Mixed-mode growth of a multicomponent precipitate in the quasi-steady state regime (original) (raw)
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Acta Materialia, 2017
The growth of an ellipsoidal precipitate has been analysed in the mixed-mode regime for a binary system. Under the assumption that the precipitate grows with constant eccentricities, an analytical solution was developed giving the time evolution of the size of the precipitate and the non-equilibrium concentration of the solute in the matrix. The mathematical analysis revealed that the evolution of the growth is characterized by a constant k called the interface migration coefficient. This coefficient was found to be equal to 1 2 ffiffiffiffiffiffiffiffiffiffiffi y c =a c p , where a c is the critical size of nucleation and y c is the maximum growth velocity attainable with the applied driving force. This velocity, which was found to be proportional to the square root of the interface mobility, was assumed to be constant during the nucleation stage, making a c =y c to be the nucleation time. This finding suggests that there is a close link between the nucleation time and the mobility of the interface separating the nucleus from the matrix.
Analytical treatment of diffusion during precipitate growth in multicomponent systems
Acta Materialia, 2008
We propose an approximate growth rate equation that takes into account both cross-diffusion and high supersaturations for modeling precipitation in multicomponent systems. We then apply it to an Fe-alloy in which interstitial C atoms diffuse much faster than substitutional solutes, and predict a spontaneous transition from slow growth under ortho-equilibrium to fast growth under the non-partitioning local equilibrium condition. The transition is caused by the decrease in the Gibbs-Thomson effect as the growing particle becomes larger. The results agree with DICTRA simulations where full diffusion fields are calculated.
On evaluation of the equations for the growth kinetics of ellipsoidal precipitates
Scripta Metallurgica, 1984
Grain boundary allotriomorphs are probably most accurately modeled as double spherical caps (1). However, a solution to the diffusional growth problem for this morphology has not yet been reported, A reasonable approximation of the double spherical cap model is provided by the oblate ellipsoid of revolution.
A model for evolution of shape changing precipitates in multicomponent systems
Acta Materialia, 2008
Recently the authors introduced a concept of shape factors to extend an already established model for the growth and coarsening kinetics of spherical precipitates in multicomponent multiphase environments to needle-and disc-shaped geometries. The geometry of the precipitates is kept in the original version of the concept to be self-similar with a given fixed aspect ratio. In the present treatment, the aspect ratios of individual precipitates are treated as independent evolving parameters. The evolution equations of each precipitate, described by its effective radius, mean chemical composition and the aspect ratio, are derived by application of the thermodynamic extremal principle. The driving force for the evolution of the aspect ratio of the precipitate stems from the anisotropic misfit strain of the precipitate and from the orientation dependence of the interface energy. The model is used for the simulation of the precipitation of Ti 3 AlN and Ti 2 AlN in Ti-Al-0.5 at.% N matrix.
Growth kinetics and morphological stability of precipitates in 3-D: a phase field study
arXiv: Materials Science, 2014
We have studied the growth kinetics of isolated precipitates growing from a supersaturated matrix in 3-dimensions (3-D) using phase field models; we assume isotropic interfacial energy consider both constant and variable diffusivity. We report and compare our numerical growth rates with the classic analytical solutions of Zener and Frank (ZF). The numerical results deviate from the analytical ones. These deviations can be understood in terms of the generalised Gibbs-Thomson effect. Specifically, due to the higher capillary contribution in 3-D (curvature is twice for a sphere compared to a circle), the precipitate growth kinetics deviates more from ZF in 3-D as compared to 2-D. In addition, the kinetic parameter associated with the normal velocity of the precipitate-matrix interface also modifies the deviation of the precipitate composition from its equilibrium value and hence its growth kinetics. In phase field models (such as the one used by us) which use a combination of Allen-Cah...
Precipitate growth kinetics can be modelled using a combination of Allen-Cahn and Cahn-Hilliard equations. In this paper, we describe one such phase field model which incorporates cubic anisotropy in interfacial energy as well as in atomic mobility to study the growth ki-netics of a β-phase precipitate growing from a supersaturated α-phase matrix. The model is implemented using a semi-implicit Fourier spectral technique. We show that the morphologies which result from atomic mobility anisotropy can be very different from those which result from interfacial energy anisotropy. We also show that the combined effect of both these anisotropies leads to very interesting microstructural features. Finally, we indicate the extension of these models to incorporate hexagonal anisotropies.
