co-doping influence on the solid solubility of the ith component.In addition, α i(1) ¼ 1 for an ideal solution and (X 1 ) Àβ i (1) approaches to unity for a dilute solution.
Next, let me illustrate the general character of type I HEGBs with a simplified case of a "symmetric" ideal solution (i.e., with identical properties for (N À 1) binary 1-i systems, i = 2, … , N].Assuming f i ¼ 1 and all Δg (where "S" in the subscript represents solute i = 2, … , N) is a constant for all i ≠ 1 for simplicity, the following expression can be obtained: Here, Δh should be in the range of ∼0.10 ± 0.06 eV/atom.Thus, kT where , N À 1)
Figure 5 shows an analysis of type I HEGBs based on Eq. ( 53) for a hypothetic saturated multicomponent alloy with one principal component and (N À 1) segregating minor components, where (N À 1) precipitated secondary phases pin the chemical potentials.Specifically, computed GB energy reduction vs temperature curves for Δh ads:Àppt: S(1) ¼ 0:1, 0.05, and 0.15 eV/atom, respectively, are shown in Fig. 5.In all cases, GB energy decreases with increasing temperature (producing a positive effective GB entropy, Àdγ GB /dT) in the saturated multicomponent alloy.Furthermore, the effective GB entropy (the negative slope of the curve) increases with increasing number of components.In Fig. 5, the reductions in GB energies are plotted in normalized parameter (γ GB À γ (0) GB )/Γ 0 to represent more general results.For a reference value, if the Γ 0 value for the Ni (100) twist GB is adopted, (γ GB À γ (0) GB )/Γ 0 ¼ À0:18 eV corresponds to a (γ (0) GB À γ GB ) value of ∼0.969 J/m 2 , which represents a substantial reduction in GB energy.
An isothermal section of ternary phase diagram displaying the saturated composition for binary (N À 1 ¼ 1) and ternary (N À 1 ¼ 2) alloys are also shown in Fig. 5.This ternary phase diagram illustrates the increased total solid solubility of all solutes in the primary bulk phase from a binary alloy (represented by the black dots) to a ternary alloy (represented by the red dot), which results in the increased total GB adsorption (segregation amount) to reduce GB energy.The same mechanism also exists and is expected to provide HEGB effects for N .3.
The general characters of type I HEGBs are well illustrated in Fig. 5 for the simplified cases of identical binary subsystems in normalized parameters.For type I HEGBs, Eqs. ( 49) and ( 50) can be applied to forecast multicomponent alloys with different Δh or Δg values for different segregating solutes and precipitates.More realistic modeling of real multicomponent alloys and type I HEGBs can be conducted by applying Eq. ( 22) directly using the maximum solvus composition as the bulk composition, which can be obtained by bulk CALPHAD methods.If the multicomponent maximum solvus composition cannot be determined (due to lacking CALPHAD data), the following approximation (that is rigorously held for ideal solutions) can be adopted as a first-order approximation: It should be noted that in this case that X Bulk i (i = 1) is different from (smaller than) the overall (initial) composition due to the precipitation.This approach (i.e., applying Eq. ( 22) directly using the maximum solvus composition as the bulk composition or adopting Eq. ( 55) if the multicomponent solubility data is not available) enables us to use an ideal solution with the bonding energies and segregation enthalpies mimicking the real multicomponent alloy, along with real binary phase diagrams, to predict useful trends.¼ (a) 0.1 eV/atom, (b) 0.05 eV/atom, and (c) 0.15eV/atom, respectively.In all cases, GB energy decreases with increasing temperature (producing a positive effective GB entropy) in the saturated multicomponent alloy, where the chemical potentials are pined by the (N À 1) precipitated secondary phases.Furthermore, the effective GB entropy (Àdγ GB /dT) increases with increasing number of components.If Γ 0 value for the Ni (100) twist GB is adopted as a reference, (γ GB À γ (0) GB )/Γ 0 ¼ À0:18 eV corresponds to a GB energy reduction of ∼0.969 J/m 2 .An isothermal section of ternary phase diagram displaying the saturated composition for binary (N À 1 ¼ 1) and ternary (N À 1 ¼ 2) alloys are also shown.This ternary phase diagram illustrates the increased total solid solubility of all solutes in the primary bulk phase from a binary alloy (represented by the two black dots) to a ternary alloy (represented by the red dot), which results in the increased total GB adsorption to reduce GB energy.The same mechanism is also expected to provide HEGB effects for N .3.

