Thermodynamic Stabilization of Precipitates through Interface Segregation: Chemical Effects

TL;DR

This paper develops a thermodynamic model to understand how interface segregation, especially of zinc, stabilizes nanoscale precipitates in Mg-Sn-Zn alloys, enhancing high-temperature strength and resisting coarsening.

Contribution

It introduces a thermodynamic framework to predict chemical segregation effects and precipitate stabilization in ternary alloys, aiding alloy design for improved high-temperature performance.

Findings

Zn segregation stabilizes Mg$_2$Sn precipitates at nanoscale.

Precipitate coarsening is suppressed by increased Zn concentration.

Model predicts stabilization across various compositions and temperatures.

Abstract

Precipitation hardening, which relies on a high density of intermetallic precipitates, is a commonly utilized technique for strengthening structural alloys. Structural alloys are commonly strengthened through a high density of small size intermetallic precipitates. At high temperatures, however, the precipitates coarsen to reduce the excess energy of the interface, resulting in a significant reduction in the strengthening provided by the precipitates. In certain ternary alloys, the secondary solute segregates to the interface and results in the formation of a high density of nanosize precipitates that provide enhanced strength and are resistant to coarsening. To understand the chemical effects involved, and to identify such systems, we develop a thermodynamic model using the framework of the regular nanocrystalline solution model. For various global compositions, temperatures and thermodynamic parameters, equilibrium configuration of Mg-Sn-Zn alloy is evaluated by minimizing the Gibbs free energy function with respect to the region-specific (bulk solid-solution, interface and precipitate) concentrations and sizes. The results show that Mg$_2$Sn precipitates can be stabilized to nanoscale sizes through Zn segregation to Mg/Mg$_2$Sn interface, and the precipitates can be stabilized against coarsening at high-temperatures by providing a larger Zn concentration in the system. Together with the inclusion of elastic strain energy effects and the input of computationally informed interface thermodynamic parameters in the future, the model is expected to provide a more realistic prediction of segregation and precipitate stabilization in ternary alloys of structural importance.

Figures

Tables

Table 1. Table 1 : Number of bonds, bond probabilities and bond energies of various bond-types specific to the different bonding-regions in the binary system.

Region	Number of bonds in region $r$, $N_r^{bonds}$	Bond type, $kl$	Bond energy	Bond probability, $P_r^{kl}$
Bulk, $b$	$N_{\circ }\left(1-f_i-f_p\right)\frac{z_b}{2}$	AB	$E_b^{AB}$	$2\left(1-x_b\right)x_b$
Interface, $i$	$N_{\circ }{f}_i\frac{z_{ii}}{2}$	AA	$E_i^{AA}$	${\left(1-x_i\right)}^2$
		BB	$E_i^{BB}$	$x_i^2$
		AB	$E_i^{AB}$	$2\left(1-x_i\right)x_i$
Interface-bulk transition, $ib$	$N_{\circ }{f}_i\frac{z_{ib}}{2}$	AA	$E_i^{AA}$	$\left(1-x_i\right)\left(1-x_b\right)$
		BB	$E_i^{BB}$	$x_i{x}_b$
		AB	$E_i^{AB}$	$\left(1-x_i\right)x_b+\left(1-x_b\right)x_i$
Interface-precipitate transition, $ip$	$N_{\circ }{f}_i\frac{z_{ip}}{2}$	AA	$E_i^{AA}$	$\left(1-x_i\right)\left(1-x_p^{i/f}\right)$
		BB	$E_i^{BB}$	$x_i{x}_p^{i/f}$
		AB	$E_i^{AB}$	$\left(1-x_i\right)x_p^{i/f}+\left(1-x_p^{i/f}\right)x_i$

Table 2. Table 2 : Number of bonds, bond probabilities and bond energies of various bond-types specific to the different bonding-regions in the ternary system.

Region	Number of bonds in region $r$, $N_r^{bonds}$	Bond type, $kl$	Bond energy	Bond probability, $P_r^{kl}$
Bulk, $b$	$N_{\circ }\left(1-f_i-f_p\right)\frac{z_b}{2}$	AB	$E_b^{AB}$	$2\left(1-x_b-y_b\right)x_b$
		BC	$E_b^{BC}$	$2x_b{y}_b$
		AC	$E_b^{AC}$	$2\left(1-x_b-y_b\right)y_b$
Interface, $i$	$N_{\circ }{f}_i\frac{z_{ii}}{2}$	AA	$E_i^{AA}$	${\left(1-x_i-y_i\right)}^2$
		BB	$E_i^{BB}$	$x_i^2$
		CC	$E_i^{CC}$	$y_i^2$
		AB	$E_i^{AB}$	$2\left(1-x_i-y_i\right)x_i$
		BC	$E_i^{BC}$	$2x_i{y}_i$
		AC	$E_i^{AC}$	$2\left(1-x_i-y_i\right)y_i$
Interface-bulk transition, $ib$	$N_{\circ }{f}_i\frac{z_{ib}}{2}$	AA	$E_i^{AA}$	$\left(1-x_i-y_i\right)\left(1-x_b-y_b\right)$
		BB	$E_i^{BB}$	$x_i{x}_b$
		CC	$E_i^{CC}$	$y_i{y}_b$
		AB	$E_i^{AB}$	$\left(1-x_i-y_i\right)x_b+\left(1-x_b-y_b\right)x_i$
		BC	$E_i^{BC}$	$x_i{y}_b+x_b{y}_i$
		AC	$E_i^{AC}$	$\left(1-x_i-y_i\right)y_b+\left(1-x_b-y_b\right)y_i$
Interface-precipitate transition, $ip$	$N_{\circ }{f}_i\frac{z_{ip}}{2}$	AA	$E_i^{AA}$	$\left(1-x_i-y_i\right)\left(1-x_p^{if}\right)$
		BB	$E_i^{BB}$	$x_i{x}_p^{i/f}$
		AB	$E_i^{AB}$	$\left(1-x_i-y_i\right)x_p^{i/f}+\left(1-x_p^{i/f}\right)x_i$
		BC	$E_i^{BC}$	$y_i{x}_p^{i/f}$
		AC	$E_i^{AC}$	$y_i\left(1-x_p^{i/f}\right)$

Table 3. Table 3 : Default values are presented for: bulk interaction ( ω b subscript 𝜔 𝑏 \omega_{b} ), interface interaction ( ω i subscript 𝜔 𝑖 \omega_{i} ) and interface penalty ( δ i subscript 𝛿 𝑖 \delta_{i} ) parameters; coordination numbers ( z 𝑧 z ); temperature ( T 𝑇 T ); global concentration of B 𝐵 B ( x ∘ subscript 𝑥 x_{\circ} ); and precipitate B 𝐵 B concentration adjacent to the interface ( x p i / f superscript subscript 𝑥 𝑝 𝑖 𝑓 x_{p}^{i/f} ).

