The mixing and its inverse, separation/demixing, that occur when forming a solid solution are mostly understood through the heuristic, semi-empirical Hume-Rothery rules. In his Ph.D. thesis and following publications, Hume-Rothery coined the term “electron compounds”, pointing to the violation of the prevailing “valence rule” in the mixing/reactions of inorganic compounds. (1) This followed the efforts of Clendinnen, Rivett, and Alkins, who established the basis of the formation of solid solutions using the phase rule. (2) Hume-Rothery invoked the role of noncovalent interactions of valence electrons and their associated volumetric changes to provide insight beyond the phase rule. The culmination of these studies was the Hume-Rothery rules, a set of criteria that have since dominated the discourse on binary solid solution formation. The Hume-Rothery rules for solid solution formation are described as follows: (i) a 15% similarity in size and (ii) a similarity in electrochemical nature must exist, and (iii) the low valent metal dissolves the higher valent metal. Except for (i), these rules are qualitative with the role of valency deemed invalid by some. (3) Advances in understanding the electronic structure (4) and bonding (5) of solid solutions, however, validated and expanded the application of these rules. Darken and Gurry enabled the quantification of (ii) by introducing electronegativity (φ* ), (phi star) thus enabling their translation into two-dimensional maps. (6) Hume-Rothery and Darken–Gurry plots, however, failed to establish reliable solubility rules for both interstitial solid solutions and substitutional solid solutions, (7−9) there is a need for better tools. Analogous Chelikowsky (Miedema coordinates) plots (10) focus on the electrochemical nature of solid solutions, as they use φ* and the electron density at the boundary of the bulk atomic cell (n_b), hence have improved predictions compared to Darken–Gurry plots. Given the limitations of Chelikowsky plots and the simplicity of Darken–Gurry plots, Alonso and Simozar captured the contribution of atomic size, φ* (phi star), and n_b based on the relationship of the latter two parameters on the heat of formation (ΔH_c). (DelH_c) (11) In essence, Alonso–Simozar plots relate the Wigner–Seitz radius to the heat of formation, giving them a slightly improved prediction power compared to the previously mentioned plots. In search of a more predictive criterion, the Gschneidner electronic crystal structure Darken–Gurry (ECSDG) and the electronic and crystal structures size (ECS2) methods were developed. (7) In the latter method, crystal structure was brought to the foreground with its correlation to the electronic structure of the metal. In developing the ECS2 method, Gschneider points to the potential redundancy that exists in Darken–Gurry plots, since electronegativity is dependent on both the radius (size) and the valency. However, is size in Hume-Rothery and Darken–Gurry plots a proxy for something other than just the electronic properties of the metal?

In creating these semi-empirical rules, no reference to size of the material is made, even though solid solutions are needed from the nano- to the macroscale. In many metal processing techniques, especially with recent advances in additive manufacturing, metallic powders are required. How will these rules apply with associated surface-to-bulk constraints at ambient conditions? Surface oxides and curvature-dictated pressure jump conditions must be considered. Herein, we revisit solubility/mixing in metals, particularly considering surface oxidation and the associated speciation. Fundamentally, because all of these rules have not utilized broader thermodynamics considerations (PV (π ) (pi)and non-PV (τ )(tau) work, see Figure 1), there is a need for a different viewpoint...

...Thermodynamics, however, is a tool that allows us to determine what is likely to happen or not. In a simplified approach, the evolution in ΔG_mix (del G_mix) is often evaluated against stoichiometry (Figure 2a). Considering kinetics, we realize that the mixing materials are in different phases at some point, which should in turn perturb free diffusion and affect the overall free energy due to the generation of new interfaces (Figure 2b). (22) Extending the existence of phases to include surfaces/interfaces work (ΔG_mix,τ ) (Del) (tau) ( leads to a complex free energy space (Figure 2c) that is akin to one we highlighted before. Given that surfaces evolve with their surroundings, such an energy landscape is dynamic and an easier entry point to engineering the entire materials state. The surroundings-dictated non-PV work (τ ) (tau) affects not only the kinetics of a reaction (diffusion) but also its thermodynamic state (Figure 2b). Although including this dimension in the free energy landscape adds complexity, it represents a more accurate depiction of miscibility in a real system, especially in powders...

Figure 1. A three-dimensional (3D) schematic of the mixing free energy landscape based on both PV (π ) (pi) and non-PV (τ ) (tau) work. Emergent multiple relaxation pathways are revealed, implying that a set of combinations of elements can relax into multiple potential wells.

Figure 2. (a) A 2D energy landscape for mixing. (b) A 3D energy landscape for mixing with effects of the external work function τ manifesting as proportion of segregated phases. (c) A hypothetical 3D energy landscape for mixing capturing dependency on the tensors from PV (π ) (pi) and non-PV (τ ) (tau) work. (d) A cuboctahedral 55-atom cluster, where the atomic positions in a core–shell system and the matrix of segregation energies for 12 late transition metal binary system are indicated. Reproduced with permission from ref (22). Copyright 2009 American Chemical Society. (e) XPS depth analysis of the EGaIn passivating oxide layer. Reproduced with permission from ref (24). Copyright 2012 American Chemical Society. (f) Scanning electron microscopy and energy dispersive X-ray spectroscopy analysis of a heat-treated GaInSn particle...

Figure 3. A redox-based solubility and surface speciation 3D miscibility plot (preferential interactivity parameter)