C. Type II HEGBs
Next, let me consider a HEA or MPEA with (N À 1) principal elements (components 1, 2, … , N À 1) and one strongly segregating solute element (component N).For this case (again assuming Δg ads:(i) ¼ Δh (0) ads:(i) for an ideal solution for simplicity), Eq. ( 22) can be rewritten as Here, Δh (0) ads:(N) is significant for a strong segregation component N and cannot be neglected.
To illustrate bulk high-entropy effects to reduce GB energy with increasing temperature and increasing number of components in type II HEGBs, a saturated (N À 1)-component ideal HEA solution with a strongly segregating Nth component (assuming X Bulk N ,, 1) can be considered.Here, the solvus line for each i-N binary system can be expressed as X Binary Solvus N(i) % exp[Δh ppt: N(i) /(kT)] for simplified ideal solutions (ignoring precipitation entropy).As derived in Appendix B for a multicomponent ideal solution, the corresponding bulk solid solubility of the Nth component in the HEA limited by the precipitation of any binary line compound in any i-N system (for any i < N) is given by The maximum effects to reduce GB energy with increasing temperature for type II HEGBs can be achieved for a HEA of specific bulk composition that can be in equilibrium with a (N À 1) precipitates (so that there is no thermodynamic degree of freedom according to the Gibbs phase rule).Assuming, for simplicity, precipitates are (N À 1) binary intermetallic line compounds, the (optimal) composition of the HEA with a maximum reduction of GB energy can be determined by the chemical potentials of the (N À 1) precipitates, by finding the bulk composition {X Bulk i } that simultaneously satisfies (N À 1) equations for i = 1, 2, … , (N À 1) for a simplified ideal solution, In general, for a non-ideal solution, the (temperaturedependent) maximum solid solubility and the corresponding bulk composition that produce maximum HEGB effects can be calculated via CALPHAD.
For a theoretical analysis to illustrate the key characteristics of type II HEGBs, I will consider a highly simplified case of ideal solution, where all (N À 1) binary subsystems are further assumed, for simplicity, to have identical thermodynamic properties.Thus, β N(i) ¼ x/y ; β, Δh (0) ads:(i) ¼ Δh (0) ads:(P) , and Δg ppt: (where "P" in the subscript represents Principal component i = 1, … , N À 1) are taken to identical values for all i ≠ N. In this simplified case, (N À 1) secondary phases will precipitate simultaneously.In this symmetric case with ) and for all i ≠ N. Thus, Δh ppt: N(P) kT Then, the first expression in Eq. ( 56) can be rewritten using the relation Δh (0) ads:(N!P) ¼Δh (0) ads:(N) À Δh (0) ads:(P) , as Here, the approximation is obtained assuming Δh (0) ads:(P) % 0, X Bulk N ,, 1, and exp[ÀΔh (0) ads:(N!P) /(kT)] .. 1 for a strong segregating component N. In this case,X Bulk N is maximized in the HEA saturated with the Nth component, with the composition moving along the multicomponent solvus line (in equilibrium with (N À 1) precipitates, which will appear simultaneously in this simplified symmetric system; noting that in a real asymmetric system, however, the first precipitate pins the chemical potential of the Nth component).Plugging Eq. ( 59) into Eq.( 60) and using Δh ads:Àppt: N(P) ¼ Δh (0) ads:(N!P) À Δh ppt: N(P) produce the following expression for the simplified "symmetric" ideal solution: Thus, the adsorption contribution to the effective GB entropy for type II HEGBs is given by /(kT)] is a dimensionless factor that increases with increasing (N À 1).Thus, s eff:(adsorption, saturated) GB is positive and increases with increasing (N À 1), the number of principal elements in the HEA/ MPEA. Figure 6 further shows an analysis of type II HEGBs based on Eq. ( 61) for a hypothetic saturated MPEA or HEA with (N À 1) principal components plus one strongly segregating minor component.Here, it is assumed that the precipitate is in equilibrium with (N À 1) simultaneously precipitated secondary phases of binary intermetallic compounds of identical M x S y (β ¼ x/y) stoichiometry and all binary subsystems have identical thermodynamic properties in this symmetric system for simplicity.
