Grain Boundary Segregation Transitions and Critical Phenomena in Binary Regular Solutions: A Systematics of Complexion Diagrams with Universal Characters

A systematics of grain boundary (GB) segregation transitions and critical phenomena has been derived to expand the classical GB segregation theory. Using twist GBs as an example, this study uncovers when GB layering vs. prewetting transitions should occur and how they are related to one another. Moreover, a novel descriptor, normalized segregation strength (phi_seg), is introduced. It can represent several factors that control GB segregation, including strain and bond energies, as well as misorientation for small-angle GBs (in a mean-field approximation), which had to be treated separately in prior models. In a strong segregation system with a large phi_seg, first-order layering transitions occur at low temperatures and become continuous above GB roughing temperatures. With reducing phi_seg, the layering transitions gradually merge and finally lump into prewetting transitions without quantized layer numbers, akin to Cahn's critical-point wetting model. Furthermore, GB complexion diagrams with universal characters are constructed as the GB counterpart to the classical exemplar of Pelton-Thompson regular-solution binary bulk phase diagrams.


Introduction
Grain boundary (GB) segregation (a.k.a. adsorption in thermodynamics) is an important phenomenon for materials science because it can critically influence microstructural evolution and a broad range of materials properties [1,2]. Over decades, several classical statistical thermodynamic models have been developed. In 1957, McLean proposed a GB segregation model [3] analogous to the Langmuir surface adsorption model [4], which considered fixed segregation sites and assumed ideal adsorbate-adsorbate interactions (here, GB adsorbates = segregating solute atoms). Various researchers have studied GB segregation experimentally and proposed GB segregation models [5][6][7][8][9][10][11][12]. Notably, Hondros and Seah [13] suggested the existence of first-order segregation transition with strong adsorbate-adsorbate attraction at GBs, in an analogy to the Fowler-Guggenheim surface adsorption model [14]; this segregation transition occurs when the effective GB regular-solution parameter is >2R (R is gas constant), thereby suggesting a GB "phase separation" [15]. More rigorous models of GB segregation transitions have been proposed recently.
On the one hand, analogous to Cahn's critical-point wetting model [16], Wynblatt and Chatain predicted a prewetting transition using a lattice adsorption model [17,18]; it should be noted that similar coupled GB prewetting and premelting transitions had been suggested by phase-field (diffuse-interface) models by Tang et al. [19] and Mishin et al. [20]. On the other hand, Rickman et al. suggested first-order GB layering transitions (for a system resembling Cu-Ag) mostly based on elastic interactions in a micromechanical framework [21]. However, it is not yet known when GB prewetting vs. layering transitions should occur and how they relate to each another, which represents the first motivation of this study.
In a broader context, Hart first proposed to treat GBs as 2D interfacial phases in 1968 [22], which were more recently named as "complexions" [15,[23][24][25] to differentiate them from thin layers of bulk (3D) phases precipitated at GBs. Notably, Dillon et al. discovered a series of six discrete GB complexions with increasing total adsorption levels in Al2O3 based ceramics [15,26,27]. These Dillon-Harmer complexions [15,[26][27][28] resemble the characters of layering and prewetting transitions. However, they were found in ceramic systems with different dopants; it is yet unknown when particular types of complexions should form. Moreover, ceramic GBs are generally complex with interfacial electrostatic and London dispersion interactions [15,[29][30][31].
Various GB complexions and segregation-induced GB transitions have also been observed in simpler metallic alloys [32][33][34][35][36][37][38][39][40][41]. In addition to the lattice [18,42] and phase-field [19,20] models, 5 [21]. Notably, this model uncovers the new physics of GB segregation transitions by revealing the evolution from the layering to prewetting transition and how it depends on a small number of (mainly three) normalized thermodynamic variables. Moreover, we aim at establishing a unified and general framework to understand GB segregation transitions as a basis to develop GB complexion diagrams.