${\omega }_b$ (kJ/mol)

${\omega }_i$ (kJ/mol)

${\delta }_i$ (kJ/mol)

$z$

$T$

$x_{\circ }$

$x_p^{i/f}$

${\omega }_b^{AB}$

${\omega }_b^{BC}$

${\omega }_b^{AC}$

${\omega }_i^{AB}$

${\omega }_i^{BC}$

${\omega }_i^{AC}$

${\delta }_i^{AA}$

${\delta }_i^{BB}$

${\delta }_i^{CC}$

$z_b$

$z_{ii}$

$z_{ib}$

$z_{ip}$

(K)

(at.%)

(at.%)

-2.03

+2.54

-0.11

-2.03

+2.54

-10

3.5

3.5

3.5

12

6

3

3

300

2.2

33.33

Equations

$$x_{\circ }=x_b\left(1-f_i-f_p\right)+x_i{f}_i+x_p{f}_p.$$

$$\begin{array}{rl}\Delta {\bar{G}}_b^{mix}=& \{{\omega }_b^{AB}\left(1-x_b\right)x_b{z}_b\\ & +RT\left[\left(1-x_b\right)\text{ln}\left(1-x_b\right)+x_b\text{ln}{x}_b\right]\}\left(1-f_i-f_p\right),\end{array}$$

$${\omega }_b^{AB}$$

$$N_r^{kl}=N_r^{bonds}{P}_r^{kl},$$

$$\begin{array}{rl}\Delta {\bar{G}}_i^{mix}=& \{{\omega }_i^{AB}\left(1-x_i\right)x_i{z}_{ii}+\left[{\delta }_i^{AA}\left(1-x_i\right)+{\delta }_i^{BB}{x}_i\right]\frac{z_{ii}}{2}\\ & +RT\left[\left(1-x_i\right)\text{ln}\left(1-x_i\right)+x_i\text{ln}{x}_i\right]\}{f}_i.\end{array}$$

$${\omega }_i^{AB}$$

$${\delta }_i^{AA}=E_i^{AA}-\frac{z_b}{z_i}{E}_b^{AA}\text{and}{\delta }_i^{BB}=E_i^{BB}-\frac{z_b}{z_i}{E}_b^{BB}.$$

$$\begin{array}{rl}\Delta {\bar{G}}_{ib}=& {\omega }_i^{AB}\left[\left(1-x_b\right)x_i+\left(1-x_i\right)x_b\right]f_i\frac{z_{ib}}{2}\\ & +\left[{\delta }_i^{AA}\left(1-x_i+1-x_b\right)+{\delta }_i^{BB}\left(x_i+x_b\right)\right]f_i\frac{z_{ib}}{4},\end{array}$$

$$\begin{array}{rl}\Delta {\bar{G}}_{ip}=& {\omega }_i^{AB}\left[\left(1-x_p^{i/f}\right)x_i+\left(1-x_i\right)x_p^{i/f}\right]f_i\frac{z_{ip}}{2}\\ & +\left[{\delta }_i^{AA}\left(1-x_i+1-x_p^{i/f}\right)+{\delta }_i^{BB}\left(x_i+x_p^{i/f}\right)\right]f_i\frac{z_{ip}}{4}.\end{array}$$

$$\begin{array}{rl}\Delta {\bar{G}}_{b,i}^{ref}=& \left[\left(1-x_b\right)\Delta {\bar{G}}_{ref(b)}^A+x_b\Delta {\bar{G}}_{ref(b)}^B\right]\left(1-f_i-f_p\right)\\ & +\left[\left(1-x_i\right)\Delta {\bar{G}}_{ref(b)}^A+x_i\Delta {\bar{G}}_{ref(b)}^B\right]f_i.\end{array}$$

$$\Delta {\bar{G}}_p=\Delta {\bar{G}}_f^{A_m{B}_n}{f}_p$$

$$\Delta {\bar{G}}_{bin}=\Delta {\bar{G}}_b^{mix}+\Delta {\bar{G}}_i^{mix}+\Delta {\bar{G}}_{ib}+\Delta {\bar{G}}_{ip}+\Delta {\bar{G}}_{b,i}^{ref}+\Delta {\bar{G}}_p.$$

$$\begin{array}{r}{y}_{\circ }=y_b\left(1-f_i-f_p\right)+y_i{f}_i.\end{array}$$

$$\Delta {\bar{G}}_{tern}=\Delta {\bar{G}}_b^{mix}+\Delta {\bar{G}}_i^{mix}+\Delta {\bar{G}}_{ib}+\Delta {\bar{G}}_{ip}+\Delta {\bar{G}}_{b,i}^{ref}+\Delta {\bar{G}}_p,$$

$$\Delta {\bar{G}}_b^{mix}$$

$$+RT\left[\left(1-x_b-y_b\right)\text{ln}\left(1-x_b-y_b\right)+x_b\text{ln}{x}_b+y_b\text{ln}{y}_b\right]\}\left(1-f_i-f_p\right),$$

$$\Delta {\bar{G}}_i^{mix}=$$

$$+\left[{\delta }_i^{AA}\left(1-x_i-y_i\right)+{\delta }_i^{BB}{x}_i+{\delta }_i^{CC}{y}_i\right]\frac{z_{ii}}{2}$$

$$+RT\left[\left(1-x_i-y_i\right)\text{ln}\left(1-x_i-y_i\right)+x_i\text{ln}{x}_i+y_i\text{ln}{y}_i\right]\}{f}_i,$$

$$\Delta {\bar{G}}_{ib}=$$

$$+{\omega }_i^{AC}\left[\left(1-x_b-y_b\right)y_i+\left(1-x_i-y_i\right)y_b\right]\}{f}_i\frac{z_{ib}}{2}$$

$$+\left[{\delta }_i^{AA}\left(1-x_i-y_i+1-x_b-y_b\right)+{\delta }_i^{BB}\left(x_i+x_b\right)+{\delta }_i^{CC}\left(y_i+y_b\right)\right]f_i\frac{z_{ib}}{4},$$

$$\Delta {\bar{G}}_{ip}=$$

$$+{\omega }_i^{AC}\left(1-x_p^{i/f}\right)y_i\}{f}_i\frac{z_{ip}}{2}$$

$$+\left[{\delta }_i^{AA}\left(1-x_i-y_i+1-x_p^{i/f}\right)+{\delta }_i^{BB}\left(x_i+x_p^{i/f}\right)+{\delta }_i^{CC}{y}_i\right]f_i\frac{z_{ip}}{4},$$