Computed GB energy reduction vs temperature curves for Δh  61) for a hypothetic saturated MPEA (or HEA) with (N À 1) principal components plus one segregating minor component.It is assumed, for simplicity, that the primary phase is in equilibrium with (N À 1) precipitated secondary phases of binary intermetallic compounds of identical M x S y (β ¼ x/y) stoichiometry and all (N À 1) binary subsystems have identical thermodynamic properties.Computed GB energy reduction vs temperature curves for Δg ads:Àppt: N(P) % Δh ads:Àppt: N(P) ¼ 0:1 eV/atom and β ¼ (a) 1, (b) 2/3, and (c) 3/2, respectively.In all cases, GB energy decreases with increasing temperature (producing a positive effective GB entropy) in saturated MPEAs (HEAs).Furthermore, the effective GB entropy (Àdγ GB /dT) increases with increasing number of components in the MPEA/HEA primary phase.If the Γ 0 value of the NbMoTaW (110) twist GB in this lattice model is adopted as a reference, (γ GB À γ (0) GB )/Γ 0 ¼ À0:18 eV corresponds to a GB energy reduction of ∼1.55 J/m 2 .An isothermal section of the ternary phase diagram displaying the saturated compositions for binary (N À 1 ¼ 1) and ternary (N À 1 ¼ 2) alloys are also shown.This ternary phase diagram illustrates the increased solid solubility of the segregating element (always set to be component N) with an increasing number of components in the primary bulk phase, which results in the increased total GB adsorption.The same mechanism is also expected to provide HEGB effects for N .3.
are shown in Fig. 6.In all cases, GB energy decreases with increasing temperature (with a positive effective GB entropy) in the saturated HEAs and MPEAs.Furthermore, the effective GB entropy (Àdγ GB /dT) increases with increasing number of components in the MPEA/HEA primary phase.Here, the reduction in GB energy is also plotted in normalized parameter (γ GB À γ (0) GB )/Γ 0 for generality.If the Γ 0 value for the NbMoTaW (110) twist GB in this lattice model is adopted, (γ GB À γ (0) GB )/Γ 0 ¼ À0:18 eV corresponds to a GB energy reduction of ∼1.55 J/m 2 as a reference value, which represents a substantial reduction in GB energy.
An isothermal section of ternary phase diagram displaying the saturated composition for binary (N À 1 ¼ 1) and ternary (N À 1 ¼ 2) alloys are also shown in Fig. 6.This ternary phase diagram illustrates the increased solid solubility of the segregating element (always set to be the Nth component) with increasing number of the principal components in the primary bulk phase, which results in the increased GB adsorption of the Nth component.This mechanism is also expected to provide HEGB effects for N .3. Here, the bulk high-entropy effect lowers the bulk chemical potentials (with respect to binary intermetallic precipitates) so that more adsorptions can be accommodated at GBs before the occurrence of precipitation.
In other words, HEGBs form from a competition between GB adsorption and precipitation.If the precipitation is inhibited by the bulk high-entropy effects that lower the chemical potentials of the HEA phase with respect to precipitation, it promotes the accommodation of GB adsorption to reduce GB energy.
It should be noted that this symmetric ideal solution model (with identical Δh ads:Àppt: N(i) and β N(i) values for all binary subsystems) shows a special case with is no thermodynamic compositional degree of freedom according to the Gibbs phase rule, as the (N À 1) precipitated secondary phases pin the bulk chemical potential of the Nth component simultaneously.This is, however, not a general case.Assuming ideal mixing and X Bulk N ,, 1 for simplicity, for a system with different Δh ads:Àppt: N(i) and β N(i) values for different binary subsystems, Eq. ( 61) can be generalized (still for a simplified ideal solution) to Here, the chemical potential of the Nth component is pinned by the first precipitate of the i-N compound with the minimum value of (X Bulk i N(i) /(kT)], which corresponds to component i = i min in Eq. ( 63) and determines the actual X Bulk N in the HEA phase saturated with the Nth component.
Similar to type I HEGBs, more realistic modeling of real HEAs with type II HEGBs can be conducted by applying Eq. ( 22) directly using the actual bulk solvus composition (the temperature-dependent solid solubility of the segregating Nth component) obtained by bulk CALPHAD methods.If the multicomponent solvus composition cannot be determined because of lacking thermodynamic data, the following approximation can be adopted to estimate the bulk solid solubility of the Nth component from binary solvus line (albeit the existence of the ternary and multicomponent precipitates, if any, can further limit the bulk solid solubility of the segregating Nth component) using a simplified (ideal solution) approximation, Here, it is assumed that Δh (0) ads:(P) % 0 and P P j¼1 X Bulk j % 1.The above equation suggests strong HEGB effects to reduce GB energy with increasing temperature, where the GB energy reduction increases with increasingS Á P β , where P is the number of principal (solvent) elements and S is the number of segregating solute elements.This is a more pronounced effect than type I and type II HEGBs.