An Ising-Type Model for GB Segregation
Here, we adopt an Ising-type lattice model to represent the segregation near twist GBs (to represent either general or small-angle GBs) in a binary A-B regular solution. As shown in Fig.   1(a), each lattice site is represented by a "spin" variable nk = 0 (or 1) for the occupation of a solvent atom A (or a solute atom B). The flow chart of the model development is shown in Fig. 1(b). The Hamiltonian of a microstate is expressed as: where Vk represents the potential energy and Jkl is the Ising-type pairwise interaction: Jkl = ω  , where U is internal energy, S is entropy, A and A are chemical potentials (per atom), and NA and NB are total numbers of A and B atoms, respectively. For a symmetric twist GB, the (relative) change in the grand potential per unit area after adding the solute B atoms can be expressed using the Bragg-Williams approximation [80] in a mean-field approach as: where Xi is the solute fraction on the i th layer from the GB core, Vi represents the GB potential on the i th layer (assumed to be identical within the same layer in the mean-field model), AGB is the GB area, N0 is the number of the lattice points per unit area in each atomic layer, and the prefactor 2 corresponds to the summation of two sides of a symmetric twist GB. Here, i J and i S (defined 6 below) are the averaged pair-interaction energy and mixing configurational entropy, respectively, per atom on the i th layer. We neglect the difference in the vibrational entropy before and after mixing (by assuming them to be identical). Since we need to find the Xi profile that minimizes the grand potential, it is convenient to rewrite Eq. (2) and define a relative grand potential per unit area (by removing the Xi-independent constant terms) as: where = − is the chemical potential difference (upon replacing a solvent A atom with a solute B atom). For convenience, we can select the reference states so that   = 0 at the bulk two-phase co-existence for our numerical calculations. Here, the averaged pairwise interaction energy (i.e., the bond energies per atom referenced to the pure elements) can be expressed as: where z the coordination number (number of bonds per atom) and zv is the number of bonds per atom between two adjacent layers (so that (z − 2zv) is the number of bonds within the plane). In Eq. (4), the first term represents the interaction between atoms within the same layer and the second term corresponds to the interaction between adjacent layers. From Eq. (4), we can further obtain: where the first term is the homogenous interaction energy (corresponding the layer composition Xi) and the two other terms represent the extra energy due to compositional gradients.
The ideal mixing configurational entropy (per atom) on the i th layer is: where kB is the Boltzmann constant. By combining Eqs. (1-6), we can obtain: In the above equation, the first term represents the change in the grand potential upon adding solute B atoms, the second term arises from the pair interaction, and the third term is the ideal configurational entropy of mixing. Only Xi-dependent terms are kept in this referenced  .
At a given temperature T and bulk composition Xbulk = X (that defines ), the equilibrium composition profile (Xi) that minimizes GB energy ( = Ω ̅ + constant) can be obtained by taking /0 i X   = , leading to a set of McLean-type equations: Here, Δ is the segregation energy of solute B in the i th layer (noting X0 = X1 for the twist GB).

The GB Potential
A merit of this Ising-type model is that it can serve as a generic platform to represent various GB segregation models with the choice of the different GB potential functions. In this section, we derive a GB potential based on the Wynblatt-Chatain model [18]. This will be used as the primary example for a systematic numerical analysis of GB segregation transitions and critical phenomena in this work. Similar GB potentials can also be derived for other models. Several examples are given in Table 1.
Let us consider a lattice model with broken bonds crossing the GB plane. The GB energy for a symmetric twist GB can be written as: where ex U , ex S , and B  (= A − in this lattice model) are the GB excesses of internal energy, entropy, and solute B, respectively. Assuming (for simplicity) that bonds can only form between the adjacent layers across the twist plane (corresponding to the cases with Jmax = 1 in Refs. [18,77]), we can derive: In the above equation, (1 − ) is the fraction of broken bonds across the twist plane on each side.
In such a lattice model, the (0 K) interfacial energies for a surface or a GB of pure A are: where h i is the distance of the i th layer from the plane i = 1, Here, B K is the bulk modulus of B and A G is the shear modulus of A. See Ref. [82] for elaboration.
We also have the expressions for the (ideal) GB excess entropy of mixing: and the GB adsorption (GB excess of solute): We can plug in Eqs. (10), (15) and (16) into Eq. (9) and rearrange/simplify it to obtain: Comparing Eq. (17) with Eq. (7), we find the following relations: Here, Eq. (18) defines the GB potential for the Wynblatt-Chatain model [18].
Other forms of GB potential functions (see, e.g., Table 1) can also be adopted, which will produce similar general trends. More complicated GB potential functions can be developed in future studies to represent different types of GBs (e.g., asymmetric GBs) and the effects of GB structural changes (with the introduction of additional structural order parameters). In this study, we adopt Eq. (18) for symmetric twist GBs to establish a baseline of GB segregation transitions and critical phenomena in this regular-solution type GB segregation model.

