The Grain Size Distribution in Crystallization Processes With Anisotropic Growth Rate
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The Grain Size Distribution in Crystallization Processes with Anisotropic Growth Rate Kimberly S. Lokovic1, Ralf B. Bergmann2 and Andreas Bill1 1 California State University Long Beach, Department of Physics & Astronomy, 1250 Bellflower Blvd., Long Beach, CA 90840, U.S.A. 2 Bremen Institute for Applied Beam Technology (BIAS), Klagenfurter Str. 2, 28359 Bremen, Germany.
ABSTRACT The grain size distribution allows characterizing quantitatively the microstructure at different stages of crystallization of an amorphous solid. We propose a generalization of the theory we established for spherical grains to the case of grains with ellipsoidal shape. We discuss different anisotropic growth mechanisms of the grains in thin films. An analytical expression of the grain size distribution is obtained for the case where grains grow through a change of volume while keeping their shape invariant. The resulting normalized grain size distribution is shown to be affected by anisotropy through the time-decay of the effective growth rate.
INTRODUCTION Recently, we developed a theory of the grain size distribution (GSD) N(g,t) for the crystallization of an amorphous solid [1-3]. N is the normalized count of the number of grains of a certain size g found at time t in a sample. The quantity g is, for example, the diameter of the grain, the number of atoms contained in the grain, its volume, etc. As discussed below, g can also be replaced by a vector quantity. The interest in determining the GSD lies in the fact that physical properties of a solid such as their electrical conductivity, magnetization or optical absorption may be substantially affected by the degree of crystallization and the microstructure of a material. Thus, it is important to determine the grain size distribution during the crystallization of a sample. In previous work [1,3], we obtained a closed analytical form of the grain size distribution for spherical grains. The expression was derived for any dimension d of the crystallization process and can easily be used for the description of experimental data on bulk or thin film systems, as shown in the case of solid-phase crystallization of silicon in Ref. [2]. The theory is developed for constant microscopic nucleation and growth rates I 0 and v 0 and incorporates the well-known Kolmogorov-Avrami-Mehl-Johnson (KAMJ) expression for the fraction of crystallized material during a random nucleation and growth (RNG) crystallization process [4-6]. This leads to a time-dependent decay of the effective nucleation rate I( t ) [see r below]. In addition, we introduced a new effective time-dependent growth rate v (t) that accounts for the fact that a specific grain stops growing once a sufficient number of other grains impinge on it [3]. Coalescence is not considered in the theory. The aim of the present work is to generalize the theory of Refs. [1-3] beyond the spherical grains assumption. We thus allow for the existence of grains with different shapes and sizes in the crystallization process. It is worth pointing out t
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