ON SIZE AND SHAPE COARSENING OF NEEDLE-SHAPED PRECIPITATES

A coarsening theory is proposed which incorporates both size and shape changes for particles having a constant interfacial energy. The theory describes time evolution of the mean volume of needle-shaped precipitates. Deviations from the cubic coarsening law of the classical LSW theory are predicted with initial coarsening rates being significantly larger compared to those of spherical particles. Because of simultaneous shape coarsening the initially needle-like precipitates approach gradually the spherical shape and the associated lower coarsening rates. The theory can be simply extended to disc-shaped precipitates. Introduction In a two-phase precipitate-matrix system after a first-order phase transformation, a large amount of interfacial area can be present which increases the total free energy of the system. In order to lower the free energy, diffuse fluxes between the precipitates occur which result in a dissolution of small precipitates and the growth of large ones. As a result, the mean particle size increases while the number of particles decreases. This process, originally discovered at the beginning of this century by W. Ostwald, is called coarsening or Ostwald ripening. According to Gibbs-Thomson equation, the driving force for the coarsening process is related to the curvature of the interface. The higher the curvature of the precipitate-matrix interface, the higher the driving force for atoms to flow from regions of high curvature to regions of low curvature. Thus, the coarsening kinetics is generally controlled by volume diffusion through the matrix. The first attempt to describe the Ostwald ripening quantitatively was proposed by Greenwood in 1956 [1]. A generally accepted coarsening theory was proposed in 1961 by Lifshitz and Slyozov [2] and Wagner [3], now simply known as LSW theory. Considering the limiting case of zero volume fraction of spherical precipitates with the mean radius and a particle size distribution (PSD) function obeying the equation of continuity, the LSW theory predicts a cubic coarsening law . (1) where is the initial mean particle radius before the coarsening process, K is the coarsening rate constant and t is time. The cubic coarsening law (1) is derived from the Gibbs-Thomson equation and the Fick's law by assuming a small degree of supersaturation and constant coarsening rate K [1]. The assumption of constant coarsening rate requires constant temperature and interfacial energy per unit surface during coarsening. Based on the above-mentioned assumptions, the LSW theory predicts the coarsening kinetics, i.e. the mean particle size and the PSD function. In spite that a cubic coarsening law is frequently observed in experimental studies, the details of the predictions by the LSW theory are usually in disagreement with experimental results. Consequently, numerous attempts have been made to propose improved theories by considering finite volume fractions of the second phase [4-12] or elastic stress interactions either between precipitates or on the precipitate-matrix interface [13-18]. Although these modifications usually result in the change of the PSD shape, the coarsening kinetics of the precipitates obeys in most cases the cubic law; deviations from the cubic law are predicted when elastic stress interactions occur between particles [15]. Surprisingly small attention was paid to the effect of non-spherical shape of precipitates. Due to the anisotropy of surface energy in many systems, the precipitates grow in a shape with lower symmetry than spherical [19-21]. When the coarsening starts at the end of the growth process, precipitates have still non-spherical morphology and thus the basic assumption of the LSW theory is not valid. Since the curvature along the surface of a non-spherical particle vary, volume diffusion to and from the surface will vary too. For growing particle, according to the Gibbs-Thomson equation, the volume diffusion of atoms will be lower to the regions of high curvature and higher to the regions of low curvature. In addition, new diffuse fluxes occur along the particle surface and this interface diffusion will exist together with the volume diffusion. The volume diffusion, therefore, changes both the shape and the size of a non-spherical particle while interface diffusion changes only the particle shape. A few theories of shape coarsening of an isolated plate-shaped precipitate have been developed [22-25]. Shiflet et al. [22] considered only volume diffusion and, in addition, a special case of precipitate growth by the ledge mechanism, which require coherent or semi-coherent precipitates. Merle and Doherty [23] developed an alternative model for interface controlled diffusion, in which the volume diffusion is neglected, and thus, the model describes only shape coarsening or, strictly speaking, only the shape changes. Zwillinger [24] analysed the coarsening of a collection of nonspherical particles of general shape. He introduced a shape function, which described the shape of each particle but in the same time, the shape function was assumed to be time independent, i.e. each particle was constrained to remain in its original shape during coarsening. By this restriction, the model describes only size coarsening and, as one could expect, the coarsening kinetics obeys the cubic law. Recently, Marsh and Glicksman [25] proposed a more general approach to size coarsening in nonspherical morphologies by describing time evolution of small convex interfacial patches. The patches were, again, constrained to have a fixed shape in time, and the model did not take into account diffuse fluxes along surface of the patches. However, due to the general approach, the model is applicable to arbitrarily curved surfaces. Thus, its results are used as starting point in the present work. As experiments show, size and shape coarsening processes occur simultaneously and the diffuse fluxes associated with shape coarsening are often comparable in magnitude to the interparticle fluxes [26]. Therefore, an attempt is presented in the following to develop a simple coarsening theory for incoherent needle-shaped precipitates by incorporating both size and shape changes. Size coarsening The needle-shaped precipitate can be mathematically modelled by a prolate symmetrical ellipsoid with two independent variables describing its shape, i.e. two principal axes or one of them and the aspect ratio. If a is the major and b the minor axis, the eccentricity ε of the prolate symmetrical ellipsoid is defined as