[Paper Review] Precipitation in Al-Zr-Sc alloys: a comparison between kinetic Monte Carlo, cluster dynamics and classical nucleation theory

Emmanuel Clouet, Maylise Nastar+3 more|arXiv (Cornell University)|Jul 12, 2005

Aluminum Alloy Microstructure Properties48 references13 citations

TL;DR

This study develops a multiscale modeling approach for precipitation in Al-Zr-Sc alloys, combining kinetic Monte Carlo (KMC) simulations with cluster dynamics (CD) and an extended classical nucleation theory (CNT). It demonstrates that accounting for short-range order in CNT enables accurate prediction of nucleation rates, showing that Zr addition enhances Sc precipitation in Al-Sc alloys by increasing nucleation density, with good agreement between KMC, CD, and experimental data across varying supersaturations and temperatures.

ABSTRACT

Zr and Sc precipitate in aluminum alloys to form the Al\\_3Zr\\_xSc\\_{1-x} compound which, for low supersaturations of the solid solution, exhibits the L1\\_2 structure. The aim of the present study is to model at an atomic scale the kinetics of precipitation and to build mesoscopic models so as to extend the range of supersaturations and annealing times that can be simulated up to values of practical interest. In this purpose, we use some ab initio calculations and experimental data to fit an Ising type model describing thermodynamics of the Al-Zr-Sc system. Kinetics of precipitation are studied with a kinetic Monte Carlo algorithm based on an atom-vacancy exchange mechanism. Cluster dynamics is then used to model at a mesoscopic scale all the different stages of homogeneous precipitation in the two binary Al-Zr and Al-Sc alloys. This technique correctly manages to reproduce both the kinetics of precipitation simulated with kinetic Monte Carlo as well as experimental observations. Focusing on the nucleation stage, it is shown that classical theory well applies as long as the short range order tendency of the system is considered. This allows us to propose an extension of classical nucleation theory for the ternary Al-Zr-Sc alloy.

Motivation & Objective

• To model precipitation kinetics in Al-Zr-Sc alloys at atomic and mesoscopic scales for practical heat treatment conditions.
• To bridge atomic-scale KMC simulations with mesoscopic models (CD and CNT) to extend accessible simulation ranges.
• To assess the validity of classical nucleation theory in ternary Al-Zr-Sc systems and extend it to account for short-range order effects.
• To quantify how Zr addition influences Sc precipitation kinetics and nucleation rates in Al-Sc alloys.
• To derive mesoscopic parameters (e.g., interface energy, nucleation rate) from ab initio and experimental data for use in predictive models.

Proposed method

• Developed an Ising-type lattice model with first- and second-nearest-neighbor interactions to describe thermodynamics of the Al-Zr-Sc system, fitted to ab initio data and solubility limits.
• Performed kinetic Monte Carlo (KMC) simulations using an atom-vacancy exchange mechanism to simulate early-stage precipitation at high supersaturation.
• Applied cluster dynamics (CD) to model mesoscopic cluster size distributions and reproduce KMC kinetics and experimental observations.
• Extended classical nucleation theory (CNT) by incorporating short-range order effects via free energy expressions dependent on cluster composition and size.
• Derived analytical expressions for nucleation rate $ J^{st}(x) $, critical nucleus size $ n^*(x) $, and Zeldovitch factor $ Z(x) $, using second derivatives of cluster free energy.
• Calculated interface free energy $ \bar{\sigma} $ and chemical potentials from KMC and ab initio data to parameterize the extended CNT model.

Experimental results

Research questions

• RQ1 How does the addition of Zr affect the nucleation rate of Sc-rich precipitates in Al-Sc alloys?
• RQ2 Can classical nucleation theory be extended to accurately describe precipitation in the ternary Al-Zr-Sc system?
• RQ3 To what extent do short-range order effects influence the free energy barrier and critical nucleus size in Al-Zr-Sc precipitation?
• RQ4 How well do cluster dynamics and extended CNT models reproduce kinetic Monte Carlo and experimental data across varying supersaturations and temperatures?
• RQ5 What is the role of solute diffusion coefficients in precipitate inhomogeneity and nucleation rate enhancement in the ternary system?

Key findings

• The extended CNT model, which accounts for short-range order, successfully predicts the increase in nucleation rate upon Zr addition to Al-Sc alloys, matching KMC and experimental trends.
• The critical nucleus size $ n^*(x) $ in the ternary system depends on the composition $ x $, with a maximum nucleation rate observed at intermediate Sc content and Zr concentration.
• For $ x^0_{\mathrm{Sc}} = 0.5 $ at.% at $ 450^\circ\mathrm{C} $, the nucleation rate $ J^{st} $ increases significantly with Zr concentration, peaking when Zr and Sc concentrations are comparable.
• Cluster dynamics (CD) accurately reproduces KMC-simulated precipitation kinetics and experimental data, validating its use for extended simulations.
• The Zeldovitch factor $ Z(x) $ and rate constant $ \beta^*(x) $ are derived analytically, enabling quantitative prediction of nucleation rates as a function of cluster composition.
• The model shows that precipitate inhomogeneity arises primarily from differences in Sc and Zr diffusion coefficients, even though the final precipitate composition is not known a priori.