Precipitate growth kinetics under inhomogeneous concentration fields using a phase-field model
Physical Review Materials, 2021
We investigate precipitation dynamics in the presence of a local solute gradient using phase-field simulations. During the homogenization heat treatment of the solidified Inconel 718 alloy, high Nb concentration within the Laves phases or at the core of the secondary arms results in Nb diffusion into the γ matrix. The volume fraction and spatial distribution of precipitation during subsequent annealing can be controlled by tailoring the Nb concentration gradient in the matrix during homogenization. We use a surrogate Ni-Fe-Nb alloy for Inconel 718 to explore the growth dynamics of δ precipitates related to the local Nb concentration levels. The simulations indicate that in the presence of a Nb concentration gradient the growth rate of δ precipitates is higher than in a matrix of uniform average Nb concentration. The higher growth rate is a result of the higher local thermodynamic driving force at the interface between the solute-rich matrix and the δ interface. We propose a phenomenological model to describe the diffusion-controlled growth kinetics of the δ phase under a solute concentration gradient.
Phase field study of precipitate growth: Effect of misfit strain and interface curvature
Acta Materialia, 2009
We have studied diffusion controlled growth of an isolated, misfitting precipitate in a supersaturated matrix using a phase field model. Treating our simulations as computer experiments, we have critically compared our simulation results with those from Zener-Frank and Laraia-Johnson theories for the growth of non-misfitting and misfitting precipitates, respectively. The agreement between simulations and the ZF theory is very good for 1D systems. In 2D systems with interfacial curvature, we still get good agreement between simulations and both ZF and LJ theories, but only for large supersaturations. At small supersaturations, the growth coefficient from our simulations does converge towards that from theory, but a large gap does remain when the simulations end due to overlap of diffusion fields. An interesting finding from the simulations is the less complete realization of the Gibbs-Thomson effect during growth, particularly in more supersaturated alloys. Thus, even at the same precipitate size, the curvature effects are less severe in more supersaturated alloys.
3D Growth Kinetics of Precipitates with Anisotropic Interfacial Free Energy: A Phase-Field Study
Transactions of the Indian Institute of Metals, 2015
In this study, we report on the 3D growth kinetics of an isolated β precipitate growing from a supersaturated α matrix. The α-β interfacial energy is assumed to be cubic anisotropic; specifically, either (100) or (111) interfaces are preferred. The diffusivity is maintained a constant in our model; this constant diffusivity is achieved through a combination of Cahn-Hilliard and Allen-Cahn equations (with appropriate choice of model free energy parameters). Our results indicate that depending on the matrix supersaturation, the growth modes can either be shape preserving or non-shape preserving. In the case of shape preserving growth, we are able to verify, that as indicated by the earlier theoretical studies, the classical Zener-Frank theories can be used to predict the growth rates of precipitates.
Journal of Materials Science, 2007
We present a mathematical model to describe competitive growth of spherical precipitates in reactioncontrolled systems. In this model the flux of solute atoms through the interface depends on the interface migration velocity and on the differences of chemical potential at the interface. The growth-rate obtained is dependent on the precipitate radius, much like in the diffusion-controlled case. Numerical simulations were performed using a modified finite-difference approach where the time-step increase changes during evolution to avoid dissolution of more than one precipitate each step. By using the continuity equation we obtained an analytical function that represents the self-similar shape of the precipitate-size distribution dependent of the growth-parameter m. The effect of m on the coarsening evolution was investigated. Our results show that the precipitate size distribution obtained from the numerical simulations agrees well with the analytical solution. As predicted by the theory, we obtained the growth parameter (m = 4) and the temporal dependence of the mean-radius (t 1/2) different of the diffusion case, m = 6.75 and t 1/3. We also show that the self-similarity of the PSD is independent of the initial PSD.