An analytical model with interaction between species for growth and dissolution of precipitates

Abstract

An analytical model for growth in a semi-infinite matrix with cross-diffusion between species is presented. Application is given for precipitation of the -phase in the -matrix during isothermal holding at 600 °C in the Ni -7.56 at.% Al -8.56 at.% Cr alloy. The exact time-dependent solutions for the solute profiles and the growth kinetics are validated with a numerical front-tracking simulation. The simulation of cross diffusion terms in a multicomponent alloy is thus demonstrated. Extension of the analytical solution is given for growth in a matrix of finite size. The driving force is then based on a mathematical estimation of the far-field composition. The Gibbs-Thomson effect is also accounted for to consider the effect of curvature on the equilibrium tie-lines. Comparison of analytical solution with the numerical front-tracking simulation shows excellent agreement. Results also point out the detrimental approximation of using the average composition of the matrix for computing the driving force as well as the limitation of the solution proposed by Chen et al. [Acta Mater. 56 (2008) 1890. A detailed discussion is finally given on the origin of oscillations observed for the time evolution of the precipitate radius which alternates between growth and dissolution regimes, pointing out the combined role of solute fluxes and tie-lines compositions at the precipitate/matrix interface.

A review of microstructural changes occurring during FSW in aluminium alloys and their modelling

“…12) for any current temperature, . The growing kinetics is based on Aaron et al (1970) solution extended to multicomponent alloys by Guillemot and Gandin (2017) as detailed in previous part (Eq. 20).…”

“…An improvement of the Laplace approximation dedicated to multicomponent alloys has been recently proposed by Legrand (2015a) corresponding to the extension of the exact solution developed by Aaron et al (1970). This approach follows the work of Guillemot and Gandin (2017) and is based on the exact solution in the unsteady growth regime when temporal evolutions of the solute field are considered. This approach is also useful for large oversaturation (i.e.…”

Morphological stability of spherical particles - Extension of the Mullins-Sekerka criteria to multi-component alloys under a non-stationary diffusive regime

“…So Eqs (3) and (4) extend the description of the phase diagram by adding the Gibbs-Thomson effect due to curvature κ. The authors previously investigated the development of a spherical particle in a multicomponent alloy with cross diffusion and under unsteady growth regime [1,2]. They demonstrated that the previous set of equations has a single solution when neglecting the effect of curvature and the solute flux in the particle.…”

“…Morphological stability of spherical particles formed by solute diffusion in a matrix phase is of paramount interest in materials science. Common situations are the growth of solid grains in its surrounding liquid during solidification processing or the precipitation of intermetallic particles in its solid solution during heat treatment [1,2,3]. Depending from the applying conditions, the particles may evolve with spherical (or globular) shape if stability is preserved, cauliflower and dendritic shape otherwise.…”

Analytical Modeling of the Mixed-Mode Growth and Dissolution of Precipitates in a Finite System

“…As an advanced model, TC-PRISMA and PanPrecipitation use the solution of the quasi-stationary mass conservation equation proposed by Chen et al [4], who applied a binary exact solution to the multicomponent case, assuming that the diffusion length of each specie depends only on the specie's own supersaturation. This assumption was proven to have some limitations and inaccuracies by Guillemot and Gandin [5], who provided a more exact treatment of the quasi-stationary mass conservation problem in a multicomponent system.…”