Phase Boundary Segregation in Multicomponent Alloys: A Diffuse-Interface Thermodynamic Model

We invoke the concept of an interfacial phase or complexion for the interphase boundary (IB). That is, a compositionally-homogeneous layer of IB phase $i$ is governed by a fundamental thermodynamic equation distinct from that of the adjacent matrix $m$ and precipitate $p$ phases. This description follows from the classical treatment of solute segregation to free surfaces and grain boundaries (GB) [11, 28, 29, 30, 13], and the non-classical treatment of diffuse segregation to GBs in phase-field models [31, 32, 33, 34, 35].

A non-conserved phase field $\phi(\bm{x})$ variable is used to represent the phases: $m$ at $\phi=0$, $i$ at $\phi=0.5$, and $p$ at $\phi=1$. Any infinitesimal volume around the spatial location $\bm{x}$ is defined as a hypothetical thermodynamic mixture of $m$, $i$ and $p$ phases. The alloy is composed of the primary components $1$ and $2$, which act as solvents to from the two-phase system, and additional components $3,4,\ldots,\mathcal{N}$ which act as solutes. The mole fractions corresponding to a phase are represented by the phase concentration variables as $c_{\theta m}(\bm{x})$, $c_{\theta i}(\bm{x})$ and $c_{\theta p}(\bm{x})$; where, $\theta=1,2,3,\ldots,\mathcal{N}$.

We introduce interpolating functions $M(\phi)$, $I(\phi)$ and $P(\phi)$ for each of the phases $m$, $i$ and $p$. These functions serve as local phase fractions to define effective properties at $\bm{x}$ as a mixture of the phase properties. The effective concentrations $c_{\theta}(\bm{x})$ are defined by the mixture rule as

The local effective free energy density $f(\bm{c},\phi)$ is defined using the local phase-specific free energy densities $f^{\psi}(c_{2\psi},c_{3\psi},\ldots,c_{\mathcal{N}\psi})$ as

where $\bm{c}=\{c_{2},c_{3},\ldots,c_{\mathcal{N}}\}$. Under the assumption of substitutional solution and identical molar volumes of the components across the alloy, the effective concentrations satisfy $c_{1}(\bm{x})=1-\sum_{\theta=2:\mathcal{N}}c_{\theta}(\bm{x})$.

The phase concentrations $c_{\theta\psi}(\bm{x})$ at any given point are constrained by the condition of equal diffusion potential between the phases as

where ${\mu}_{\theta 1}\equiv\mu_{\theta}-\mu_{1}$ ($\theta=2:\mathcal{N}$) are the diffusion potentials. The above conditions follow from the KKS (Kim-Kim-Suzuki) phase-field formulation—originally developed for two-phase alloy solidification [36], later applied for GB segregation in two-grain alloy with GB phase [31, 32, 35], and recently developed for IB segregation in two-phase alloy with IB phase [37, 38, 39].

The functional forms of the interpolating functions are chosen to satisfy the definition of exclusive phases as $M(\phi=0)=1$, $I(\phi=0.5)=1$ and $P(\phi=1)=1$, and the phase-mixture rule $M(\phi)+P(\phi)+I(\phi)=1$. For the sake of continuity in the derivatives, $M(\phi)$ and $P(\phi)$ $\in C^{1}[0,1]$ are chosen to be differentiable, piecewise functions. A convenient choice of interpolating functions constructed from the double-well form $I(\phi)=16\phi^{2}(1-\phi)^{2}$ is shown in Fig. 2. Note that for $\phi\in[0,0.5)$, $M(\phi)=1-I(\phi)$ and $P(\phi)=0$; for $\phi\in(0.5,1]$, $M(\phi)=0$ and $P(\phi)=1-I(\phi)$.

Given the local microstructure and its properties thus far, the overall free energy of the domain of volume $V$ can be defined by the functional,

where the second term in the integrand is the gradient energy density due to $\phi$, with $\varepsilon^{2}$ representing the gradient energy coefficient.

