Precipitation of Non-Spherical Particles in Aluminum Alloys Part I: Generalization of the Kampmann–Wagner Numerical Model

Abstract

Particles precipitated during aging treatments often have non-spherical shapes, e.g. needles or plates, while in the classical Kampmann-Wagner Numerical (KWN) precipitation model it is assumed that the particles are of spherical shape. This model is here generalized resulting in two correction factors accounting for the effects induced by the particles' non-spherical shape on their growth kinetics. The first one is for the correction of the growth rate. It is derived from the approximate solution of the diffusion problem on spheroidal coordinate and verified by the three-dimensional numerical solutions for cuboid particles. The second factor is for the energetic correction due to the particle surface curvature. It is derived from chemical potential equality (or Gibbs energy minimization principle) at equilibrium for non-spherical particles and provides a correction factor for the Gibbs-Thomson effect. In the accompanying paper the two correction factors are implemented into a multi-component KWN modelling framework and the resulting improvements on the model's predictive power are demonstrated.

An Extended Age-Hardening Model for Al-Mg-Si Alloys Incorporating the Room-Temperature Storage and Cold Deformation Process Stages

“…Therefore, although physically based yield strength models capable of handling rod-shaped particles already exist, [72,73] they will not be implemented in NaMo-Version 2 because of the additional complications that such a refinement creates when it comes to modifying the existing nanostructure models to allow for the particle shape effect. However, the work on extending their applicability has now started, [74,75] which, in turn, may clear the way for a full treatment of the non-spherical particle case as well within the framework of the Kampmann-Wagner formalism if this is deemed to be necessary in the future. showing how the r p r /r p s ratio varies with the particle aspect ratio u and the dislocation locking mechanism (characterized by m = 0 and m = 1, respectively).…”

Modeling over-ageing in Al-Mg-Si alloys by a multi-phase CALPHAD-coupled Kampmann-Wagner Numerical model

“…Using the CALPHAD-coupled multi-component version of the KWN model reported in [16,30,31] as a starting point, the extension required to treat concurrent nucleation, growth and coarsening of multi-phase particles is adding one more interfacial phase composition relation equation for each extra precipitation phase. Before proceeding to the multi-phase KWN model description, it is useful to summarize the assumptions adopted below.…”

“…Based on the assumptions listed above and the correction factor f introduced in [30,31], the changing rate of the equivalent radius could be calculated from the diffusion of alloying component i:…”

“…This modeling framework and its CALPHAD-coupled multi-component extensions have been seen as a key microstructure chain model in an Integrated Computational Materials Engineering (ICME) modeling framework to optimize alloy chemistry and heat treatment parameters for many industrial metallic materials [14][15][16][17][18][19][20][21][22][23][24][25][26]. In the most recent extension of the KWN approach, the assumption of the precipitate particles being spherical has been released enabling a better treatment of needle-shaped particles' precipitation kinetics in aluminum alloys [30,31]. Another extension is reported in [32], where the KWN model is extended to accommodate the dislocation generation, the cluster formation and the competitive nucleation of β'' and β' precipitates during cold deformation, natural ageing and artificial ageing of aluminum alloys, respectively.…”

“…In this paper, we will extend the multi-component KWN model reported in [16,30,31] to treat concurrent nucleation, growth and coarsening of multi-phase precipitate particles. The coupling of the extended model with the reported CALPHAD databases [27,28] will be implemented.…”