Effect of elastic anisotropy on phase separation in ternary alloys: A phase-field study

TL;DR

This study uses a phase-field model to explore how elastic anisotropy influences microstructural evolution and phase separation in ternary alloys, revealing its critical role in precipitate morphology and alignment.

Contribution

It introduces a phase-field approach incorporating elastic anisotropy to analyze microstructure development in ternary alloys, highlighting the impact of elastic moduli anisotropy on precipitate morphology.

Findings

Elastic anisotropy affects precipitate shape and alignment.

Misfit strains influence phase equilibria and pathways.

Elastic moduli anisotropy determines inclusion morphology.

Abstract

The precipitate shape, size and distribution are crucial factors which determine the properties of several technologically important alloys. Elastic interactions between the inclusions modify their morphology and align them along elastically favourable crystallographic directions. Among the several factors contributing to the elastic interaction energy between precipitating phases, anisotropy in elastic moduli is decisive in the emergence of modulated structures during phase separation in elastically coherent alloy systems. We employ a phase-field model incorporating elastic interaction energy between the misfitting phases to study microstructural evolution in ternary three-phase alloy systems when the elastic moduli are anisotropic. The spatiotemporal evolution of the composition field variables is governed by solving a set of coupled Cahn-Hilliard equations numerically using a semi-implicit Fourier spectral technique. We methodically vary the misfit strains, alloy chemistry and elastic anisotropy to investigate their influence on domain morphology during phase separation. The coherency strains between the phases and alloy composition alter the coherent phase equilibria and decomposition pathways. The degree of anisotropy in elastic moduli modifies the elastic interaction energy between the precipitates depending on the sign and magnitude of relative misfits, and thus determines the shape and alignment of the inclusions in the microstructure.

Tables

Table 1. Table 1 : Alloy systems used for simulations (prescribed alloy composition and magnitude and sign of misfit strains)

System	$c_B$	$c_C$	${ϵ}_{\alpha \beta }$	${ϵ}_{\alpha \gamma }$
$X_1$	0.15	0.15	0.01000	0.01000
$X_2$	0.15	0.15	0.01000	-0.01000
$X_3$	0.15	0.15	0.01265	-0.00632
$Y_1$	0.40	0.40	0.01000	0.01000
$Y_2$	0.40	0.40	0.01000	-0.01000
$Y_3$	0.40	0.40	0.01265	0.00632

Equations

$$F=N_{v}\int_{v}\Big{(}\frac{1}{2}\sum_{i\neq j}\chi_{ij}c_{i}c_{j}+\sum_{i}c_{i}\ln{c_{i}}+\sum_{i}\kappa_{i}|\nabla c_{i}|^{2}\Big{)}dV,$$

$$F_{el}=\frac{1}{2}\sum _{p,q=0}^1\int \frac{d^3\text{k}}{(2\pi {)}^3}{B}_{pq}(\text{n})\widetilde{{\theta }_p}(\text{k}){\widetilde{{\theta }_q}}^{\ast }(\text{k}),$$

$$\begin{split}\frac{\partial c_{B}}{\partial t}=M_{BB}&\nabla^{2}\Bigg{(}g_{B}-2\kappa_{BB}\nabla^{2}c_{B}-2\kappa_{BC}\nabla^{2}c_{C}+\frac{\delta F_{el}}{\delta c_{B}}\Bigg{)}+\\ M_{BC}&\nabla^{2}\Bigg{(}g_{C}-2\kappa_{BC}\nabla^{2}c_{B}-2\kappa_{CC}\nabla^{2}c_{C}+\frac{\delta F_{el}}{\delta c_{C}}\Bigg{)},\end{split}$$

$$\begin{split}\frac{\partial c_{C}}{\partial t}=M_{BC}&\nabla^{2}\Bigg{(}g_{B}-2\kappa_{BB}\nabla^{2}c_{B}-2\kappa_{BC}\nabla^{2}c_{C}+\frac{\delta F_{el}}{\delta c_{B}}\Bigg{)}+\\ M_{CC}&\nabla^{2}\Bigg{(}g_{C}-2\kappa_{BC}\nabla^{2}c_{B}-2\kappa_{CC}\nabla^{2}c_{C}+\frac{\delta F_{el}}{\delta c_{C}}\Bigg{)},\end{split}$$