Following classical linear irreversible thermodynamics, the concentration fields evolve temporally under mass conservation as

where $\theta=2:\mathcal{N}$. The variational derivatives take the simplified definition as $\delta\mathcal{F}/\delta c_{\theta}=\partial f/\partial c_{\theta}={\mu}_{\theta 1}$. $\mathbf{\mathcal{M}}$ is the diffusion mobility matrix with components $\mathcal{M}_{\theta\xi}=\sum_{j=1:\mathcal{N}}(\delta_{k\xi j}-c_{\xi})(\delta_{j\theta}-c_{\theta})c_{j}\mathcal{M}_{j}$. $\mathcal{M}_{\theta}$ is the atomic mobility of component $\theta$ ($=1:\mathcal{N}$); $\mathcal{M}_{\theta}$ can be expressed in terms of the phase-dependent atomic mobilities as $\mathcal{M}_{\theta}=M(\phi)\mathcal{M}_{\theta}^{m}+I(\phi)\mathcal{M}_{\theta}^{i}+P(\phi)\mathcal{M}_{\theta}^{p}$.

Following Allen-Cahn dynamics, the non-conserved field $\phi$ evolves temporally as

where $\mathcal{L}$ is the kinetic mobility parameter for the interface.

We now consider a one-dimensional, planar IB at stationary equilibrium. The microstructure fields become invariant with time and will be denoted by $\phi_{e}(x)$ and $\bm{c}^{e}(x)$. Eqs. 5 and 6 reduce to

where $\mu_{\theta 1}^{e}$ are the equilibrium diffusion potentials. $\mu_{\theta 1}^{e}$ are constant across $x$ as required by the exchange of atoms between substitutional sites [40]. The conditions (Eq. 3) for the equality of diffusion potentials at a point now read

The above relations imply that the phase-concentrations become spatially constant, i.e. $c_{\theta\psi}(x)=c_{\theta\psi}^{e}$. Therefore, the effective concentration fields (Eq. 1) are given by

Multiplying Eq. 8 with $d\phi_{e}/dx$ and integrating piecewise with respect to $x$ from $-\infty$ to $0$ and $0$ to $+\infty$ yields the following equilibrium relations between $m$, $i$ and $p$ (see Appendix A for derivation). Here, the limits of integration $-\infty$, $0$, and $+\infty$ refer to the far-field matrix, the exclusive IB and the far-field precipitate phases, respectively.

and

Eq. 9 and  12 constitute the well-known common tangent hyperplane condition for equilibrium between the bulk phase free energy hypersurfaces $f^{m}$ and $f^{p}$. Eq. II.2 and Eq. 9 constitute the parallel tangent hyperplane conditions for the equilibrium of the IB phase $f^{i}$ with respect to the bulk phases. For known convex functions of $f^{m}$, $f^{i}$ and $f^{p}$, the above conditions uniquely determine the equilibrium phase concentrations $c^{e}_{\theta\psi}$. $W_{e}$ represents the vertical distance between the two parallel tangent hyperplanes of $i$ and $m/p$. Physically, $W_{e}$ describes the free energy for the formation of a unit volume of equilibrium IB phase from the equilibrium matrix phase (or equivalently from the equilibrium precipitate phase) [41].

Substituting Eqs. 2 and II.2 in Eq. 8, we obtain the expression for the stationary phase-field profile $\phi_{e}(x)$ as (see Appendix A for derivation)

which can be integrated to obtain

The above expression depends only on the functional form of $I=16\phi^{2}(1-\phi)^{2}$. $W_{e}$ depends on the equilibrium phase concentrations, which are constants across the microstructure. Therefore, $W_{e}$ is a constant and the above expression can be readily integrated to obtain the well-known closed-form solution

The width $\lambda$ of the diffuse IB (defined by the bounds $\phi_{e}=0.1$ and $\phi_{e}=0.9$) is given by

We can now obtain expressions for the classical excess quantities from the equilibrium diffuse IB. The excess IB energy per unit IB area, $\gamma$, is the excess grand potential evaluated as (see Appendix B for derivation)

where $\pm l$ are locations within the bulk phases far-field of the interface. The integrand $2W_{e}I(\phi_{e}(x))\equiv\Omega(x)$ describes the energy density across the system in excess of the thermodynamic mixture of the equilibrium bulk phases. The integrand vanishes within the bulk phases since $I\left(\phi_{e}(\pm l)\right)\rightarrow 0$, and therefore, the integral converges. This property makes $\gamma$ an invariant thermodynamic quantity that is independent of the specific choice of the layer thickness [42].