Normalization of the Model
We define a normalized segregation strength as: We further define a dimensionless parameter to characterize the decaying of the GB potential: By the definition: ν1 = 1 and νi→∞ → 0. We note that Eqs. (19) and (20) are general definitions that are applicable to any GB potentials. In this study, we focus on the region of > 0, where the critical temperature for the bulk phase separation is: Then, we can rewrite Eq. (7) by using dimensionless parameters , νi, and zv/z, and normalized thermodynamic variables T/TC and Δμ/(zω), as: In Eq. (18), the GB potential of the first layer (V1) depends on the composition of that layer (X1) so it is not a constant. However, the third term in Eq. (18) is usually significantly smaller than the differential bonding and strain energies. Thus, we can neglect this X1-dependent term (for simplicity) to define a normalized segregation strength based on Eq. (18), as: 10 For an average general GB in a pure metal, the following empirical relation is well known: Combining Eqs. (11), (12), (23) and (24), we can obtain and adopt the following expression to represent an "average" general twist GB: An expression of for small-angle GBs as a function of misorientation angle () is derived later and given in Eq. (36). Subsequently, we will show that this normalized segregation strength dominates the GB adsorption behaviors.
We further define the ratio of the strain energy contribution to as: Combining Eqs. (13), (20), and (26), we can obtain and adopt the following expression for a (100) twist GB of a hypothetic lattice constant 0.361 nm (as an example for our subsequent analytical and numerical analyses):  [21,83]. We will show that fstrain is the second most important material parameter (after ) that affect the GB adsorption and transition behaviors.
We again note that the current model is generic, where different forms of the GB potentials can be adopted. This regular-solution type GB model and the complexion diagrams derived in the subsequent section provide a basis to understand the systematics of the GB segregation transitions and critical phenomena, akin to their bulk counterparts of regular solutions and the Pelton-Thompson phase diagrams [73]. 11

Grand States of GB Adsorption
Let us first analyze the ground states at T = 0 K. We use a non-negative integer n to represent the number of adsorption layers on one side of the GB (so that the absorption Γ = 2nN0 for the symmetric twist GB for a perfect "quantum" state at 0K). For n = 0, Ω ̅ 〈0〉 = 0 represents a GB state without segregation. At T = 0 K, we can simplify Eq. (22) for the n th GB ground state (n  1), denoted as "〈 〉", as: Here, a larger enables the transition to occur at a more negative Δμ/(zω) (lower B  ).
For a medium segregation system ( 〈1〉 ≥ > 〈∞〉 , as shown in Fig. 2 The 〈0 ↔ + 1〉 "skip" transition line (n  1) is solved by letting 01 0 n+  =  = , as: For weak segregation systems ( < 〈∞〉 ), the 〈0〉 state is stable at all chemical potentials at 0 K. This threshold can be obtained by letting n →  in Eq. (30), as: Eqs. (29)-(32) can be used to construct GB segregation diagrams at 0 K. One primary example is shown in Fig. 2(a) for an average general (100) twist GB in an FCC metal (zv/z = 1/3) with fstrain = 0.5 from the Wynblatt-Chatain model [18]. More instances derived from different GB potential functions can be found in Suppl. Fig. S2 in the SM for different models or systems.