$$M_{BB}=(1-c_B{)}^2{M}_B+c_B^2(M_A+M_C),$$

$$M_{CC}=(1-c_C{)}^2{M}_C+c_C^2(M_A+M_B),$$

$$M_{BC}=M_{CB}=c_B{c}_C{M}_A-c_B(1-c_C)M_C-c_C(1-c_B)M_B,$$

$$B_{pq}(\text{n})={\lambda }_{ijkl}{ϵ}_{ij}^{(p)}{ϵ}_{kl}^{(q)}-n_i{\widehat{{\sigma }_{ij}}}^{(p)}{\omega }_{jk}(\text{n}){\widehat{{\sigma }_{kl}}}^{(q)}{n}_l,$$

Full text

††journal: Scripta Materialia

Elastic anisotropy significantly influences the microstructures in multicomponent, multiphase alloys with coherent elastic misfit. The elastic interactions arising due to misfit between the phases, anisotropy in elastic moduli, and elastic homogeneity produce changes in the morphology and alignment of phases [1, 2]. The studies on the influence of elastic anisotropy on morphological evolution in binary alloys was pioneered by Cahn in his seminal paper on spinodal decomposition in cubic crystals [3]. He showed that the coherency strain between coexisting phases and anisotropy in elastic moduli leads to decomposition occurring by sinusoidal composition modulations along elastically preferred orientations depending upon the degree of elastic anisotropy.. He predicted the alignment of domains along elastically soft $\langle 100\rangle$ directions when $A_{z}>1$ and along $\langle 111\rangle$ directions when $A_{z}<1$.

There is substantial experimental evidence on the modulated arrangement of precipitates in two-phase coherent alloys [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. On investigating the modulated structures observed in $Ni$-$Al$ alloys [4], Ardell and Nicholson postulated that the alignment is caused by elastic interactions between precipitates. They also reported that the degree of alignment depends on volume fraction of the precipitates (supersaturation), misfit strains and elastic anisotropy.

There have been several attempts at modeling the microstructural evolution in elastically anisotropic two-phase alloy systems. Some of these studies focused on the equilibrium shape of misfitting particle in an elastically anisotropic medium [17, 18]. Computational models were also employed to investigate coarsening in elastically anisotropic solids with two or multiple coherent precipitates [19, 20, 21]. Several researchers simulated the development of modulated structures during both spinodal decomposition and nucleation and growth process in coherent binary alloys [22, 23, 24, 25, 26].

With the above discussed motivation, the purpose of our work is to the extend the understanding of the role of elastic anisotropy in the emergence of modulated structures in binary alloys to ternary three-phase alloys. In this regard, we develop a diffuse interface model to perform systematic simulations to investigate the combined effect of anisotropy in elastic moduli and the sign and degree misfit between coexisting phases on ternary phase separation. The presence of a third phase increases the complexity of elastic interactions and thus affects the direction of alignment of three phases differently.

We model a ternary substitutional alloy containing three atomic species A, B and C. The concentration of $i^{\prime}$th species $c_{i}(\textbf{r},t)$ $(i=A,B,C)$ is a function of position r and time $t$, where $c_{A}+c_{B}+c_{C}=1$ according to conservation condition. We modify an existing diffuse-interface ternary Cahn-Hilliard model  [27, 28, 29] by introducing additional terms describing elastic interactions.

The chemical energy contribution to the total energy for the isotropic, compositionally inhomogeneous system expressed as a function of composition field variables is

[TABLE]

where $i,j=A,B,C$, $N_{v}$ is the number of molecules per unit volume (assumed to be independent of composition and position), $\chi_{AB}$, $\chi_{AC}$ and $\chi_{BC}$ are the pair-wise interaction parameters, and $\kappa_{i}$ are the gradient energy coefficients associated with composition fields.