The extensive solute excess quantities are evaluated by adapting Cahn’s layer approach [11] to the phase-field model (see Appendix C for derivations). The generalized Gibbs adsorption equation relating the change in $\gamma$ to the independent variations of intensive quantities, temperature $T$ and $\mu^{e}_{\theta}$, is obtained as [11, 13]

where $\Gamma^{(1,2)}_{S}$ is the relative entropy excess and $\Gamma^{(1,2)}_{\xi}$ is the relative solute excess for $\xi$, defined per unit interface area. The chemical potentials of the base components $1$ and $2$ have been eliminated with the aid of equilibrium conditions between the two bulk phases, and thus, $\Gamma^{(1,2)}_{1/2}=0$. The excess quantity $\Gamma^{(1,2)}_{S/\xi}$ describes the difference in the amount of entropy/component-$\theta$ between the chosen layer—containing the diffuse interface—and that of a hypothetical system. The hypothetical system is constructed from the two homogeneous bulk phases in such a proportion that the total amount of $1$ and $2$ is equal to that in the original layer [11, 13]. Analytic expressions for these quantities were realized from the equilibrium phase and concentration fields as

where $\theta=1:\mathcal{N}$, $c^{e}_{1\psi}=1-\sum_{k=2:\mathcal{N}}c^{e}_{k\psi}$, and

$C^{xs}_{\xi}$ is the Gibbs solute excess evaluated from the effective concentration fields using the Gibbs dividing surface at $x=0$. Since the equilibrium concentrations are generally unequal in the two bulk phases, $C^{xs}_{\xi}$ depends on the specific location of the dividing surface within the diffuse IB. Note that $\Gamma^{(1,2)}_{\xi}$ is independent of the position of the dividing surface, and is an invariant thermodynamic quantity like $\gamma$. A similar expression is obtained for the $\Gamma^{(1,2)}_{S}$ (Eq. C).

The Gibbs adsorption isotherm can be written in terms of the derivatives of the matrix-phase concentrations $c^{e}_{\theta m}$ as

where $\mu^{e}_{\theta}=f^{m}_{e}+\sum_{k=2:\mathcal{N}}\left(\delta_{\theta k}-c^{e}_{km}\right)\mu^{e}_{k1}$ ($\delta_{\theta k}$ is the Kronecker delta) [43].

We assume regular solution behavior for the bulk and IB phases. The free energy density-composition dependence characteristic to $\psi$ is given by

Here, the energetics of $\theta$ in all the phases $\psi$ are defined from the same pure component reference state $\{\pi\theta\}$ [44]. The reference states can have crystal structures distinct from that of the solid-solution $\psi$. $G^{\psi}_{\theta}\{\pi\theta\}$ accounts for the pure component energy of $\theta$ in $\psi$ with reference to ${\pi\theta}$. For mixing (second term), the structure of each component must be identical to that of the final solid-solution $\psi$. $L^{\psi}_{\theta\xi}\{\psi\theta,\psi\xi\}$ accounts for the non-ideal interaction energy between distinct components $\theta$ and $\xi$ in $\psi$, with reference to the pure components in $\psi$. Therefore, the parameters $G^{\psi}_{\theta}$ and $L^{\psi}_{\theta\xi}$ account for the distinct chemical and structural energetics of the bulk and interfacial phases. The final term in the equation is the ideal configurational entropy for mixing and $v_{m}$ is the molar volume.

Using Eq. II.4, the equilibrium compositions in the bulk phases are established by Eqs. 9 and  12. These constitute $\mathcal{N}$ equations in $2\left(\mathcal{N}-1\right)$ bulk-phase concentrations and allow $\mathcal{N}-2$ compositional degrees of freedom. We choose the matrix-phase solute concentrations $c^{e}_{3m},\ldots,c^{e}_{\mathcal{N}m}$ as the control variables, while the rest are to be uniquely determined.