$$\Delta {\bar{G}}_{b,i}^{ref}=$$

$$+\left[\left(1-x_i-y_i\right)\Delta G_{ref(b)}^A+x_i\Delta G_{ref(b)}^B+y_i\Delta G_{ref(b)}^C\right]f_i,$$

$$\Delta {\bar{G}}_p=$$

$$r_p=\frac{3t}{\varphi }.$$

$$\frac{\partial \Delta {\bar{G}}_{tern}}{\partial x_i}=0,\frac{\partial \Delta {\bar{G}}_{tern}}{\partial y_i}=0,\frac{\partial \Delta {\bar{G}}_{tern}}{\partial \varphi }=0\text{and}\frac{\partial \Delta {\bar{G}}_{tern}}{\partial f_p}=0.$$

$${\Gamma }_i^C=\frac{1}{N_{avg}{\Omega }^{2/3}}\left[y_i-y_b\frac{x_i-\left(1-y_i\right)x_p}{x_b-\left(1-y_b\right)x_p}\right],$$

$$U_{sys}=\sum _{reg=b,i,p}\left(N_{reg}^A{E}_{reg}^A+N_{reg}^B{E}_{reg}^B\right).$$

$$U_{std}=N_{total}^A{E}_{std}^A+N_{total}^B{E}_{std}^B,$$

$$N_{total}^A=$$

Full text

Keywords

Statistical thermodynamics, Intermetallic precipitatation, Coarsening, Solute segregation, Heterophase interface

††preprint: APS/123-QED

I Introduction

In structural alloys, one of the most commonly utilized strengthening mechanism involves the precipitation of ordered intermetallic compounds. For example, the precipitation of L12 Al3Sc hyland1992homogeneous; royset2013scandium and Al3Li gregson1985microstructural; flower1987solid; noble1997microstructural intermetallics in aluminum has led to significant improvements in yield strength. Similar effects were observed in certain maraging floreen1968physical; vasudevan1990precipitation and stainless steels judy1973properties. The beneficial properties of intermetallic precipitation are not just confined to fcc alloy systems but have also been observed in hexagonal alloys of titanium madsen1995separating; ramos2006peritectoid and, more recently, in magnesium alloys antion2003hardening; apps2003precipitation; mendis2006refinement. Even in iron-based bcc alloys, L21 precipitates have resulted in significant improvements in yield strength jack1975low; dimiduk1988structural; perrier2011precipitation.

Precipitation hardening is usually attributed to two primary mechanisms - (i) dislocation cutting-through or bowing-around dispersed precipitate particles (dispersion hardening hazzledine1974coplanar) and (ii) dislocations interacting with the coherency strains of precipitates (coherency strain hardening mott1940attempt; gladman1999precipitation).

In addition to mechanical properties, thermal stability of certain fcc alloys can be dramatically improved through the precipitation of ordered intermetallics. For example, precipitation of cubic L12 intermetallics, in nickel kear1970stacking; timmins1986negative; condat1987shearing; sarosi2005imaging; sarosi2006direct and cobalt-based alloys chinen2007new; suzuki2007flow, results in an improved high-temperature yield strength pope1984mechanical. While the improved mechanical properties have a non-trivial dependence on the size, density and crystallography of the precipitates, it is generally true that a higher density of smaller precipitates that are resistant to coarsening improves both the strengthening characteristics and high-temperature stability. For example, while the separate addition of Zr or Sc to Aluminum alloys results in increased tensile strength and resistance to recrystallization by forming ordered precipitates, the combined effect is considerably larger yelagin1985influence; davydov1996alloying; toropova1998advanced; fuller2003mechanical; riddle2004study; fuller2005temporal1; fuller2005temporal2. This is due to the formation of a higher density of small precipitates driven by Zr segregation to Al/Al3Sc hetero-phase interfaces clouet2006complex.

Zr segregation in Al-Sc-Zr alloys has been attributed primarily to kinetic effects - the low diffusion coefficient of Zr compared to Sc restricts the partitioning of Zr to just the interface layer rather than the core of Al3Sc clouet2006complex. However, in general, both kinetic and thermodynamic factors may promote solute segregation to an interface. Some observed examples include: segregation of Mg marquis2005coarsening, Zr tolley2005segregation and Mg+Ag kang2014determination to $\alpha$-Al/Al3Sc interface; Sc yang2016influence, Si biswas2011precipitates, Ag rosalie2012silver and Si+Mg biswas2010simultaneous to $\alpha$-Al/Al2Cu interface; Gd+Zn nie2008solute to Mg/Mg5Zn interface; Zn liu2016interphase; liu2017zn to Mg/Mg2Sn interface. The thermodynamic driving force for solute segregation to the interface is attributed to a reduction in the interfacial free-energy weissmuller1993alloy; weissmuller1994alloy; kirchheim2002grain; wynblatt2006anisotropy; chookajorn2012design; murdoch2013stability; murdoch2013estimation, which is attributed primarily to two factors: (a) favorable chemical interactions of solute element at the interface over that in the bulk; and (b) reduction of solute size misfit strain energy voorhees2006alloys. The reduction in the interfacial free-energy $\gamma$ further results in a decrease in the coarsening kinetics at elevated temperatures kuehmann1996ostwald.

The afore-mentioned examples are but a few among a large number of binary alloy systems that favor intermetallic precipitation. For example, other commonly observed binary intermetallics in structural alloys include: Cu3Al, Cu3Sn, Cu3Ti, Cu3Au, Ni3Al, Ni3Ti, Ni3Nb, Ni3Si, MgZn, Mg2Si, Mg3Nd, Mg2Cu, Mg2Ni. By carefully introducing ternary atoms that will segregate to the the precipitate/matrix interface, it will be possible to stabilize much smaller precipitates that are resistant to coarsening at high-temperatures. To identify such systems, in section II, we develop a thermodynamic model that describes the energetics of simple ternary alloys (A-B-C), where the binary A-B system favors the precipitation of the ordered compound $A_{m}B_{n}$ and the impurity C atoms may segregate to the interface between the matrix and the $A_{m}B_{n}$ precipitate.