The elastic energy contribution to the total energy for the system is expressed in reciprocal space using Khachaturyan’s microelasticity theory [1]:

[TABLE]

where k denotes the Fourier wave vector, $\textbf{n}=\frac{\textbf{k}}{|\textbf{k}|}$ is the unit vector in reciprocal space, $\tilde{\theta_{p}}(\textbf{k})$ represents the Fourier transform of $\theta_{p}(\textbf{r})$. $B_{pq}(\textbf{n})=\lambda_{ijkl}\epsilon_{ij}^{(p)}\epsilon_{kl}^{(q)}-n_{i}\hat{\sigma_{ij}}^{(p)}\omega_{jk}(\textbf{n})\hat{\sigma_{kl}}^{(q)}n_{l}$ is the elastic interaction energy between $\theta_{p}$ and $\theta_{q}$ and $\omega_{il}^{-1}(\textbf{n})=\lambda_{ijkl}n_{j}n_{k}$ is the inverse Green tensor. $\tilde{\theta^{*}}$ denotes the complex conjugate of $\tilde{\theta}$.

The temporal evolution of the conserved field variables $c_{B}(\textbf{r},t)$ and $c_{C}(\textbf{r},t)$ ($\because c_{A}+c_{B}+c_{C}=1$) is governed by a set of Cahn-Hilliard equations:

[TABLE]

[TABLE]

where $\frac{\delta F_{el}}{\delta c_{B}}$ and $\frac{\delta F_{el}}{\delta c_{C}}$ are variational derivatives of elastic energy with respect to composition fields, $g_{B}$ and $g_{C}$ are bulk driving forces with respect to composition fields, $\kappa_{BB}=\kappa_{A}+\kappa_{B},\kappa_{BC}=\kappa_{CB}=\kappa_{A},\kappa_{CC}=\kappa_{A}+\kappa_{C}$. $M_{BB}$, $M_{BC}$ and $M_{CC}$ are Onsager kinetic coefficients defined as follows [30]:

[TABLE]

[TABLE]

[TABLE]

where $M_{A}$, $M_{B}$, and $M_{C}$ are the mobilities of species $A$, $B$, and $C$, respectively.

We use a semi implicit Fourier spectral method [31, 32] to solve the coupled evolution equations (Eqns. 3 and 4). All parameters used in the simulation are rendered non-dimensional using characteristic length, time and energy values. We perform two dimensional simulations of microstructural evolution on a square grid of size $1024\times 1024$ with dimensionless grid spacing $\Delta x=\Delta y=1$ employing periodic boundary conditions.

The simulations start with a homogeneous alloy of a prescribed composition. A sustained Gaussian noise of strength $0.5\%$ is added for the initial 10000 steps to mimic thermal fluctuations. We choose a non-dimensional time step $\Delta t=0.05$ for evolving the phase-field variables. We choose the bulk free energy coefficients and the gradient energy coefficients to be equal ($\chi_{AB}=\chi_{AC}=\chi_{BC}=3.5$, $\kappa_{A}=\kappa_{B}=\kappa_{C}=4$) to ensure a symmetric ternary miscibility gap with equal interfacial energies between the equilibrium phases ($\Gamma_{\alpha\beta}=\Gamma_{\alpha\gamma}=\Gamma_{\beta\gamma}$).

We introduce anisotropy in elastic moduli by fixing the Zener anisotropy ratio ($A_{z}=3$) and systematically vary the alloy composition and the sign and magnitude of relative misfit between coexisting phases to understand the role of anisotropy in elastic moduli on microstructural evolution during ternary phase separation. We have chosen six different alloy systems categorized depending on their chemical composition ($c_{B},c_{C}$) and misfit strains ($\epsilon_{\alpha\beta},\epsilon_{\alpha\gamma}$) as listed in Table 1.

We follow a particular naming convention to specify an alloy system: an uppercase letter denotes the composition of the alloy and an integer subscript represents the state of misfit strain. Integer $`1^{\prime}$ represents the case where the lattice expansion coefficients, $\epsilon_{\alpha\beta}$ and $\epsilon_{\alpha\gamma}$, associated with concentrations of B and C, respectively, have equal magnitude and same sign. On the other hand, integer $`2^{\prime}$ represents the case where $\epsilon_{\alpha\beta}=-\epsilon_{\alpha\gamma}$, and integer $`3^{\prime}$ represents the case where $|\epsilon_{\alpha\beta}|\neq|\epsilon_{\alpha\gamma}|$.