The equilibrium composition in the IB phase $\psi=i$ is then established using Eq. II.4 in $\partial f^{i}/\partial c^{e}_{\theta i}=\partial f^{m}/\partial c^{e}_{\theta m}$ of Eq. 9. These constitute $\mathcal{N}-1$ segregation equations relating $\mathcal{N}-1$ unknowns $(c^{e}_{2i},\ldots,c^{e}_{\mathcal{N}i})$ to the known $(c^{e}_{2m},\ldots,c^{e}_{\mathcal{N}m})$. They can be written as

where

$\Delta E^{\theta 1}_{\text{seg}}$ describes the segregation energy resulting form the exchange of $\theta$ in $m$ with component $1$ in $i$. The above coupled and non-linear equations must be solved simultaneously to determine the equilibrium IB phase composition.

The segregation equations obtained in our model are equivalent to the generalized multicomponent grain boundary segregation isotherms of Guttmann [45]. For an ideal alloy ($L=0$), the equations reduce to the decoupled multicomponent versions of Langmuir-McLean [28, 46]. For a non-ideal ($L\neq 0$) alloy, the second term in Eq. 24 is identical to that of the Fowler-Guggenheim isotherm [47, 46] representing the interaction between $\theta$ and $1$. The third and following terms are the cross terms relating segregation of $\theta$ with segregation of $\xi$. For IB segregation, these interactions become important for a quaternary (or higher-order) alloy as shown in the next section.

In this section, we apply the phase-field model to explore segregation behavior in quaternary alloys. The model is non-dimensionalized by setting $x=l_{o}\tilde{x}$, $t=t_{o}\tilde{t}$, $T=T_{o}\tilde{T}$ and $f^{\psi}=\left(RT_{o}/v_{m}\right)\tilde{f}^{\psi}$ ($\psi=m,i,p$). $l_{o}$, $t_{o}$ and $T_{o}$ are the characteristic length, time and temperature, respectively, and $RT_{o}/v_{m}$ is the characteristic energy. The dimensionless quantities are denoted by the tilde symbol. The free energy density for a quaternary alloy now takes the form, $\tilde{f}^{\psi}=\tilde{G}^{\psi}_{1}c_{1\psi}+\tilde{G}^{\psi}_{2}c_{2\psi}+\tilde{G}^{\psi}_{3}c_{3\psi}+\tilde{G}^{\psi}_{4}c_{4\psi}+\tilde{L}^{\psi}_{12}c_{1\psi}c_{2\psi}+\tilde{L}^{\psi}_{13}c_{1\psi}c_{3\psi}+\tilde{L}^{\psi}_{23}c_{2\psi}c_{3\psi}+\tilde{L}^{\psi}_{14}c_{1\psi}c_{4\psi}+\tilde{L}^{\psi}_{24}c_{2\psi}c_{4\psi}+\tilde{L}^{\psi}_{34}c_{3\psi}c_{4\psi}+\tilde{T}\left(c_{1\psi}\ln{c_{1\psi}}+c_{2\psi}\ln{c_{2\psi}}+c_{3\psi}\ln{c_{3\psi}}+c_{4\psi}\ln{c_{4\psi}}\right)$. The equilibrium equations in Sec. II.1 are replaced with the dimensionless parameters: $\tilde{\varepsilon}^{2}=\varepsilon^{2}/\left[l_{o}^{2}\left(RT_{o}/v_{m}\right)\right]$ and $\tilde{\mu}^{e}=\mu^{e}/\left(RT_{o}/v_{m}\right)$.

For the parametric study of the model, we assume $(\tilde{\varepsilon}^{2},\tilde{T})=(1,1)$. The dimensionless results can be dimensionalized as $\gamma=\left(RT_{o}l_{o}/v_{m}\right)\tilde{\gamma}$ and $\Gamma_{\xi}^{(1,2)}=(l_{o}/v_{m})\tilde{\Gamma}_{\xi}^{(1,2)}$. For example, setting the characteristic scales to be $l_{o}=1$ nm, $T_{o}=300$ K and $v_{m}=7\times 10^{-6}$ m³/mol yields $\gamma=0.36\tilde{\gamma}$ J/m² and $\Gamma_{\xi}^{(1,2)}=\left(1.43\times 10^{-4}\right)\tilde{\Gamma}_{\xi}^{(1,2)}$ mol/m².