Systematics of Complexion Diagrams for Average General GBs
With a clear understanding of the ground states at 0 K, we can now investigate the GB segregation transitions and critical phenomena at finite temperatures. We can express our model with five independent dimensionless variables or parameters: (a) normalized temperature T/TC, (c) normalized segregation strength , (d) fraction of the strain contribution fstrain, which determines the decaying of GB potential νi in the Wynblatt-Chatain model [18] according to (Eq. 27), and (e) the fraction of the one-side out-of-plane bonds zv/z (which is a crystallographic parameter dependent on the crystal structure and the orientation of the twist plane).
Here, (a) and (b) are thermodynamic variables; (c) is the dominating, while (d) and (e) are secondary, material or crystallographic parameters. In the numerical analysis, we adopt Eqs. (25) and (27) as the GB potential for an "average general GB" (1 − = 1/6 and zv/z = 1/3 to represent general (100) twist GBs in FCC alloys, unless noted otherwise). 13 By solving the equilibrium Xi profile via minimizing Ω ̅ in Eq. (7) (or the normalized Eq. (22)), we can compute GB adsorption  (Eq. (16)) as a function of two thermodynamic variables (T/TC and Xbulk or Δμ/(z)) for given material parameters ( , fstrain, and zv/z). system ( = 1) shows a complete series of first-order layering transitions with discontinuous increases in  ( Fig. 3(b)), mimic its 0 K character. Here, the  in the <n> state corresponds to nominally 2n monolayers (albeit not exact integer numbers at T > 0 K due to entropic effects) for the twist GB. With increasing temperature, each first-order transition vanishes at a GB critical (roughening) temperature ( GB crit. 〈 ↔ +1〉 ), above which the transition becomes continuous ( Fig. 3(b)).
On the other hand, a weak segregation system ( = 0.4) shows a single prewetting (thinthick) segregation transition vanishing at a prewetting critical (PWC) point at GB crit. , where the equilibrium segregation states do not correspond to any integer numbers of monolayers ( Fig. 3(d)).
This case is analogous to the Cahn critical-wetting model [16,18]. In the corresponding T/TC-Xbulk GB segregation diagram (Fig. 3(e)), the prewetting transition line starts (at the low-temperature end) from TW (on the boundary of the bulk miscibility gap, which corresponds to a first-order wetting transition, i.e., wetting at T > TW, in the phase separation region), and it terminates (at the high-temperature end) at GB crit. PWC .
After the merging, the 〈1〉 state is no longer stable; with increasing temperature, the actual adsorption deviates from the nominal values (i.e., the "quantum" numbers become fuzzy) until the first-order 〈0 ↔ 2〉 transition ends at a merged GB critical point at GB crit. 〈0↔2〉 . The complete merging of these two layering transitions at 〈0↔2〉 corresponds to the onset of the medium segregation region in the GB ground state (at 0 K) shown in Fig. 2(a). With further reduction of , the GB layering transitions further merge, producing a series of merged transition lines starting from 0K at 〈 〉 and ending at GB crit. 〈0↔ +1〉 . In the weak segregation region ( < 〈〉 ), all layering transitions lump into a prewetting transition [16,18] (Fig. 4(f)). In the 3D adsorption diagram ( Fig.   4(a)), the prewetting vanishes at -dependent GB crit. PWC = GB crit. 〈0↔〉 . The prewetting is divergent at Δμ = 0 to (complete) wetting. Fig. 2(a) is the T = 0 K cross section of the 3D adsorption diagram shown in Fig. 4(a), which sets the behaviors of strong, medium, and weak segregation systems at the ground states. ( ) and ( ) 1 GB crit.
In the current example, zv/z = 1/3 so that 〈0↔1〉  0.67TC and 〈1↔2〉  0.33TC, which agree with the actual values shown in Fig. 3  T  , at least for the two cases fstrain = 0.5 and 1 (Fig. 3(b), Fig. 4 (b, c), and Suppl. Here, we suspect that the 〈2 ↔ 3〉 transition is stabilized with a high critical temperature than that of the 〈1 ↔ 2〉 transition due to a strain effect. In summary, all the cases discussed in this section exhibit universal characters in computed GB complexion diagrams. These results collectively suggest that is the dominant factor controlling the segregation transitions and critical phenomena, while fstrain and zv/z play secondary roles to influence the specific positions of the first-order transition lines and critical points.