The microstructures of the alloys are represented using an RGB (red-green-blue) color map indicating the local compositions of components $A$, $B$, and $C$. As per the color map, blue shade represents $A-$rich $\alpha$ phase, green represents $B-$rich $\beta$ phase, and red represents $C-$rich $\gamma$ phase. We specify the interfaces by the linear combination of three terminal colors depending upon the compositions.

The time snapshots of the microstructures of $A-$rich $X$ alloys and $A-$poor $Y$ alloys and power spectra corresponding to the microstructures are shown in Fig. 1. Power spectrum is the measure of the power content of a signal as a function of it’s frequency. The spectrum of the microstructure will contain information about morphological anisotropy. The power spectrum analysis of a simulated microstructure is analogous to the small angle X-ray scattering (SAXS) data obtained from an experimental microstructure.

In $A-$rich $X$ alloys, the initial phase separation takes place along the median perpendicular to $BC$ line leading to the formation of $A-$poor domains in an $A-$rich matrix. The ‘$BC$’-rich domains undergo secondary decomposition forming $\beta$ and $\gamma$ phases. As per the Zener ratio, $\langle 10\rangle$ directions are the elastically soft directions for these alloys.

Since $\alpha$ is associated with non-zero misfit, primary decomposition into $A-$poor and $A-$rich domains is sluggish in $X_{1}$. But the secondary decomposition of the $A-$poor domains to $\beta$ and $\gamma$ phases takes place quickly because of the absence of any misfit between the two phases ($\epsilon_{\beta\gamma}=0$). The power spectrum of the microstructure has its peaks at $\langle 10\rangle$ directions indicating alignment of $\beta$ and $\gamma$ domains along these elastically favorable directions (Fig. 1(a)). This orientational preference is attributed to the equal and positive misfit of these phases with matrix phase $\alpha$. Since the $\beta\gamma$ interface is strain-free ($\epsilon_{\beta\gamma}=0$), the system prefers more $\beta\gamma$ interfaces for the minimization of elastic energy. Thus, we observe chains comprising alternating beads of $\beta$ and $\gamma$ aligned along $\langle 10\rangle$ directions in this alloy.

The initial phase separation begins quickly in $X_{2}$ and $X_{3}$ since the misfits of $\beta$ and $\gamma$ phases with the matrix phase $\alpha$ are opposite in sign. But the secondary decomposition of the $A-$poor domains is delayed because of high misfit between $\beta$ and $\gamma$. The strain is highest in alloy $X_{2}$ ($\epsilon_{\beta\gamma}=0.02$). Therefore both domains appear much later. Since the $\beta\gamma$ interfaces have large misfit in both alloys, the domains remain isolated in the matrix.

In alloy $X_{2}$, the domains show a weak inclination towards $\langle 10\rangle$ directions since misfits are equal in magnitude, but opposite in sign ($\epsilon_{\alpha\beta}=-\epsilon_{\alpha\gamma}=1.0$). Therefore, the power spectrum is diamond shaped indicating the weak alignment of isolated $\beta$ and $\gamma$ particles along $\langle 10\rangle$ directions (Fig. 1(b)). In $X_{3}$, $\beta$ and $\gamma$ have unequal misfits of opposite sign with the matrix $\alpha$ phase. This leads to shape changes of the isolated domains from cuboidal to rhombohedral/diamond-like. The power spectrum is diffused and has circular symmetry (Fig. 1(c)). This is because of the weak alignment of the isolated domains along $\langle 10\rangle$ and interconnected domains along $\langle 11\rangle$.

The $A-$poor $Y$ alloys phase separate initially to $B-$rich and $C-$rich domains and $\alpha$ phase appears later during secondary decomposition. These alloys also have Zener anisotropy ratio $A_{z}=3$ and hence the $\langle 10\rangle$ directions are elastically soft.