We choose an ideal binary base alloy ($1$-$2$) that forms a two-phase coexistence region in the bulk phase diagram (see Table 1 and Ref. [37]). The alloy microstructure and the equilibrium concentration profiles are shown in Fig. 3 and represented by the schematic with the label B. The matrix phase $m$ is rich in $1$ ($c^{e}_{2m}\approx 0.27$) while the precipitate phase $p$ is rich in $2$ ($c^{e}_{2p}\approx 0.73$). The IB phase $i$ has been chosen to have an equilibrium concentration as the average of $m$ and $p$ phases ($c^{e}_{2m}\approx 0.5$). The phase-field calculations yield a relative excess entropy of $\tilde{\Gamma}^{(1,2)}_{S}=0$ and an excess IB energy of $\tilde{\gamma}_{B}=1.2$ units.

Next we consider the addition of solutes $3$ and $4$ individually to form the ternaries ($1$-$2$)-$\xi$ ($\xi=3/4$). Two types of solute behaviors are considered as references–ternary non-segregating (TN) and ternary segregating (TS). The alloy microstructures of TN and TS are shown via representative schematics and equilibrium concentration profiles in Fig. 3. TN alloys are realized for $L^{i}_{\theta\xi}$ = $L^{m/p}_{\theta\xi}$ ($\theta=1,2$), i.e. when the regular solution interactions are identical or ideal across all phases in the alloy. For these cases, $\tilde{G}^{i}_{\theta}\approx 1.62$ units was derived from Eq. 23 such that $c^{e}_{\xi i}=c^{e}_{\xi{m/p}}$ is satisfied. Calculations of excess quantities yields $\tilde{\Gamma}^{(1,2)}_{\xi}=0$ and $\tilde{\gamma}=\tilde{\gamma}_{B}$. TS alloys, on the other hand, are realized for $L^{i}_{\theta\xi}<L^{m/p}_{\theta\xi}$, i.e. when the solute $\xi$ interacts more favorably with $\theta$ in $i$. These can further be noted for the ternary parameters $L^{i}_{\theta\xi}$ and $L^{m/p}_{\theta\xi}$ listed in Table 2. Calculation of excess quantities yield a positive segregation $\tilde{\Gamma}^{(1,2)}_{\xi}>0$ and a reduced IB energy $\tilde{\gamma}<\tilde{\gamma}_{\text{B}}$. For simplicity, we omit segregation cases arising from variations in the pure energies $\tilde{G}^{\psi}_{\xi}$. We also simplify the parameter space by considering that the solutes interact identically within the bulk phases $m$ and $p$, and with solvent component $1$ and $2$. This condition is captured in Table 2 by $L^{m/p}_{\theta 3}$ and $L^{m/p}_{\theta 4}$, and lead to vanishing cross terms between the solute and solvent in Eq. 24, $\tilde{L}^{\psi}_{12}+\tilde{L}^{\psi}_{1\xi}-\tilde{L}^{\psi}_{2\xi}=0$.

For the sake of completeness, we note here that the variation of excess quantities for the ternaries can be read from the limits $c^{e}_{\xi m}\rightarrow 0$ of $\tilde{\Gamma}^{(1,2)}_{\xi}$ and $\tilde{\gamma}$ contour plots presented for quaternary alloys in Fig. 7. As the matrix-phase concentration $c^{e}_{\xi m}$ is increased, TN alloys (in I–III) exhibit constant values of $\tilde{\Gamma}^{(1,2)}_{\xi}=0$ and $\tilde{\gamma}=\tilde{\gamma}_{\text{B}}$. TS alloys (in V–VIII) exhibit a positive and increasing $\tilde{\Gamma}^{(1,2)}_{\xi}$, with a correspondingly decreasing $\tilde{\gamma}$ from that of $\tilde{\gamma}_{\text{B}}$. The model parameters used for the reference alloys here were inspired by our earlier works on phase boundary segregation in binary [37] and ternary [38] systems. Kadambi et al. [38] rigorously demonstrated the validity of the current phase-field approach with the Gibbs adsorption isotherm $d\tilde{\gamma}=-\tilde{\Gamma}^{(1,2)}_{\xi}d\mu^{e}_{\xi}$.

In the case studies presented below, we examine the effect of combined solute addition in the quaternary ($1$-$2$)-$3$-$4$ alloy. The interaction between solutes governed by $\tilde{L}^{m/p}_{34}$ and $\tilde{L}^{i}_{34}$ forms the basis for understanding the segregation behavior in ($1$-$2$)-$3$-$4$ relative to ($1$-$2$)-$3$ (TN or TS) and ($1$-$2$)-$4$ (TN or TS). The different cases examined here are labeled I through IX in Table 2.