The Equivalence of General and Small-Angle GBs and Influences of the Misorientation
In addition to general GBs, this model can also represent small-angle twist GBs following an approach by Wynblatt et al. [82]. This is a mean-field approximation to obtain the effective GB potential averaged over the plane parallel to the GB. Specifically, the general and small-angle GBs have virtually identical GB segregation diagrams if the normalized segregation strength is identical (achieved by selecting different strain and/or differential bonding energies, as well misorientation angle  for small-angle GBs, to match the same ).
Here, we use small-angle (100) twist GBs in an FCC alloy as an illustrating example. In this mean-field approximation, the relation between twist angle Δθ and broken bond fraction can be expressed as [82]: Moreover, the elastic energy is also proportional to the GB free volume or (1 − ) [82]: Plugging in Eqs. (34) and (35)   Each computed GB segregation diagram can represent three equivalent cases (two small-angle GBs and a general GB). These comparisons of three series of equivalent GBs have been conducted for both fstrain = 0, as shown in Fig. 9(a), and fstrain = 1, as shown in Fig. 9(a). Again, fstrain has a secondary influence here.

Normalized Segregation Strength
The normalized segregation strength defined in this study is a useful descriptor that can consider the combined effects of bonding and strain energies for both general and small-angle GBs, as well as misorientation for small-angle GBs. These factors had to be treated separately in prior studies [ We should note that this equivalence is held under the approximations adopted in this meanfield model and affected by the accuracies of GB potentials. For example, for small-angle GBs, periodical GB dislocations will lead to 2D segregation patterns within the GB plane, which is not captured in the mean-field approach. In addition, the GB potential functions adopted are only firstorder approximations (which can be improved via developing more realistic GB potentials).
In summary, the normalized segregation strength can be used to forecast the systematic trends in GB segregation transitions and critical phenomena. It can help to establish the equivalence between small-angle and general GBs.

GB Counterparts to Pelton-Thompson Phase Diagrams
In most materials thermodynamic textbooks, we start to construct binary alloy phase diagrams using two (a solid and a liquid) regular solutions, following Pelton and Thompson's foundational article in 1975 [73]. They can produce a systematics of binary regular-solution phase diagrams with most common features in real binary alloy phases. These Pelton-Thompson regular-solution phase diagrams serve as a basis to understand the real alloy phase diagrams that can often be more complex. Moreover, they can forecast useful trends for real alloys as an approximation.
Based on a regular-solution GB model, this study establishes GB counterparts to the Pelton-Thompson regular-solution bulk phase diagrams [73]. Via using normalized variables (T/TC and The bulk regular-solution model can be subsequently extended to the full calculation of phase diagram (CALPHAD) models to construct more accurate bulk phase diagrams. Likewise, the current regular-solution type GB model and analysis can also be further improved for developing more accurate GB complexions diagrams (as elaborated in §4.6).

Comparison with Experiments
To our best knowledge, no systematic experimental data exist to validate the temperature and chemical potential dependent GB complexion transitions in binary alloys, but several individual cases have been characterized and can be compared with our model. On the one hand, bilayer complexions (i.e., the 〈1〉 state) are pervasive in strong segregation systems like Ni-Bi [33,34], Cu-Bi [85], and Si-Au [51], with > 1 (estimated to be ~1.3, ~8.3, ~3.1, respectively, based on a Miedema type model [86]). On the other hand, nanometer-thick amorphous-like interfacial films consistent with prewetting layers were found in a weak segregation system Cu-Zr (estimated  0.01) [36]. These observations support the current model. We also note that W-Ni [37] and Mo-Ni [32], which exhibit coupled GB premelting and prewetting (with experimental characterization of general GBs formed at high temperatures), also have large estimated of 1.31 and 3.41, respectively. However, the high-temperature premelting (interfacial disordering) effects (represented in the experiments [32,37]) are not considered in the current model.
Notably, the current model revealed a systematics of GB segregation transitions and critical phenomena (as discussed above in §4.3). It serves as a basis to develop more realistic GB complexion diagrams, where the effects of structural transitions (reconstruction [34], broken symmetry [46], or disordering [29,32,37]) can be further considered.

Other GB Segregation Models
As we have noted earlier, the current lattice GB model can also represent other GB segregation models by selecting different GB potential functions (e.g., those shown in Table 1).
Notably, Rickman et al. and GB complexion diagrams with the same universal characters. The current model can represent 20 not only the general high-angle GB, but also small-angle GBs. In general, we expect similar trends and universal characters in GB segregation diagrams regardless the GB potential adopted, while the specific details can vary (being more or less accurate for different GB segregation models).