It is assumed that the segregation is promoted just through favorable chemical interactions at the interface. The important contribution of elastic strain is ignored, for now, so that we can build a simple thermodynamic model for the ternary alloy system. Future work will focus on incorporating the strain energy effects. Under these simplifying conditions, the model is assumed to be applicable to precipitating systems with incoherent interfaces (e.g. incoherent equilibrium $\theta$ phase in Al-Cu alloys biswas2011precipitates). In this article, we present the thermodynamic model using the ternary Mg-Sn-Zn alloy system, where recent experimental studies liu2016interphase; liu2017zn have shown the segregation of Zn to all Mg/Mg2Sn interfaces irrespective of interface structure and orientation relationship and that the segregation is not limited by the kinetics of Zn diffusion. This observation suggests the presence of a chemical driving force for heterophase interface segregation. In section III, the variations in equilibrium precipitate sizes and the Gibbs interfacial excess of the solute atoms, for the Mg-Sn-Zn ternary alloy, as a function of different interaction energy parameters and temperature are presented.

II ANALYTICAL TREATMENT

In this section, we present a thermodynamic model to study solute segregation to the interface between a solid-solution bulk phase and a binary ordered-intermetallic-compound. The Gibbs free energies of the bulk and the interface regions of the system are described using a statistical-thermodynamic framework involving the regular solution assumptions of random mixing and nearest-neighbor pairwise interaction dehoff2006thermodynamics. This follows the regular nanocrystalline solution model developed by Trelewicz and Schuh (trelewicz2009grain) and extended to ternary systems by Saber et al. saber2013thermodynamic. Energy contribution of the precipitate region is described using the Gibbs free energy of formation of the intermetallic compound; this avoids the complexity involved in sublattice models. The model for a binary alloy system is presented first. The binary system consists of a solvent element, $A$, and a solute element, $B$, that favor precipitation of an intermetallic compound of the type $A_{m}B_{n}$ from solid-solution. The model is subsequently extended to a ternary alloy system, wherein a secondary solute element, $C$, is considered to be soluble in the bulk and the interface solid-solutions but assumed to be insoluble in the precipitate. The bulk and interface regions are provided with distinct descriptions of energetic parameters and compositions. This feature allows equilibrium segregation of solute to the interface for energetic parameters favoring the reduction in interfacial energy, and thus in the system free energy, on solute segregation. The free energy function, important relations and definitions pertaining to the model are presented below; the complete derivation is given in section S1 of Supplemental Materials.

II.1 Binary Model

The alloy system, consisting of a heterogeneous distribution of $A$ and $B$ atoms, is divided into three distinct regions—the bulk, the precipitate and the bulk/precipitate interface regions, denoted by $b$, $p$ and $i$, respectively. The global concentration of $B$ in the system, $x_{\circ}$, can be expressed as a function of region-specific concentrations, $x_{b}$, $x_{i}$ and $x_{p}$, and volume fractions, $f_{b}$, $f_{i}$ and $f_{p}$, using the mass balance relation as:

[TABLE]

Here, $f_{b}=1-f_{i}-f_{p}$, and since the precipitate is an intermetallic compound, concentration in $p$ is stoichiometric with $x_{p}=n/m$. For a closed system of given $x_{\circ}$, this relation imposes constraint on the values that the variables $x_{b}$, $x_{i}$, $f_{i}$ and $f_{p}$ can take simultaneously.

The assumptions of random mixing and nearest-neighbor pairwise interaction between the atoms of bulk solid-solution result in the following expression for the Gibbs free energy of mixing of $b$, $\Delta\bar{G}_{b}^{\,mix}$:

[TABLE]

where, $z_{b}$ is the coordination number and and $\omega_{b}^{AB}$ is the regular solution interaction parameter, specific to region $b$. The bar over $\Delta\bar{G}_{b}^{mix}$ defines the quantity per mole of atoms in the system and applies to all quantities represented this way in this paper; $\omega_{b}^{AB}$ is defined per mole of atoms in $b$. This expression is essentially the free energy of mixing obtained in the regular solution model, scaled by the size of the bulk region relative to the system (i.e. $f_{b}=1-f_{i}-f_{p}$). The first term in the expression corresponds to the excess enthalpy of mixing, obtained from the internal energy of mixing assuming negligible volume change during mixing. Here, $\omega_{b}^{AB}$ accounts for the energy difference involved in the formation of unlike $AB$ bonds from like $AA$ and $BB$ bonds having energies characteristic of the bulk region; this is given by:

[TABLE]

The part of term multiplied to $\omega_{b}^{AB}$, along with the bulk volume fraction $f_{b}$, represents the number of $AB$ bonds, $N_{b}^{AB}$, in the bulk region. The first term thus corresponds to the enthalpy of forming unlike bonds from like bonds. The second term in Eq. (2) represents the ideal entropy of mixing in $b$.

In general, the number of bonds of type $kl$ in region $r$, $N_{r}^{kl}$, is obtained as:

[TABLE]

where, $r$ refers to the bonding regions of $b$ and $i$, and the transition regions $ib$ and $ip$, occurring between atoms of $i$ and $b$, and atoms of $i$ and $p$, respectively. $N_{r}^{bonds}$ is the total number of bonds in $r$; $P_{r}^{kl}$ is the statistical probability that the bond in $r$ is of type $kl$, and is obtained as a function of the composition in each region $r$. For the binary system with a total number of $N_{\circ}$ atoms of $A$ and $B$, expressions for various $N_{r}^{bonds}$ and $P_{r}^{kl}$ are presented in Table 1.

The Gibbs free energy of mixing for the formation of interface solid-solution, $\Delta\bar{G}_{i}^{\,mix}$, is obtained similar to $b$ following the regular solution model as:

[TABLE]

Here, the interface interaction parameter, $\omega_{i}^{AB}$, captures the energetics of formation of unlike bonds in region $i$ from like bonds in pure element states characteristic of $i$, and is given by:

[TABLE]

The formation of the interface solid-solution is considered from the same reference state as that for the formation of the bulk; i.e. the reference state is pure $A$ and pure $B$ with bond-energies and bond-coordination characteristic of $b$. Therefore, in addition to the enthalpy of mixing (first term), an additional term (second term) arises—this corresponds to the energy associated with a change of state of components $A$ and $B$ from a state characteristic of $b$ to one characteristic of $i$. This change of state is captured by the parameters $\delta^{AA}_{i}$ and $\delta^{BB}_{i}$, and is defined per mole of $A$ and $B$, respectively, as,

[TABLE]

These parameters represent the difference between like bond energies characteristic of the interface and the bulk and can be directly related to the interface free energies of $A$ and $B$ (see Supporting Materials S3); $z_{b}/z_{i}$ accounts for the different atomic coordination of $b$ and $i$. Analogous to Eq. 2, the terms multiplying the energy parameters in Eq. 5 are obtained form Table 1. The term coupled to $\omega_{i}^{AB}$ is $N_{i}^{AB}$; and terms coupled to $\delta_{i}^{AA}$ (i.e. $f_{i}(1-x_{i})z_{ii}/2$) and $\delta_{i}^{BB}$ (i.e. $f_{i}x_{i}z_{ii}/2$) represent the number of $AA$ bonds ($N_{i}^{AA}+N_{i}^{AB}/2$) and $BB$ bonds ($N_{i}^{BB}+N_{i}^{AB}/2$), respectively, in the pure element reference state of $i$. The terms coupled to the parameters $\omega_{b}$, $\omega_{i}$ and $\delta_{i}$ in subsequent equations can be interpreted in a similar manner.