Similar to the prior treatments, mixed HEGBs can also be modeled by applying Eq. ( 22) directly using the actual bulk solvus composition obtained by bulk CALPHAD methods.If the multicomponent solvus compositions are not available, the following approximation can be adopted for the precipitated secondary phase that pins the chemical potential of the jth component (P þ 1 j P þ S) from binary solvus curve read in relevant binary phase diagrams (that are usually available): Similar to prior cases, X Bulk j (j .P) is different from (smaller than) the overall fraction of the jth component in the system due to the precipitation.The actual X Bulk i (i P) in the bulk phase should be determined based on the mass conservation, if the amounts of precipitated phases are not negligibly small.If the total fraction of secondary phases is negligibly small, it can be assumed that the relative ratios of principal elements remain unchanged after the precipitation.

E. Further discussion
As a further note, the total effective GB entropy can be greater than the GB adsorption contribution derived here for all types of HEGBs, with the additional contributions from GB disorder and GB free volume, as well as the coupling effects that possibly enhance both GB adsorption and GB disordering in a positive feedback loop.
As discussed previously, one limitation of the lattice models is that they do not consider GB free volume and GB structural transitions (including GB disordering).A recent study clearly showed that the GB energy is linearly correlated with the free volume. 69GB free volume can alter adsorption and GB entropy.As discussed in Sec.V A, GB disordering at high temperatures, 5,34,36,44,[46][47][48][49][50][51] which is also not considered in the current lattice models, can increase effective GB entropy.In unary systems, GB disordering and free volume are known to produce a positive effective GB entropy based on Eq. ( 43), which was supported by a prior atomistic simulation that showed reduced GB energy with increasing temperature in pure Ni. 66 In multicomponent systems, interfacial disordering and widening, particularly with the formation of liquid-like GBs in the coupled premelting and prewetting regions, [34][35][36]44,[46][47][48][49][50][51] can significantly affect (often enhance) GB adsorption [Fig. 4(b)]. TThus, it is likely that s eff :(structural) GB .0 in Eq. ( 44) for multicomponent GBs and the coupling of interfacial disordering and adsorption can further promote the formation of HEGBs (possibly even in the fixed bulk composition scenario as proposed in Fig. 4(b), albeit not yet proven rigorously by modeling or experiments).The effects of GB free volume and GB disordering, which can often be significant, should be considered in refined models in future studies.
Here, I should also point out that GBs have five macroscopic (crystallographic) degrees of freedom, and their properties depend on GB character.We expect general GBs may share some similar characters (albeit large GB-to-GB variations).The analyses presented here represent the average large-angle general twist GBs, while we acknowledge the general existence of significant GB-to-GB variations (large standard deviations).The model presented here can be further generalized to assess anisotropy and variations of GB properties (e.g., with respect to twist angle or other macroscopic degrees of freedom).Special GBs (particularly low-energy GBs) behave differently (and should be modeled one by one separately).
Finally, an appealing scientific goal is to engineer polycrystalline MPEAs or HEAs via tailoring various GB properties (including GB energy, entropy, free volume, adsorption, and structure), along with precipitated secondary phases, through controlling compositions and processing recipes to influence both microstructural evolution and final materials properties.Akin to the success of GB engineering of conventional alloys, it can be envisioned to utilize HEGBs to tailor GBs and subsequently materials properties [including high-temperature stability of nanoalloys (as shown in Fig. 3) 18,19 and other thermodynamic, kinetic, mechanical, and other functional properties].

VI. CONCLUSIONS
In summary, GB segregation models have been derived for multicomponent alloys, particularly for HEAs and MPEAs.Differing from classical models where one component is taken as a solvent and others are considered solutes, the models derived in this study are referenced to the bulk composition to enable improved treatments of HEAs and MPEAs with no principal component.An ideal solution model is first formulated and solved to obtain analytical expressions to predict GB segregation and GB energy.A regular solution model is further derived.The simplicity of the derived analytical expressions makes them useful for predicting trends.It is demonstrated that the GB composition calculated using the simple analytical expression derived in this study and data readily from the Materials Project agree well with a prior sophisticated atomistic simulation, illustrating the predictability and usefulness of the simple analytical expressions.
