On evaluation of the equations for the growth kinetics of ellipsoidal precipitates (original) (raw)
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Using size distributions for determining growth mechanisms of grain boundary precipitates
In the past various single parameters such as the mean, mode or maximum of the precipitate size distribution have been used in experiments to determine growth mechanisms. In the present study the development with aging time of the size and shape distributions of bcc precipitates at grain boundaries in an fcc material (Co-20Fe at 1003oK) have been compared with possible theoretical models to determine the rate controlling process. The growth of these precipitates is initially well described by the grain boundary dependent collector plate mechanism of Brailsford and Aaron. As the precipitates grow low energy facets are formed which can move only by the propagation of ledges and growth becomes interface controlled. The precipitates’ diffusion fields soon overlap and coarsening occurs with interface control. The results demonstrate that this would not have been revealed using simpler measures of precipitate size.
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.
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...
On the coarsening of grain boundary precipitates
Acta Metallurgica, 1972
The diffusion-controlled coarsening behavior of precipitates situated on both high-angle end low-angle grain boundaries haa been investigated theoretically. The effect of volume fraotion, (li, is included in the calculations. At high-angle grain boundaries the average particle size increases 88 1"' after long aging times, irrespective of the value of 4. The coarsening rate constant inoreases, and the theoretical distribution of particle sizes broadens, &s 4 increases. These results are considered in the light of previous theoretical treatments of this problem. It is assumed that precipitate growth on low-angle grain boundaries is controlled by the diffusion of solute along the dislocation network in the boundary. In this case the average particle size increases as ill* provided that the particle radii are much larger than the dislo~tion spacing. However, when the dislooation spacing is greater than the radii of all the particles, the average particle size increases as i'f5. Theeffect of+ on the coarsening rate constants and particle size distributions is estimated semiquantitatively for the extreme cases oft 114 and 11'5 kinetics, and discussed in a qualitative manner for intermediate ratios of particle size and dislocation spacing. GROSSISSEMENT DES PRECIPITES AUX JOINTS DE GRAINS L'auteur a effect& une Etude thborique du grosaissement, contr616 par diffusion, des pruicipitis situ& sur les joints de grains it forte d&orientation et sur les joints de grains B. faible d&orientation, en tenant oompte, dans lea calmis, de la fraction de volume 4, Aux joints de grains B faible d&orientation, la taille moyenne des particules augmenta comme tlia apr2ss des vieillissements de longue dun%, quelle que soit la valeur de QI. La con&ante de vitesse de grossissement augmente, et la dist~bution th6orique de la taille des particules s'8argit quand # augmente. Ces r&&tats sent exami&s B la lumi&re des theories antirieures concernant ce probl&me. L'autaur suppose que la croissance du pr6cipit6 aux joints de grains k faible d&orientation est con-t&l&e par la diffusion du solut6 le long du reseau de dislocations dans le joint. Dans oe cas, la taille moyenne des particules augment0 comme t'l' & condition que le rayon des particules soit beaucoup plus grand que l'espacement des dislocations. Cependant, quand oelui-ci est sup&ieur aux rayons de toutes les part&lea, la taille moyenne des particules augmente comme t116. L'influence de + sur les con&antes de vitesse de grossissement et sur les distributions des tailles des particules est Qvalu& quantitativement pour les cas extrt3mes des cinbtiques en 1'14 et lllS, et discut6e qualitativement, pour les rapports intermbdiaires entre le, taille des particules et l'espaoement des dislocations. ZUR ~RGR~BER~NG VON KORNGRENZEN-AUSS~HEIDUNG~N Die d~usionskont~~ierte Vergr&erung von Ausscheidungen sowohl an GroBwinkel-als such an Kleinwinkelkorngrenzen wurde theoretisch untersucht. Der Einflul.3 des Volumenanteils + wird bei den Rechnungen beriicksichtigt. An GroDwinkelkorngrenzen nimmt der mittlere Teilchendurchmesser nach grol3en Zeiten unabh&ngig vom Wert Cp mit t114 zu. Mit zunehmendem 4 nimmt, die Geschwindigkeitskonstante der Vergriiberung zu und die theoretische Verteilung der Teilchengr+Ren verbreitert sich. Diese Ergebnisse werden im Zusammenhang mit frtiheren thaoretischen Behandlungen des Problems diskutiert. Es wird angenommen, daB das Wachstum der Ausscheidungen an Kleinwinkelkorngrenzen durch die Diffusion des Fremdstoffs entlana des Versetzungsnetzwerkes in der Hornzrenze kontrolliert wird. In diesem Fall nimmt die mittlere T&chengri%e rnittli* zu unter der Vorau~et~ung, dal3 die Teilchen~dien vie1 griiRer als die Ver~tzung~bst~de sind. Sind jedooh die Abst&nde der Versetzungen gri%er als alle auftretenden Teilchenradien, so nimmt die mittlere TeilchengrtiDe mit tl'6 zu. Fiir die beiden Extrem-f&lie der 0/P. und l'tS-Kinetik wird der Einflu5 von $ auf die Geschwindigkeitskonstante der Vergrijberung und auf die TeilchengriiOe halbquantitativ abgeschiitzt; ftir dazwischen liegende Relationen zwisohen TeilchengraDe und Versetzungsabstand wird der EinfluCi von 4 qualitativ diskutiert.
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.
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.
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.
Phase field models as computer experiments: Growth kinetics of anisotropic precipitates
Phase field models are widely used for the study of microstructures and their evolution. They can also be used as computer experiments. As computer experiments, they serve two important roles: (a) theoretical results which are hard to verify/validate experimentally can be verified/validated on the computer using phase field models; and, (b) when severe assumptions are made in a theory, they can be relaxed in the phase field model, and hence, results with wider reach can be obtained. In this paper, we discuss some such computer experiments in general, and the growth kinetics of precipitates in systems with tetragonal and cubic interfacial anisotropies in particular.
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.
A two-dimensional computer simulation of capillarity-driven grain growth: Preliminary results
Scripta Metallurgica, 1988
Inlroduction Microstructural evolution in polycrystalline thin films usually begins with nucleation and growth of islands, followed by coalesence and grain growth. In previous work (1) we considered the effect of various nucleation conditions on the geometry and topology of two-dimensional grain structures. In this paper we report preliminary results for modelling of grain boundary migration and grain growth in su'uctures previously generated. Here, we specifically consider evolution of structures resulting from constant growth of simultaneously nucleated crystals. This is contrasted with growth in structures generated by sequential packing of hard disks. These significantly different starting structures lead to very different initial grain growth behavior. After a transient period, however, grain growth in both cases appears to be similar. There has recently been considerable interest in the modelling of grain growth in two dimensions using various computer simulation techniques. Weaire and Ken'node (2,3) have reported a model for the evolution of a twodimensional soap froth. Although their model does not directly represent grain boundary migration, there are several interesting parallels. Another widely reported model was created by Anderson et al. (4,5). In their model a plane was divided into an array of incremental areas, each belonging to one grain or another. The incremental areas are allowed to switch from one grain to another, biased according to the grain membership of neighboring areas. This results in grain growth which is similar, but not identical, to that expected from capillarity driven growth. Several other models have also been reported. (6-9) Models for two-dimensional grain growth may be characterized using an average effective diameter (square root of grain area), D, which usually depends on time, t, as: D m-Do m = B t (1)