In $Y_{1}$ and $Y_{3}$, the initial decomposition to majority phases $\beta$ and $\gamma$ happens quickly due to the low coherency strains between those phases. $\epsilon_{\beta\gamma}=0$ for $Y_{1}$ alloy and $\epsilon_{\beta\gamma}=0.006$ for $Y_{3}$. Since $\epsilon_{\beta\gamma}=0$, the $\beta\gamma$ interfaces are curved in $Y_{1}$ and the domains do not show any specific directional alignment. However, in $Y_{3}$, since the $\beta$ and $\gamma$ phases have unequal misfits of same sign with the third phase $\alpha$, both the $\beta$ and $\gamma$ domains align along $\langle 11\rangle$ directions.

In both the alloys, $\alpha$ phase separates out eventually, forming elongated domains along elastically soft $\langle 10\rangle$ directions. Due to this orientational preference shown by the $\alpha$ domains, the peaks for the power spectrum of $Y_{1}$ are at $\langle 10\rangle$ directions (Fig. 1(d)). Whereas, the power spectrum of $Y_{3}$ peaks at both $\langle 11\rangle$ and $\langle 10\rangle$ directions owing to the strong alignment of minority phase $\alpha$ along $\langle 10\rangle$ and majority phases $\beta$ and $\gamma$ along $\langle 11\rangle$ directions (Fig. 1(f)).

The initial phase separation of $\beta$ and $\gamma$ domains is delayed in $Y_{2}$ since the relative misfit between them is large ($\epsilon_{\beta\gamma}=0.02$). Subsequently, after the formation of $\beta$ and $\gamma$, isolated domains of $\alpha$ appear by secondary spinodal decomposition. The $\alpha$ domains exhibit weak alignment along the elastically preferred $\langle 10\rangle$ directions. The $\beta$ and $\gamma$ phases also align along the same directions. Therefore, we observe only $\langle 10\rangle$ peaks in the power spectrum (Fig. 1(e)).

Since interfacial energies between the coexisting phases are assumed to be isotropic, the change in orientation of the domains as a function anisotropy in elastic moduli can be explained from the corresponding polar plot of elastic interaction energies as function of orientation with respect to the crystal axes. The elastic interaction energy $B_{pq}(\textbf{n})$ between two domains $p$ and $q$ is given by the expression:

[TABLE]

where $\lambda_{ijkl}$ is the elastic stiffness tensor, $\omega_{il}^{-1}(\textbf{n})=\lambda_{ijkl}n_{j}n_{k}$ is the normalized inverse Green tensor, $\epsilon_{ij}^{(p)}$ is the eigenstrain field and $\hat{\sigma_{ij}}^{(p)}$ is the stress field associated with respective domains. The inner envelop of the polar plots of $B_{pq}(\textbf{n})$ predicts the equilibrium shapes of domains that minimize the interaction energy.

Since $X_{1}$, $X_{2}$ and $X_{3}$ alloys have $\beta$ and $\gamma$ domains distributed in $\alpha$ matrix, we explain the morphological patterns according to $B_{\beta\beta}(\textbf{n})$, $B_{\gamma\gamma}(\textbf{n})$ and $B_{\beta\gamma}(\textbf{n})$, where $B_{\beta\beta}(\textbf{n})$ denotes interaction energy between $\beta$ domains, $B_{\gamma\gamma}(\textbf{n})$ denotes interaction energy between $\gamma$ domains, and $B_{\beta\gamma}(\textbf{n})$ denotes interaction energy between $\beta$ and $\gamma$ domains. As shown in Fig. 2(a), the three interaction energies are equal for alloy $X_{1}$ and they show minima when n corresponds to $\langle 10\rangle$ directions. This explains the arrangement of the domains along $\langle 10\rangle$ directions in $X_{1}$.