As an application example of using the derived models to investigate nascent interfacial phenomena in compositionally complex GBs, the derived expressions are used to further formulate a set of approximated equations to elucidate an emergent concept of HEGBs.Two types of HEGBs, as well as the mixed type, are analyzed, where HEGBs can be realized in saturated alloys with chemical potentials being pinned by precipitated secondary phases.Type I HEGBs can form in a saturated conventional multicomponent alloy, where (N À 1) precipitated secondary phases pin the chemical potentials of the primary phase.Type II HEGBs form in HEAs or MPEAs with (N À 1) principal elements and one strongly segregating elements, where one or more precipitated secondary phases pin the bulk chemical potentials of the segregating component.Such effects can be further enhanced in the mixed type I and II HEGBs, as suggested by the derived model.These HEGBs possess three distinct characters: (1) GB energy reduces with increasing temperature, (2) the total GB adsorption amount increases with increasing temperature, and (3) the positive effective GB entropy increases with increasing number of components.The effects of interfacial structural disordering and GB free volume, which are not considered in the lattice models, can further increase the effective GB entropy.Coupling of interfacial disordering and GB adsorption may further enhance HEGB effects, which should be investigated in future studies.verifies the derivation of the multicomponent regular solution model presented in this work [Eq.(33), reduced to a binary system with i = S and j = M], which can be considered a generalization of Wynblatt et al.'s binary regular solution model 6,7 to multicomponent regular solutions.
was used, and j = 1, in Eq. (5b) in Ref. 24].A broken-bond contribution term is also included in the above equation [but not in Eq. (5b) in Ref. 24].
lattice models) and do not consider the other structural (e.g., interfacial disordering) and physical terms.Thus, s eff :(structural) GB , s eff :(phyical) GB

FIG. 3 .
FIG. 3. (a)Prior experiments18,19 showing the stabilization of three nanoalloys with high-entropy grain boundaries (HEGBs) against grain growth at high temperatures, in comparison with nanocrystalline unary and binary nanoalloys.18,52,53(b) A proposed mechanism19 of stabilizing a nanoalloy against grain growth at high temperatures via HEGBs.To stabilize a nanoalloy, reduced thermodynamic driving pressure and increased critical kinetic solute drag pressure for grain growth have to achieve a balance below the solid solubility limit.Increasing temperature can destabilize the nanoalloy by inducing GB desorption, while HEGBs can counter this effect via increasing total adsorption with an increasing number of components.19It was proposed19 that HEGBs can simultaneously reduce the thermodynamic driving force and increase the kinetic solute drag, thereby increasing the high-temperature stability of nanoalloys.This figureisreprinted from a perspective article by Luo and Zhou, Commun.Mater.4, 7 (2023).Copyright 2023 The Author(s), licensed under a Creative Commons Attribution 4.0 International License.
FIG. 3. (a)Prior experiments18,19 showing the stabilization of three nanoalloys with high-entropy grain boundaries (HEGBs) against grain growth at high temperatures, in comparison with nanocrystalline unary and binary nanoalloys.18,52,53(b) A proposed mechanism19 of stabilizing a nanoalloy against grain growth at high temperatures via HEGBs.To stabilize a nanoalloy, reduced thermodynamic driving pressure and increased critical kinetic solute drag pressure for grain growth have to achieve a balance below the solid solubility limit.Increasing temperature can destabilize the nanoalloy by inducing GB desorption, while HEGBs can counter this effect via increasing total adsorption with an increasing number of components.19It was proposed19 that HEGBs can simultaneously reduce the thermodynamic driving force and increase the kinetic solute drag, thereby increasing the high-temperature stability of nanoalloys.This figureisreprinted from a perspective article by Luo and Zhou, Commun.Mater.4, 7 (2023).Copyright 2023 The Author(s), licensed under a Creative Commons Attribution 4.0 International License.

FIG. 6 .
FIG.6.Analysis of type II HEGBs based on Eq. (61) for a hypothetic saturated MPEA (or HEA) with (N À 1) principal components plus one segregating minor component.It is assumed, for simplicity, that the primary phase is in equilibrium with (N À 1) precipitated secondary phases of binary intermetallic compounds of identical M x S y (β ¼ x/y) stoichiometry and all (N À 1) binary subsystems have identical thermodynamic properties.Computed GB energy reduction vs temperature curves for Δg ads:Àppt:

TABLE II .
Calculated results for the non-equimolar Nb 0.155 Mo 0.246 Ta 0.280 W 0.319 .The same input data from the Materials Project to compare with atomistic simulation results.

TABLE III .
Comparison of the results from the analytical solution derived in this work with hybrid Monte Carlo and molecular dynamics (MC/MD) simulation of a polycrystal at 300 K in a supercell with six randomly inserted GBs and an average grain diameter of ∼7.5 nm conducted by Li et al. 1)e