We can understand the interfacial properties of the diffuse-interface formulation (Sec. II) by first considering the interface to be a GB in a two-component, single-phase alloy. $\phi(x)$ now denotes the variation in crystallographic orientation across the diffuse GB. The bulk-phase free energies, phase concentrations and local phase fractions take the following simplified forms: $f^{p}\equiv f^{m}$, $c_{\theta p}\equiv c_{\theta m}$ and $P(\phi)\equiv M(\phi)=1-I(\phi)$ for $\phi\in[0,1]$. Therefore, the local free energy density (Eq. 2) becomes $f(c_{2},\phi)=f^{m}(c_{2m})[1-I(\phi)]+f^{i}(c_{2i})I(\phi)$. This reduced model is equivalent in most aspects to existing phase-field formulations [31, 33, 35] for GB segregation. Invoking the regular solution behavior, and imposing the limit of pure component $1$, we get $f(\phi)=G^{m}_{1}+(G^{i}_{1}-G^{m}_{1})I(\phi)$. The term $(G^{i}_{1}-G^{m}_{1})I(\phi)$ is analogous to the classical, symmetric double-well potential $W_{\phi}I(\phi)$ used in standard phase-field models for grain growth, alloy solidification and antiphase boundary motion [48]. Since the pure component energies $G^{\psi}_{1}\{\pi 1\}$ are defined from the same pure component reference state $\pi 1$ (Sec. II.4), $G^{i}_{1}-G^{m}_{1}>0$ accounts for the excess structural energy of the GB phase over that of the matrix. The GB energy $\gamma$ of this pure metal is given by $\gamma_{1}\approx 0.943\varepsilon\sqrt{G^{i}_{1}-G^{m}_{1}}$ (Eq. 17). However, for a multicomponent alloy, the equilibrium between the GB phase and the bulk matrix phase is established by the parallel tangent construction, which determines the equilibrium GB phase concentrations $\textbf{c}^{e}_{i}$ and parallel tangent distance $W_{e}(\textbf{c}^{e}_{i},\textbf{c}^{e}_{m})$. $W_{e}$ then represents the free energy for formation of a unit volume of the equilibrium GB phase from the equilibrium bulk phase, and therefore, captures not only the the structural energetics but also the compositional dependence [41]. Therefore, the GB energy for a multicomponent alloy can be obtained (from Eqs. 12 and 17) by setting $p\equiv m$, giving $\gamma\approx 0.943\varepsilon\sqrt{W_{e}(\textbf{c}^{e}_{i},\textbf{c}^{e}_{m})}$.

Now consider the interface to be the IB in a two-phase alloy, i.e. $p\not\equiv m$. The local free energy density $f(\textbf{c}^{e},\phi_{e})$ (Eq. 2) will take the form of an asymmetric double-well potential as shown below. Reducing the system to the theoretical limit of a pure component $1$, we get $f(\phi)=G^{m}_{1}+(G^{i}_{1}-G^{m}_{1})I(\phi)$ for $\phi\leq 0.5$ and $f(\phi)=G^{p}_{1}+(G^{i}_{1}-G^{p}_{1})I(\phi)$ for $\phi>0.5$. In conventional phase-field formulations for two structurally-distinct phases, a symmetric double-well potential $W_{\phi}I(\phi)$ is employed to define the excess structural energy across the diffuse IB. In the current formulation, the symmetric part is inherent in the second term $(G^{i}_{1}-G^{p}_{1})I(\phi)$ of the piecewise function, while the first terms $G^{m}_{1}\neq G^{p}_{1}$ results in the asymmetry. For a multicomponent alloy, the model additionally captures the compositional dependence: for an equilibrium, planar IB, $f(\textbf{c}^{e},\phi_{e})=f^{m}(\textbf{c}^{e})+W_{e}(\textbf{c}^{e})I(\phi)$ for $\phi_{e}\leq 0.5$ and $f(\textbf{c}^{e},\phi_{e})=f^{p}(\textbf{c}^{e})+W_{e}(\textbf{c}^{e})I(\phi_{e})$ for $\phi_{e}>0.5$; where $W_{e}(\textbf{c}^{e})\equiv f^{i}(\textbf{c}^{e})-f^{m}(\textbf{c}^{e})=f^{i}(\textbf{c}^{e})-f^{p}(\textbf{c}^{e})$. Therefore, analogous to the GB energy discussed in the preceding paragraph, the IB energy is obtained as $\gamma\approx 0.943\varepsilon\sqrt{W_{e}(\textbf{c}^{e}_{i},\textbf{c}^{e}_{m})}$. Thereby $\gamma$ accounts for the structural and the chemical energetics for the formation of the interfacial phase (via $W_{e}$) and the associated diffuse regions (via $\varepsilon$). The parametric study (Sec. III) showed the possibility of realizing $\gamma\rightarrow 0$ for synergistic, induced and enhanced co-segregation cases of quaternary alloys. However, the gradient energy contribution of the diffuse interface—together with the condition of equi-partitioning of the local excess grand-potential (Eq. A33)—imposes a theoretical restriction of $\gamma>0$, via $W_{e}>0$ in Eq. 17.

