The coarsening kinetics of two misfitting particles in an anisotropic crystal

https://doi.org/10.1016/0956-7151(90)90207-WGet rights and content

Abstract

We examine the influence of elastic stress on the Ostwald ripening kinetics of two elastically and diffusionally interacting, misfitting spherical particles in an anisotropic crystal. The coupled equations of elasticity and diffusion are solved analytically in series form in a bispherical coordinate system when the matrix supersaturation is small, local equilibrium obtains at the interface, particle and matrix possess the same cubic elastic constants, and the stress engenered by compositional inhomogeneity is negligible. Expressions are obtained for the matrix composition field, the local normal interfacial velocities of the particles, the isotropic particle growth rates, and the velocity of the particles' centers of mass. Inverse coarsening, or the growth of a smaller particle at the expense of a larger particle, is predicted for particle alignments along the elasticity soft 〈100〉 directions in nickel and for the 〈110〉 and 〈111〉 directions in molybdenum. Coarsening rates are often significantly enhanced for other particle orientations with respect to those of the stress-free case. The elastic stresses also change the functional dependence of the particle growth rate on particle size suggesting that the temporal exponents observed during classical ripening may not obtain in stressed systems. These predictions indicate that elastically-induced preferential coarsening strongly influences microstructural development in two-phase coherent alloys.

Résumé

On examine l'influence de la contrainte élastique sur les cinétiques de mûrissement d'Ostwald de deux particules sphériques, en interaction élastique et diffusionnelle, en désaccord dans un cristal anisotrope. Les équations couplées d'élasticité et de diffusion sont résolues analytiquement sous forme de séries dans un système de coordonnées bisphériques lorsque la sursaturation de la matrice est faible, que l'équilibre local est réalisé à l'interface, que la particule et la matrice possèdent les mêmes constantes élastiques et que la contrainte engendrée par l'hétérogénéité de composition est négligeable. Des expressions sont obtenues pour le champ de composition de la matrice, les vitesses interfaciales normales locales des particules, les vitesses de croissance isotropes des particules et la vitesse des centres de masse des particules. Le grossissement inverse, ou la croissance d'une particule plus petite aux dépens d'une particule plus grande, sont prévus pour des alignements de particules le long des directions <100 > élastiquement molles dans le nickel et le long des directions 〈110〉 et 〈111〉 dans le molybdéne. Les vitesses de grossissement sont souvent fortement renforcées pour des orientations des particules différentes de celles qu'on observe en l'absence de contraintes. Les contraintes élastiques changent aussi la façon dont varie la vitesse de croissance des particules en fonction de la taille des particules, ce qui suggère que les exposants du temps observés pendant le mûrissement classique peuvent ne pas exister dans les systèmes sous contrainte. Ces prévisions indiquent que le grossissement préférntiel induit élastiquement a une forte influence sur le développement microstructural dans les alliages biphasés cohérents.

Zusammenfassung

Wir untersuchen den Einfluβ elastischer Spannungen auf die Kinetik der Ostwaldvergröberung zweier in einem anisotropen Kristall elastisch und diffusionsmäβig wechselwirkender fehlpassender kugelförmiger Teilchen. Die gekoppelten Elastizitäts- und Diffusionsgleichungen werden analytisch in Form von Reinhenentwicklungen in einem bisphärischen Koordinatensystem für den Fall gelöst, daβ die übersättigung in der Matrix klein ist, das lokale Gleichgewicht an der Grenzfläche gewahrt bleibt, Teilchen und Matrix dieselben kubischen elastischen Konstanten haben und die durch Inhomogenitäten der Zusammensetzung erzeugten Spannungen vernachlässigbar sind. Ausdrücke werden erhalten für das Zusammensetzungsfeld der Matrix, die lokalen Normalgeschwindigkeiten der Teilchengrenzflachen, die isotropen Wachstumstraten der Teilchen und die Geschwindigkeiten der Schwerpunkte der Teilchen. Inverse Vergröberung, oder das Wachstum kleinerer Teilchen auf Kosten der gröβeren, wird für Teilchenanordnungen entlang der elastisch weichen 〈100〉-Richtungen in Nickel und entlang den 〈110〉- und 〈111〉-Richtungen in Molybdän vorausgesagt. Vergröberungsraten sind bei anderen Teilchenorientierungen, verglichen mit denen des spannungsfreien Falles, häufig erhöht. Die elastischen Spannungen ändern auch die funktionelle Abhängigkeit der Wachstumsrate der Teilchen von der teilchengröβe; das deutet darauf hin, daβ die bei der klassischren Vergröberung beobachteten Zeitexponenten in verspannten Systemen nicht gelten. Diese Voraussagen zeigen, daβ die elastisch induzierte bevorzugte Vergröberung die Entwicklung der Mikrostruktur in zweiphasigen kohärenten Legierungen stark beeinfluβt.

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