Response-function-optimized phase field modeling of solute trapping and solute drag in rapid alloy solidification

Technical Summary: Response-Function-Optimized Phase Field Modeling of Solute Trapping and Solute Drag

Problem Statement
Quantitative prediction of microstructures during rapid alloy solidification requires phase field models that accurately capture the velocity dependence of interfacial properties, specifically solute partitioning (trapping), kinetic liquidus response, solute drag, and kinetic undercooling. While sharp-interface theories, such as the Continuous Growth Model (CGM), provide well-defined response functions for these phenomena, standard phase field models employing computationally efficient, enlarged interface widths ($W \gg W_0$) struggle to reproduce these target functions without introducing numerical artifacts. Previous approaches often rely on fixed, simple analytical forms for interfacial diffusivity interpolation, which lack the flexibility to simultaneously and accurately match prescribed velocity-dependent partition coefficients ($k(V)$) and drag-modified kinetic liquidus slopes ($m(V; \alpha)$) across a broad velocity range.

Methodology
The authors propose an optimization-based calibration strategy where the interfacial diffusivity interpolation function, $q_i(\phi)$, is treated as a constrained degree of freedom to match target sharp-interface response functions.

1. Dilute Multicomponent Formulation: The authors extend a binary far-from-equilibrium phase field model to dilute multicomponent alloys. By utilizing a grand-potential representation with component-specific local susceptibilities, the model allows for the independent specification of equilibrium partition coefficients ($k_i$) and liquidus slopes ($m_i$) for each solute species, avoiding unphysical thermodynamic constraints inherent in universal prefactor approaches.
2. Inverse Response-Matching: The core methodology involves solving an inverse problem. For a given set of target response functions (e.g., from CGM theory, atomistic simulations, or experiments), the coefficients of a smooth, endpoint-preserving expansion of the diffusivity function $q_i(\phi)$ are optimized.
   • The expansion is defined as a correction to a linear profile using Chebyshev polynomials: $q_{i,N}(\phi) = \frac{1-\phi}{2} + (1-\phi^2)\sum a_{i,n}T_n(\phi)$.
   • The optimization minimizes a loss function balancing the root-mean-square error of the phase field predictions ($k_i^{PF}(V)$ and $m_i^{PF}(V)$) against the target functions, subject to a smoothness regularization term to prevent unphysical sharp features in the diffusivity profile.
   • A tolerance-based selection rule is employed to choose the smoothest interpolation that satisfies a specified accuracy threshold (e.g., 1% relative error) over the velocity range of interest.
3. Validation and Simulation: The calibrated models are validated in one-dimensional steady-state planar-interface simulations. Subsequently, two-dimensional directional solidification simulations are conducted to observe the impact of the calibrated kinetics on microstructure selection.

Key Results

• Binary Al–Cu Calibration: The method successfully reproduced prescribed CGM target functions for solute trapping and drag-modified liquidus kinetics in a dilute Al–Cu system for intermediate solute drag coefficients ($\alpha = 0.3, 0.5, 0.7$). The optimized diffusivity functions differed significantly from standard linear interpolations, particularly within the diffuse interface region, to compensate for the enlarged interface width.
• Morphology Selection: Two-dimensional simulations revealed that increasing the solute drag coefficient ($\alpha$), while keeping the partition coefficient $k(V)$ constant, systematically alters the solidification morphology.
  • Low drag ($\alpha=0.3$) resulted in dendritic growth.
  • Intermediate drag ($\alpha=0.5$) produced mixed dendritic–banded structures.
  • High drag ($\alpha=0.7$) led to predominantly banded growth.
• Mechanistic Interpretation: A reduced stability analysis ($Q=0$ mode) demonstrated that increasing drag reduces the slope of the kinetic solidus relation ($dT_S^\infty/dV$). This reduction increases the sensitivity of the mean interface velocity to lateral displacements in the thermal gradient, providing a mechanistic explanation for the transition toward relaxation-type banding in high-drag scenarios.
• Multicomponent Extension: The framework was successfully applied to a dilute ternary Al–Si–Cu alloy. The model independently tracked the segregation of Si and Cu, demonstrating that species-specific response functions can be embedded simultaneously without reducing the system to a quasi-binary approximation.

Significance and Claims
The paper claims to provide a practical route for incorporating experimentally, theoretically, or atomistically informed nonequilibrium interface kinetics into quantitative phase field simulations. By decoupling the physical specification of response functions from their numerical realization, the framework allows researchers to:

• Explicitly control and test the influence of specific kinetic parameters (like solute drag) on microstructure selection.
• Avoid the limitations of fixed analytical interpolation forms that cannot capture complex velocity dependencies.
• Extend rapid solidification modeling to multicomponent systems with independent solute behaviors.

The authors emphasize that this approach does not determine the physical parameters from first principles but rather creates a robust numerical bridge between sharp-interface theories (or data) and diffuse-interface simulations. They note that while the current formulation uses the CGM as a target for demonstration, the method is "response-function agnostic" and can accommodate targets from local nonequilibrium models or atomistic simulations. The work is presented as a controlled demonstration of how drag-modified kinetics qualitatively alter morphology, rather than a complete processing map, and acknowledges that future work is needed to address concentrated alloys, full diffusion matrices, and latent heat effects for quantitative banding predictions.