For $X_{2}$ and $X_{3}$ alloys, $B_{\beta\gamma}(\textbf{n})$ is not equal to $B_{\beta\beta}(\textbf{n})$ and $B_{\gamma\gamma}(\textbf{n})$ due to the changes in sign and magnitude of the misfits. The interactions between $B_{\beta\beta}(\textbf{n})$, $B_{\gamma\gamma}(\textbf{n})$ and $B_{\beta\gamma}(\textbf{n})$ change the elastically soft directions for alloys $X_{2}$ and $X_{3}$. In $X_{2}$, $B_{\beta\beta}(\textbf{n})$ and $B_{\gamma\gamma}(\textbf{n})$ show minima along $\langle 10\rangle$ when $B_{\beta\gamma}(\textbf{n})$ shows minima along $\langle 11\rangle$ (Fig. 2(b)). Thus isolated $\beta$ and $\gamma$ particles align along $\langle 10\rangle$ and appear rectangular. However, there is a weak alignment of alternate $\beta$ and $\gamma$ particles along $\langle 11\rangle$. In $X_{3}$, the interactions are similar to $X_{2}$ (Fig. 2(c)). However, the alignment of alternate $\beta$ and $\gamma$ domains along $\langle 11\rangle$ seems stronger.

In $A-$poor $Y$ alloys, $\alpha$ domains always align along elastically soft $\langle 10\rangle$ directions dictated by Zener anisotropy ratio ($A_{z}=3$). In contrast, alignment and shape of $\beta$ and $\gamma$ domains change with the variations in the misfit strains. Since the sign and magnitude of misfits affect $B_{\alpha\beta}(\textbf{n})$ and $B_{\alpha\gamma}(\textbf{n})$, the inner envelope of orientation dependent elastic interaction energy can be used to predict the shape as well as the alignment of both $\beta$ and $\gamma$. $B_{\alpha\beta}(\textbf{n})$ and $B_{\alpha\gamma}(\textbf{n})$ shows similar behavior for these alloy systems. Therefore, we only analyze the interaction energy between $\alpha$ and $\beta$ domains. $B_{\alpha\beta}(\textbf{n})$ is zero for $Y_{1}$, which explains the absence of any orientational preference in $\beta$ domains of the alloy. Since $B_{\alpha\beta}(\textbf{n})$ has minimum along $\langle 10\rangle$ directions in $Y_{2}$ (Fig. 3(a)), the $\beta$ domains in the alloy are oriented along the same directions. Fig. 3(b) shows the polar plot of $B_{\alpha\beta}(\textbf{n})$ in $Y_{3}$ and it has a minimum along $\langle 11\rangle$ directions. Hence the $\beta$ domains are arranged along $\langle 11\rangle$ directions in $Y_{3}$ alloy. $\gamma$ phase shows similar orientational preference as $\beta$ in all these alloys.

It is evident from our investigation that the morphological alignment and shape of elastically coherent domains are strongly influenced by the interactions between anisotropy in the elastic moduli and the misfit strains between the coexisting phases. The elastic interaction energies between these co-existing domains modify their shape and directional alignment based on the elastically soft directions. The morphological characteristics are also dependent on the alloy composition, since the kinetic paths of decomposition can affect the alignment of domains.

We investigated the effect of elastic anisotropy on spinodal decomposition in ternary alloys by systematically varying the alloy composition $(c_{B}^{0},c_{C}^{0})$ and misfit strains ($\epsilon_{\alpha\beta},\epsilon_{\alpha\gamma}$). The variations in coherency strains between the coexisting phases and alloy chemistry change the equilibrium compositions and phase separation sequence, thereby influencing the morphological attributes of the domains. The interplay between elastic anisotropy and relative misfits between the phases modifies the elastic interaction energies between the domains depending on the sign and degree of misfit strains. The inner envelopes of the polar plots of these elastic interaction energies match with the morphological patterns of the precipitates. Thus, these interactions determine the change in shape and directional alignment of the domains in the microstructure. Therefore, we conclude that the preferential crystallographic directions of alignment of precipitates in three-phase alloys depend not only on the anisotropy in elastic moduli, but also on the sign and magnitude of the misfit strains.

Acknowledgements

The authors are grateful for the financial support from R $\&$ D and SS, Tata Steel Limited.