# An explicit integration approach for predicting the microstructures of multicomponent alloys

## TL;DR

This paper introduces a new method to predict the microstructures of complex alloys by overcoming computational limitations in existing models.

## Contribution

The novel approach integrates two thermodynamic conditions into a single explicit function, enabling simulations of up to 20-component systems.

## Key findings

- The model successfully satisfies equal diffusion and internal equilibrium conditions in simulations.
- It overcomes the curse of dimensionality, allowing efficient computation for multicomponent systems.
- The method is versatile and applicable to a wide range of practical materials.

## Abstract

Predicting the complex microstructures of practical materials has been a longstanding goal since Gibbs’s pioneering work on predictions for equilibrium of heterogeneous systems. The most promising approach for achieving this goal is integrating Calculation of Phase Diagrams (CALPHAD) with phase-field models. This CALPHAD-coupled phase-field model requires two Gibbs free energy minimisation conditions: equal diffusion potential and internal equilibrium, both grounded in the second law of thermodynamics. However, as implicit functions, these minimisation conditions suffer from the curse of dimensionality when applied to multicomponent systems, which imposes significant constraints on simulation capabilities. Here we propose an approach that incorporates the equal diffusion potential and internal equilibrium conditions into a single explicit function in phase-field equations. In simulations across various practical materials, our model achieved equal diffusion and internal equilibrium conditions. Furthermore, it overcame dimensionality limitations, enabling computations for systems with up to 20 components. Thus, the proposed approach proves highly versatile and efficient, supporting a wide range of practical applications.

Although Gibbs energy databases have been developed for many alloy systems, their integration into the phase field method remains limited. Here, authors develop an explicit integration approach to predict the microstructures of multicomponent alloys.