The interface region is considered as one layer of atoms between the bulk and the precipitate regions. Thus, of the total coordination $z_{i}$ per interface atom, $z_{ii}$ connects to neighboring interface atoms, while $z_{ib}$ and $z_{ip}$ connect to $b$ and $p$ atoms, respectively, that lie adjacent to region $i$. To account for the non-random distribution of atoms in $p$ and the dependence on the interface plane, concentration at the atomic layer of precipitate adjacent to $i$ is defined distinctly as $x_{p}^{i/f}$. Random distribution of atoms with a concentration of $x_{b}$ is assumed to be applicable to the atomic layer of $b$ adjacent to $i$. $ib$ and $ip$ bonds are defined to have bond-energies characteristic of $i$. Following these considerations, the Gibbs free energy for the formation of the transition regions $ib$ and $ip$, $\Delta\bar{G}_{ib}$ and $\Delta\bar{G}_{ip}$, respectively, are obtained as:

[TABLE]

[TABLE]

Analogous to the interpretation of Eq. 5, the first terms of Eqs. 8 and 9 represent the enthalpy change associated with the formation of unlike bonds of the corresponding transition regions from like bonds of pure $A$ and pure $B$ states characteristic of $i$. The second terms corresponds to the additional enthalpy associated with the change of state of the pure elements from energies characteristic of $b$ to $i$.

While the free energy expressions presented in Eqs. 2, 5, 8 and 9 considered the reference states as pure $A$ and pure $B$ with crystal structures characteristic of the bulk solid-solution, $A$ or $B$ or both, however, may have a crystal structure preference different from the bulk. To enable treatment of such systems, the initial or standard state is chosen as pure $A$ and pure $B$ at $T$, having crystal structures specific to their Standard Element Reference State (i.e. the most stable state of an element at $298.15$ K and $10^{5}$ Pa dinsdale1991sgte). The molar free energies involved in the conversion of $A$ and $B$ from the standard state at $T$ to the reference state characteristic of $b$, at the same $T$, are defined as $\Delta\bar{G}_{ref(b)}^{A}$ and $\Delta\bar{G}_{ref(b)}^{B}$, respectively. The total free energy change associated with this change in reference state, $\Delta\bar{G}_{b,i}^{\,ref}$, for atoms forming $b$ and $i$ regions of the system is obtained as a mole fraction-weighted average of $\Delta\bar{G}_{ref(b)}^{A}$ and $\Delta\bar{G}_{ref(b)}^{B}$, scaled by the region-sizes as:

[TABLE]

The Gibbs free energy for the formation of precipitate region, $\Delta\bar{G}_{p}$, is obtained in terms of the molar Gibbs free energy for the formation of $A_{m}B_{n}$ intermetallic, $\Delta\bar{G}_{f}^{A_{m}B_{n}}$, from the pure element standard states of $A$ and $B$, and is scaled by the precipitate size.

[TABLE]

The total Gibbs free energy, $\Delta\bar{G}_{bin}$—defined for the formation of the binary system configuration from initial pure element standard states of $A$ and $B$—is obtained as the summation over free energy contributions corresponding to the formation of $b$, $i$, $ib$, $ip$ and $p$ regions of the system from Eqs. 2, 5, 8, 9, 10 and 11; thus,

[TABLE]

II.2 Ternary Model

The binary model presented in section II.1 is now extended to a ternary system containing an additional element in secondary solute $C$. The ternary alloy system consists of a heterogeneous distribution of $A$, $B$ and $C$ atoms and, as before, the system has three distinct regions of atom occupancy, viz. $b$, $p$ and $i$, and bonding regions of $b$, $i$, $p$, $ib$ and $ip$. We consider $C$ to form ternary solid-solution with $A$ and $B$ in the bulk and the interface regions, while insoluble in the intermetallic compound. In addition to the mass balance relation for solute $B$, given by Eq. 1, mass balance relation for solute $C$ is obtained as:

[TABLE]

Each of the variables $x_{i}$, $x_{b}$, $y_{i}$, $y_{b}$, $f_{i}$ and $f_{p}$ can take values between [math] and $1$. For given values of $x_{\circ}$ and $y_{\circ}$, Eqs. 1 and 13 impose constraints on the values the above variables can take simultaneously. We will consider $x_{i}$, $y_{i}$, $f_{i}$ and $f_{p}$ as the independent variables.

The Gibbs free energy function, $\Delta\bar{G}_{tern}$, for the formation of the ternary system from the standard pure element states of $A$ and $B$ is obtained, similar to that for the binary system, as the summation over free energy contributions corresponding to the formation of $b$, $i$, $ib$, $ip$ and $p$ regions of the system. Thus,

[TABLE]

where,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the above equations, the interaction parameter for different regions is given by $\omega_{r}^{kl}=E_{r}^{kl}-\frac{E_{r}^{kk}+E_{i}^{ll}}{2}$, where $r$ refers to $b$ or $i$, $kl$ refers to unlike bonds $AB$, $BC$ or $AC$, and $kk$ or $ll$ refer to like bonds $AA$, $BB$ or $CC$. The energy penalty arising from the difference in bond energies of like bonds characteristic of the interface and that of the bulk is $\delta_{i}^{kk}=E_{i}^{kk}-\frac{z_{b}}{z_{i}}E_{b}^{kk}$. The terms multiplying $\omega_{b}$, $\omega_{i}$ and $\delta_{i}$ (including the volume fraction and the coordination number) correspond to the number of bonds of the type defined by $\omega_{b}$, $\omega_{i}$ or $\delta_{i}$, and were obtained, as described in section II.1, using Eq. 4; the number of bonds specific to each bonding region, and the bond probabilities of the various bond types are presented in Table 2. The $\Delta\bar{G}_{tern}$ function given Eq. 14 is the main result of our model.

