Nonequilibrium grain size distribution with generalized growth and nucleation rates
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Ralf B. Bergmann Institute for Applied Beam Technology (BIAS), 28359 Bremen, Germany
Andreas Billa) Department of Physics & Astronomy, California State University Long Beach, Long Beach, California 90840 (Received 30 November 2012; accepted 22 April 2013)
We determine the nonequilibrium grain size distribution (GSD) during the crystallization of a solid in d-dimensions under fixed thermodynamic conditions, for the random nucleation and growth model, and in the absence of grain coalescence. Two distinct generalizations of the theory established earlier are considered. A closed analytic expression of the GSD useful for experimental studies is derived for anisotropic growth rates. The main difference from the isotropic growth case is the appearance of a constant prefactor in the distribution. The second generalization considers a Gaussian source term: nuclei are stable when their volume is within a finite range determined by the thermodynamics of the crystallization process. The numerical results show that this generalization does not change the qualitative picture of our previous study. The generalization only affects quantitatively the early stage of crystallization when nucleation is dominant. The remarkable result of these major generalizations is that the nonequilibrium GSD is robust against anisotropic growth of grains and fluctuations of nuclei sizes.
I. INTRODUCTION
Electron microscope images acquired from annealed amorphous materials generally reveal a tessellation of crystalline grains with a distribution of sizes and shapes. The inhomogeneous end product of the crystallization process has its origin in the formation of nuclei and their growth into grains as schematically shown in Fig. 1. These can occur in a variety of ways, one of which is the random formation of stable clusters with respect to thermodynamic fluctuations within the sample and the growth of these nuclei into grains. An important example of such random nucleation and growth (RNG) process is observed in the crystallization of silicon thin films used for solar cells (see, e.g., Refs. 1 and 2 and references therein). Characterizing quantitatively the grain size distribution (GSD) is important since many physical properties directly depend on this distribution. For example, electrical, magnetic, optical, or even superconducting properties are affected by the granularity of a sample. We have developed recently a theory for the nonequilibrium GSD during the crystallization of a solid in d-dimensions3,4 and applied the theory successfully to the solid-phase crystallization of silicon.5 Theoretical studies a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2013.131 J. Mater. Res., Vol. 28, No. 11, Jun 14, 2013
discussing the GSD in various contexts use analytical tools mainly in one-dimension6–12 or numerical techniques10,13–23 to describe the formation of grains during crystallization. The time-dependent GSD derived in Refs. 3 and 4 and extended here is obtained analytically for d-dimensions as a solution of a d
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