Comparison with Multilayer Surface Adsorption
It should be noted that an analogous systematics of layering to prewetting adsorption transitions has been previously derived for multilayer adsorption of inert gas molecules on an attractive substrate by Nakanishi and Fisher [87] and Pandit et al. [88]. This study shows parallel Interestingly, analogous interfacial adsorption/segregation transitions and critical phenomena can exist in these apparently different physical systems. While analogous phenomena are well established for surface physical absorption [87,88], the systematics of the layering and prewetting transitions, roughening, and critical phenomena, along with the universal characters in GB complexion diagrams, are derived for the GB segregation in binary alloys for the first time in this work.

Limitations and Generalization of the Current Model
This current lattice model can be further generalized to represent symmetric tilt or asymmetric 21 (tilt or mixed) general GBs by introducing different GB potentials, with some additional assumptions. For asymmetric GBs, two separate GB potentials should be introduced to characterize the segregation profiles in the two different lattices, with appropriate boundary conditions to couple the two sides of segregation. In such cases, the layering and prewetting transitions at each side can occur at different chemical potentials, but they should couple and interact one another. Odd numbers of segregation, where are observed in experiments [15,26,27], can be produced for either asymmetric GBs or symmetric GBs with an atomic plane at the center.
However, simple lattice models do not consider the possible reconstruction of GBs, which can change the symmetry of GBs (e.g., breaking a mirror symmetry [46] or change to different local crystalline structures or 2D symmetries [34,64]).
The current analysis is based on regular solutions with positive pair-interaction parameters ( > 0) where GB segregation transitions will occur. For regular solutions with negative pair-interaction parameters ( < 0), superlattice ordering and disordering may occur, which represent separation physical phenomena that should be analyzed in a different study.
The current model does not consider the (premelting or liquid-like) interfacial disordering that can often occur at high temperatures [35,37] and other structural transition (e.g., reconstruction or change of symmetry) [33,34,46]. In this regard, the current model treats the chemical (adsorption or segregation) transitions only. Yet, this current study reveals rich new physics of layering and prewetting transitions and related critical phenomena. As we have briefly discussed previously, more complicated GB potential functions may be developed to represent the effects of GB disordering or reconstruction in future studies. In such case, additional order parameters need to be introduced to describe premelting-like interfacial disordering [35,37] or interfacial reconstruction [33,34,46], and the GB segregation potential will depend on such order parameters.
These further refinements are feasible, but nontrivial. They will enable further investigation of coupled GB segregation and structural transitions.

Conclusions
We have derived a systematics of GB segregation transitions and critical phenomena in binary regular-solution alloys. Notably, a normalized segregation strength is introduced to represent the overall effects of strain and bond energies for both general and small-angle GBs, as well as the 22 misorientation for small-angle GBs. We showed that strong segregation systems with large undergo a series of layering transitions, which gradually merge and lump into a prewetting transition (without quantum numbers) with reducing . We revealed universal characters in the GB segregation complexion diagrams.
This study has not only revealed a rich spectrum of GB segregation transitions and critical phenomena to enrich the classical GB segregation theory, but also established a generic and extendable model for understanding GB segregation transitions and critical phenomena. Notably, we have established the GB counterparts to the Pelton-Thompson regular-solution bulk phase diagrams [73], which can also serve as a starting point of fundamental importance to systematically understand GB transitions and develop GB complexion diagrams.

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.

Acknowledgement
We acknowledge partial supports by the UCI MRSEC Center for Complex and Active

Supplementary Material (SM)
The 51-page SM includes: • Supplementary Figures S1-S8 Wynblatt-Chatain [18] Rickman et al. [21] 1/6 0      driven by (a) differential bond energy (fstrain = 0) or (b) strain energy (fstrain = 1). Note that Fig. 3(a) represents a mixed case (with fstrain = 0.5) that is in between these two cases. Detailed computed results are shown in Suppl. Fig. S4 and Fig. S6, respectively, as well as the slide show of "General GBs Examples I and III" in the Appendix, in the SM.