II.3 Geometric Relationship

The analytical model was developed considering that the interface atomic region constitutes a single layer of atoms. Thus, the interface volume fraction, $f_{i}$, can be expressed as a fraction of the precipitate volume fraction, $f_{p}$, as $f_{i}=\phi f_{p}$. A geometric representation of the system configuration can be obtained for these variables under certain assumptions of the shape and size of the precipitates. For a spherical morphology with equi-sized precipitates of radius $r_{p}$, the interface region is a spherical shell, of thickness $t$ surrounding the precipitate; this is illustrated in Fig. 1. Here, $t$ is taken as $0.25$ nm, which is characteristic of the atomic length scale in crystals. From geometric dependence of the volumes of the sphere and the encompassing spherical shell, a relation between the precipitate size $r_{p}$ and the fraction of interface with respect to the precipitate, $\phi$, is obtained as,

[TABLE]

II.4 Equilibrium Conditions

The equilibrium configuration of the ternary system, for a given set of global compositions $(x_{\circ},y_{\circ})$, temperature (T) and parametric values $(\omega_{b},\omega_{i},\delta_{i},z,x_{p}^{if})$ can be obtained by minimizing the Gibbs free energy function for the system (Eq. 14) with respect to $x_{i}$, $y_{i}$, $f_{i}$ and $f_{p}$. The treatment of equilibrium between the the bulk, the precipitate and the interface regions is similar to that for equilibrium between three distinct phases. However, the interface region is different from a phase in that it cannot exist independent of the bulk and the precipitate phases. This constraint is incorporated by substituting the relation $f_{i}=\phi f_{p}$, and by setting minimum and maximum values that $\phi$ can take. System equilibrium is now obtained by minimizing $\Delta\bar{G}_{tern}$ with respect to $x_{i}$, $y_{i}$, $\phi$ and $f_{p}$ as:

[TABLE]

The equilibrium system configuration obtained in terms of $\phi$ and $x_{b}$, $x_{i}$, $y_{b}$ and $y_{i}$ can be represented by the equilibrium quantities, $r_{p}^{eq}$ and $\Gamma_{i}^{C}$, respectively. (Equilibrium in the binary system is similarly obtained by minimizing $\Delta\bar{G}_{bin}$ (Eq. 12) with respect to $x_{i}$, $\phi$ and $f_{p}$.) Here, $\Gamma_{i}^{C}$ is the excess concentration of $C$ at the interface and represents the segregation state in the system. $\Gamma_{i}^{C}$ for the ternary system is given by finnis1998accessing; sutton1995interfaces:

[TABLE]

where, $N_{avg}$ is a mole of interface atoms, $\Omega$ is the atomic volume, and $N_{avg}\Omega^{2/3}$ represents the molar interface area. Defining the interface solute excess per interface area allows the quantity to be used to compare different configurational states of the system having different interface volume fractions.

III PARAMETRIC STUDY

With the thermodynamic model and the conditions for equilibrium established, we can solve for the equilibrium precipitate size $(r_{p}^{eq})$ and interfacial excess $(\Gamma_{i}^{C})$. For the binary alloy system, the equilibrium precipitate size as a function of global solute (B) concentration is obtained at either the largest or the smallest precipitate size allowed by the limits imposed on $\phi$. The equilibrium states obtained at these limits do not represent a true equilibrium between the three regions as, in the absence of the limits, the equilibrium would be between just two regions of the system. In one case, equilibrium configuration tending towards just the bulk and precipitate regions arises when the interface energy is unfavorable compared to other two regions—this is representative of ideal binary precipitating systems where the precipitates coarsen to reduce the interfacial energy, provided it is kinetically feasible. In the other case, equilibrium between bulk and the interface occurs when the interface energy is favorable; this is obtained when the interface interaction energy, $\omega_{i}^{AB}$, is set to a large negative value.

These equilibrium configurations obtained from the binary model can be rationalized by considering the different regions of the binary system as phases and invoking the Gibbs phase rule. For a two component system at constant temperature and pressure, the phase rule states the the equilibrium between three phases has zero degrees of freedom in the intensive thermodynamic variables (i.e. chemical potentials of the components, which in the present model relates to the concentrations). This means that true equilibrium between the three regions, represented by a precipitate size within the imposed limits, can only be obtained at a unique global composition. This is of limited interest to us, and hence we turn our attention to the ternary system. For a three component system, as per the phase rule, equilibrium between three phases is possible with one degree of freedom in the intensive variables (i.e. concentrations). This additional degree of freedom is due to the presence of $C$ in the system. Indeed, we obtain a range of equilibrium precipitate sizes over a range of global compositions ($x_{\circ}$ or $y_{\circ}$) from the ternary model; these results are presented below.

The equilibrium system configurations, $r_{p}^{eq}$ and $\Gamma_{i}^{C}$, for the ternary alloy system are presented in this section as functions of $T$, $x_{\circ}$ and $y_{\circ}$, and the interface energy parameters, $\omega_{i}^{AB}$, $\omega_{i}^{BC}$, $\omega_{i}^{AC}$, $\delta_{i}^{AA}$, $\delta_{i}^{BB}$ and $\delta_{i}^{CC}$. Equilibrium values of $x_{i}$, $y_{i}$, $f_{p}$ and $\phi$ are obtained by minimization of $\Delta\bar{G}_{tern}$ (Eq. 14) using an interior-point optimization routine matlab2014; fmincon_url. The optimized variables for a given set of parameters are then expressed in terms of $\Gamma_{i}^{C}$ and $r_{p}^{eq}$ through relations given in Eqs. 17 and 15, respectively.

We take Mg-Sn-Zn as an example system for this study as recent work by Liu et al. liu2016interphase; liu2017zn suggests a thermodynamic basis for Zn segregation to the ternary solid-solution/Mg2Sn interface. Accordingly, the values for bulk interaction parameters defining the Mg-Sn-Zn solid-solution phase are used, and $\Delta\bar{G}_{f}^{A_{m}B_{n}}$ corresponds to the formation energy of Mg2Sn of $-24.5$ kJ/mol meng2010thermodynamic (Supplemental Materials S2). In the following study, except for the particular parameter whose values are varied, all other parameters are set to default values listed in Table 3. The values of interface interaction parameters, $\omega_{i}^{kl}$, are chosen to favor the presence of Zn at the interface, and the interface penalty parameters, $\delta_{i}^{kk}$, are chosen to be positive to represent the energy penalty associated with an interface. The magnitude of $\delta_{i}^{AA}$ is based on an estimate of the average interface energy of Mg/Mg2Sn interface (see Supplemental Materials S3).