The regular solution assumption (Sec. II.4) for the bulk and IB phase thermodynamics allowed a variety of segregation/de-segregation behaviors to be realized. Parameterization was readily possible based on the physical significance afforded by regular solution thermodynamic parameters. Compared to IBs, multicomponent segregation in GBs have been well studied, both experimentally and theoretically [49, 50, 51, 46, 52, 53, 54]. IB segregation in quaternary, two-phase alloys can be found to be similar in many aspects to GB segregation in ternary, single-phase alloys. For example, the addition of certain secondary solutes is well known to enhance GB segregation of the primary solute due to mutually attractive interaction between the solutes at the GB [55]. This is analogous to case VI. Another example is the addition of a secondary solute which is also known to mitigate GB embrittlement caused by the undesired segregation of the primary solute–i.e. the solutes compete for GB sites and the strongly-segregating solute preferentially segregates by desegregating the weaker solute [56]. This is analogous to case VII. Therefore, based on the nature of mutual interaction between the solutes, their combined addition in the ternaries causes GB segregation behaviors that are fundamentally different from that of the individual solutes in their respective binaries. Xing et al. [53] presented a classification of the GB segregation/de-segregation behaviors in ternary single-phase alloys. The segregation mechanisms observed in current study are analogous to those observed for GBs.

The case studies presented in Sec. III are qualitatively representative of the observed mechanisms in the literature. As mentioned in Sec. I, numerous cases of IB co-segregation has already been reported. Several experimental and first-principles studies demonstrate enhanced high-temperature stability of the quaternary alloys due to IB co-segregation and its associated reduction in IB energy $\gamma$. Moreover, some studies suggest the need to desegregate certain solute from the IB: for example, Si at Al(matrix)/$\theta^{\prime}$-Al₂Cu in (Al-Cu)-Mn-Zr-Si is found to be deleterious to the microstructural stability in larger amounts [24]; Ca at Mg(matrix)/Al₁₁La₃ in (Mg-Al)-La-Ca-Mn is found to destabilize the strengthening precipitate [57] by by nucleating Al₂Ca. Therefore, it is also necessary to understand the de-segregation mechanisms (cases VII and VIII) so that alloys could be designed to favor beneficial solutes to segregate in preference to the undesired impurities.

The diffuse-interface thermodynamic formulation was shown to be consistent (see Sec. C) with the classical Gibbs interface phase rule [13]. At a given temperature, the number of components ($n_{\mathcal{N}}$) and the number of coexisting bulk phases ($n_{\psi}$) dictates the number of compositional degrees of freedom ($d_{\theta}^{f}=n_{\mathcal{N}}-n_{\psi}$). Thereby, two compositional degrees of freedom in solute $3$ and $4$ concentrations were available to modulate the properties of the quaternary IB in Sec. III. Analogously, if the model is applied to a ternary GB (where the common tangent constraint between phases is removed), two compositional degrees of freedom in solutes $2$ and $3$ will be available. Therefore, in addition to multicomponent IB segregation, the current approach has the potential to be employed for recently observed GB segregations in multi-principal element/high entropy alloys [58].