Phase-field simulations of nucleation, growth, and coarsening of $\beta_1$ precipitates in Mg-Nd alloys

Problem Statement
The precipitation-hardening response of magnesium–rare-earth (Mg–RE) alloys, particularly Mg–Nd systems, is governed by the spatial distribution and morphology of precipitates formed during aging. While phase-field simulations have advanced the understanding of precipitate evolution, predictive modeling remains challenging due to the uncertainty or lack of key thermodynamic and kinetic parameters. A specific difficulty lies in comparing simulation results with experimental data: experiments typically provide two-dimensional (2D) cross-sectional data (e.g., from transmission electron microscopy, TEM), whereas accurate physical representation often requires three-dimensional (3D) modeling. However, full 3D phase-field simulations for systems with highly anisotropic precipitates like $\beta_1$ in Mg–Nd alloys are computationally prohibitive. Furthermore, few studies have successfully modeled the evolution of precipitate number density across all aging stages (nucleation, growth, and coarsening) to optimize aging conditions for desired mechanical properties.

Methodology
The authors developed an integrated experimental and computational workflow to parameterize a 2D phase-field model for the stochastic nucleation, growth, and coarsening of $\beta_1$ precipitates in Mg–Nd alloys aged at 523 K.

• Experimental Characterization:
  • Precipitate Number Density: High-resolution TEM images were acquired at various aging times (3 h to 9 h). Since TEM provides 2D cross-sections, a stereological method (Sonderegger's correction) was applied to estimate 3D number densities ($N_V$) from the 2D counts and foil thickness. These 3D densities were then converted to "effective" 2D number densities ($N_A$) by preserving the characteristic interparticle distance ($\lambda$), assuming $N_A = N_V^{2/3}$.
  • Matrix Composition: Atom Probe Tomography (APT) was used to measure the Nd composition in the matrix at different aging times, providing data on solute depletion.
• Phase-Field Modeling:
  • A 2D phase-field model was implemented using the PRISMS-PF framework. The model utilizes the Kim–Kim–Suzuki (KKS) formulation to couple composition and structural order parameters.
  • Three order parameters represent the three correspondence variants of the $\beta_1$ phase, capturing the lattice deformation and elastic strain energy.
  • A stochastic nucleus seeding method, based on classical nucleation theory, was coupled to the model to simulate concurrent nucleation and growth. The nucleation rate depends on local supersaturation.
• Parameterization Workflow:
  • Due to uncertainties in nucleation parameters (Zeldovich factor, attachment frequency, interfacial energy), a multi-step optimization was employed.
  • Step 1: Effective 2D number densities were derived from experimental 3D data.
  • Step 2: Nucleation rate parameters ($\rho_1, \rho_2, \tau_{in}$) were initially optimized using a simplified uniform-supersaturation model to match experimental number density trends.
  • Step 3: Diffusivity was estimated based on literature values and refined to match the rate of coarsening.
  • Step 4: A time-scale correction factor ($\xi$) was applied to the kinetic parameters ($\rho_1, D, \tau_{in}$) to fine-tune the agreement between the full phase-field simulation and experimental data, accounting for limitations in the uniform-supersaturation assumption.

Key Results

• Quantitative Agreement: The parameterized model successfully reproduced the evolution of precipitate number density, capturing the rapid increase during the nucleation stage and the subsequent decrease during coarsening. The simulated number densities showed reasonable agreement with the effective 2D experimental data.
• Microstructural Evolution: The simulations reproduced the plate-like morphology of $\beta_1$ precipitates and their six distinct growth directions (three correspondence variants with two rotation variants each), consistent with experimental observations. The model achieved this without explicit interfacial energy anisotropy, relying instead on anisotropic elastic properties.
• Matrix Composition: The simulated average Nd composition in the matrix aligned well with APT measurements, falling within statistical uncertainties for most data points.
• Kinetic Stages: The model captured the transition from nucleation to growth and finally to coarsening. It demonstrated how local solute depletion affects growth rates, where precipitates in solute-rich regions grow larger while those in depleted regions stagnate or dissolve.

Significance and Claims
The paper claims that this work demonstrates a robust strategy to overcome the challenges of parameterizing coupled phase-field and nucleation models when key physical parameters are uncertain or unknown. By using experimental data to derive effective 2D parameters from 3D reality, the authors created a computationally efficient framework that avoids the prohibitive cost of full 3D simulations while still capturing essential physical behaviors.

The authors state that this validated model enables the prediction of process-structure relationships, allowing for the optimization of aging times to achieve target precipitate number densities. They acknowledge limitations, noting that 2D simulations cannot fully capture 3D diffusional interactions and that the initial uniform-supersaturation optimization may lead to suboptimal parameter sets. However, they conclude that the workflow provides a foundation for optimizing aging conditions in Mg–Nd alloys and can be extended to other Mg–RE systems and different aging conditions. The authors suggest that future improvements could involve higher time-resolution experimental data and the incorporation of Bayesian methods for parameter space exploration.