III.1 Global solute concentrations, $x_{\circ}$ and $y_{\circ}$

The equilibrium configurations are first analyzed by varying the concentrations of components B ($x_{\circ}$) and C ($y_{\circ}$) in the ternary alloy system. Default values listed in Table 3 are used for the remaining parameters. In the range of global concentrations analyzed in this study, the equilibrium radius, $r_{p}^{eq}$, of the precipitate was found to be of the order of a few tens of nanometers. As shown in Fig. 2(a), at dilute concentrations of solute $C$, a small increase in $y_{\circ}$ results in a dramatic decrease in $r_{p}^{eq}$. Since the interaction energy parameters remain constant in this analysis, equilibrium concentrations in the bulk and the interface regions do not change by much with a change in the global solute concentrations ($x_{\circ}$ or $y_{\circ}$). This is reflected by the constant value of $\Gamma_{i}^{C}$ in Fig. 2(b).

$\Gamma_{i}^{C}$ for the default parameters corresponds to a large interfacial concentration of $C$ atoms ($\sim 53$ at.%) and a low bulk $C$ concentration ($\sim 0.15$ at.%), and thus represents a strong segregation of $C$ atoms to the interface. From an initial non-equilibrium state of uniform concentration in the bulk and the interface, segregation of $C$ to the interface reduces the interface free energy and thus the overall free energy of the system. As $\omega_{i}^{AC}$ is assigned a highly negative interaction energy, the interfacial energy is reduced by maximizing the number of energetically favorable $AC$ bonds at the interface—this is achieved at close to equiatomic interface concentration of $A$ and $C$ atoms.

As stated earlier, composition in the bulk solid-solution, the interface region and the precipitate remain almost constant as we change the solute concentration $x_{\circ}$ and $y_{\circ}$. Therefore, the volume fractions of the precipitate and the interface regions vary to accommodate for the variations in global solute compositions (this is similar to the lever-rule calculations in a binary eutectic-alloy). An increase in $y_{\circ}$ while keeping $x_{\circ}$ fixed will result in an increase in the volume fraction of interfaces $(f_{i})$ at a constant value of precipitate volume fraction $(f_{p})$. This is accomplished by reducing the size and increasing the number of precipitates, thus increasing the total interfacial volume fraction. Conversely, increasing $x_{\circ}$ at a fixed $y_{\circ}$ will result in an increase in $f_{p}$ while $f_{i}$ remains constant. This is accomplished by reducing $\phi$ since $f_{i}=\phi f_{p}$, which in-turn results in an increase in $r_{p}^{eq}$ according to Eq. 15.

III.2 Interface interaction energy parameters,

$\omega_{i}^{kl}$

To understand the influence of the interface interaction parameters, the equilibrium precipitate radius $r_{p}^{eq}$ and the interfacial solute excess $\Gamma_{i}^{C}$ are plotted as a function of the global secondary solute concentration $y_{\circ}$ as $\omega_{i}^{AB}$, $\omega_{i}^{BC}$ and $\omega_{i}^{AC}$ are varied in Figs. 3, 4 and 5, respectively. In general, an unlike bond of type $kl$ is preferred at the interface over the bulk if $\omega_{i}^{kl}+\frac{\delta_{i}^{kk}+\delta_{i}^{ll}}{2}<\omega_{b}^{kl}$. Thus, a large negative value of $\omega_{i}^{kl}$ allows a large volume fraction of interface to exist at equilibrium by promoting segregation of solutes that form $kl$ bonds to the interface. Equilibrium precipitates of nanoscale sizes are therefore achieved for $\omega_{i}^{AC}=-10$ kJ/mol through segregation of $C$ to the interface and formation of energy reducing $AC$ bonds. As shown in Figs. 3(a), 4(a) and 5(a), the equilibrium precipitate size increases with increasing $\omega_{i}^{kl}$ to account for the increase in the interfacial energy. In the following, as we justify the observed trends for $\Gamma_{i}^{C}$, it is important to note that almost all of the $B$ atoms are present in the precipitate due to the large driving force for intermetallic precipitation. With this in mind, the trends for interfacial solute excess can be understood as follows:

• •

In Fig. 3b, it is shown that $\Gamma_{i}^{C}$ increases with increasing $\omega_{i}^{AB}$. This is because $A$ atoms de-segregate out of the interface to reduce the fraction of increasingly unfavorable $AB$ bonds in the $ip$ transition region. Also, the $C$ atoms substitute interfacial $A$ atoms resulting in an increase in the favorable interfacial $AC$ bonds (in the transition region), and hence, $\Gamma_{i}^{C}$ increases with increasing $\omega_{i}^{AB}$.

• •

Similar to the trend in $\omega_{i}^{AB}$, an increase in $\omega_{i}^{BC}$ will result in a reduction of the number of interfacial $BC$ bonds. As most of the $B$ atoms are present in the precipitate, the $BC$ bonds are present in the $ip$ transition region. Therefore, the only way to reduce the number of $BC$ bonds in the $ip$ region is by removing the $C$ atoms from the interface. This results in a reduction in $\Gamma_{i}^{C}$ as shown in Fig. 4(b).

• •

An increase in $\omega_{i}^{AC}$ results in an increase in $\Gamma_{i}^{C}$ as shown in Fig. 5(b). While this trend is non-intuitive, it is not entirely surprising given the relative magnitudes of the $\omega_{i}$ parameters. The increase in $\Gamma_{i}^{C}$ with increasing $\omega_{i}^{AC}$ can be rationalized by fixing the volume fraction of the interface $f_{i}$. This fixes the total number of interface bonds (of type $ii$, $ip$ and $ib$) in the system. When $\omega_{i}^{AC}$ is increased, the system requires a larger number of interfacial $AC$ type bonds to stabilize the same volume fraction $f_{i}$. This is because the default values for $\omega_{i}^{AB}=-2.03$ kJ/mol and $\omega_{i}^{BC}=+2.54$ kJ/mol are much larger than the values of $\omega_{i}^{AC}$ explored in this study ($-14$, $-10.5$, $-10$ and $-9.7$ kJ/mol). Therefore, in order to stabilize the required volume fraction of interfaces in the system, $\Gamma_{i}^{C}$ has to be increased if $\omega_{i}^{AC}$ is increased.

In summary, increasing $\omega_{i}^{kl}$ displaces $r_{p}^{eq}$ vs. $y_{\circ}$ curves to larger precipitate sizes; this is presented in Figs. 3(a), 4(a) and 5(a). As discussed above, the interface stability corresponding to the new parametric value of $\omega_{i}^{kl}$ can be achieved by the system through a reduction in the unfavorable bond-types and an increase the favorable bond-types. However, in the absence of the required concentration of $C$ in the system to partition to the interface, a larger $\Gamma_{i}^{C}$ (for $\omega_{i}^{AB}$ and $\omega_{i}^{AC}$ variation) is achieved by decreasing the interface volume fraction; this is seen by the increase in precipitate size (note that $f_{p}$ remains constant but the number density of precipitates decreases).

III.3 Interface energy penalty parameters, $\delta_{i}^{kk}$

An energy penalty of $\delta_{i}^{kk}$ is associated with interface and transition bonds connected to interface atoms of type $k$. Each like-bond and unlike-bond has an energy penalty of $2\delta_{i}^{kk}$ and $\delta_{i}^{kk}+\delta_{i}^{ll}$, respectively, and each atom of type $k$ occupying the interface site has an energy of $\delta_{i}^{kk}\frac{z_{i}}{2}$ in excess of the bulk site (other energies being equal). To evaluate the effect of the interface energy penalty parameters on the the equilibrium system configuration, $\delta_{i}^{kk}$ for a specific like bond-type is varied, while the other two like bond-types are fixed at their default value of $3.5$ kJ/mol.

With an increase in $\delta_{i}^{kk}$, interface stability is maintained by rejecting from the interface atoms that contribute to excess interface energy through $kk$-type bonds, and by the segregation to the interface, atoms that reduce the interface energy through favorable $kl$ interactions. Thus, an increase in $\delta_{i}^{CC}$ leads to the de-segregation of $C$ from the interface, as shown by the decreasing $\Gamma_{i}^{C}$ in Fig. 6b, to reduce the overall number of interface $AC$, $CC$ and $BC$ bonds. Increasing $\delta_{i}^{AA}$ leads to rejection of $A$ atoms to reduce the fraction of $AA$ (mostly) bonds, and segregation of $C$ to increase the fraction of favorable $AC$ bonds (there is a balance here between the favorable interaction energy and the energy penalty associated with $AC$ bonds); this is shown in Fig. 7b. Since the concentration of $B$ at the interface is negligible, the increase in interface energy with $\delta_{i}^{BB}$ corresponds mainly to $AB$ and $BC$ bonds in the $ip$ transition region. In this case, the interface energy is reduced by substituting atoms of $A$ with $C$, thereby increasing $\Gamma_{i}^{C}$ (as shown in Fig. 8b) and the fraction of interface energy reducing $AC$ bonds.

The segregation state of $C$ (i.e. $\Gamma_{i}^{C}$) changes with $\delta_{i}^{kk}$ to maintain the interface in equilibrium as discussed above. However, in the absence of sufficient concentration of $C$ in the system, the equilibrium interface volume fraction decreases as shown by the shift in $r_{p}^{eq}$ vs. $y_{\circ}$ curves to larger precipitate sizes (Figs. 6a, 8a and 7a). The precipitate can be retained to the same size (i.e. same $\phi$) by increasing $y_{\circ}$ at a fixed $x_{\circ}$.

III.4 Temperature

Dependence of the system’s free energy function on temperature arises from the parameter $T$ coupled to the entropy of mixing terms corresponding to the bulk and the interface solid-solutions. The temperature dependence of $\Delta\bar{G}^{ref}_{b,i}$ is ignored and their values at $300$ K are used as input. The interaction parameters ($\omega$) are temperature-independent in the regular solution approximation. Thus the variation in the equilibrium system configuration with $T$ stems primarily from the entropic contributions.

With an increase in temperature, precipitates are stabilized at larger sizes as shown in Fig. 9(a). Correspondingly, concentration of $C$ at the interface decreases as shown by the decrease in the interfacial secondary solute excess in Fig. 9b. The results correspond to de-segregation of $C$ from the interface with temperature, leading to a reduction in the equilibrium interface volume fraction, and therefore, an increase in stable precipitate size. The cause of desegregation is the reduction in system free energy through an increase in the entropy of the bulk solid-solution region. While the total entropy of the system is a combination of configurational entropy of mixing of bulk and interface solid-solutions, since the volume fraction of the bulk is significantly greater than that of the interface ($f_{b}>0.9$), the system entropy is dominated by the contribution from the bulk (second term in Eq. 2). Therefore, with increasing temperature, the free energy of the system is reduced by increasing the configurational entropy of the bulk, which is accomplished by the desegregation of $C$ from the interface to the bulk. Free energy minimization, however, is a trade-off between enthalpic and entropic contributions. A system with stronger interactions of $C$ atoms at the interface, over the bulk, will have a lower tendency to de-segregate if the enthalpic driving force for segregation to the interface is stronger than the entropic driving force for de-segregation.

IV CONCLUSIONS

We developed an analytical model that captures the thermodynamic stabilization of nano-sized precipitates, through chemically driven solute segregation to the heterophase interface, in alloy systems precipitating an intermetallic compound from the solid-solution. In binary precipitate systems like Mg/Mg2Sn, or in the absence of secondary solutes with energy reducing interface interactions, the interface always presents a positive energy penalty which can only be reduced through precipitate coarsening. While aging treatments can be optimized to produce an alloy with small-size and high number density of precipitates to provide beneficial mechanical properties at room temperature, these precipitates are only kinetically stable. At higher temperatures, the precipitates coarsen to large sizes (typically to micron length scales) and drastically degrade structural performance. On the other hand, in ternary (or quaternary) alloy systems, where secondary solutes are chosen to have strong interactions at the heterophase interface, thermodynamic stabilization through solute segregation can be effective at high temperatures. For example, results of the ternary model evaluated for Mg-Sn-Zn system show that precipitates can be stabilized through Zn segregation driven by thermodynamic interfacial energy minimization.

While this study presents a thermodynamic basis for the experimental findings liu2016interphase of Zn segregation to Mg/Mg2Sn interface and an associated refinement of Mg2Sn precipitate sizes, actual values for the interface parameters are unknown. The parametric study presented in this paper shows a reduction in the equilibrium precipitate size when the global secondary solute concentration, $y_{\circ}$, is increased or any of the interface interaction parameters are decreased. On the other hand, the equilibrium precipitate size increases with temperature due to the entropic contributions favoring a more random partitioning of concentration between bulk and interface regions. While the present model incorporates chemical interactions, extension to the model is necessary to capture segregation and precipitate stabilization resulting from elastic strain energy effects. Together with the input of interface parameters from methods like density functional theory calculations, and the inclusion of participate morphology as a variable, the model will allow a more realistic prediction of thermodynamically stable precipitate sizes and shapes in ternary alloy systems.

V ACKNOWLEDGMENTS

This work was primarily supported by the U.S. National Science Foundation under Grant No. DMR-1554270, with partial support from North Carolina State University.