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We study the time dependence of the grain size distribution N(r,t) during crystallization of a d-dimensional solid. A partial differential equation including a source term for nuclei and a growth law for grains is solved analytically for any dimension d. We discuss solutions obtained for processes described by the Kolmogorov-Avrami-Mehl-Johnson model for random nucleation and growth (RNG). Nucleation and growth are set on the same footing, which leads to a time-dependent decay of both effective rates. We analyze in detail how model parameters, the dimensionality of the crystallization process, and time influence the shape of the distribution. The calculations show that the dynamics of the effective nucleation and effective growth rates play an essential role in determining the final form of the distribution obtained at full crystallization. We demonstrate that for one class of nucleation and growth rates the distribution evolves in time into the logarithmic-normal (lognormal) form discussed earlier by Bergmann and Bill [J. Cryst. Growth 310, 3135 (2008)]. We also obtain an analytical expression for the finite maximal grain size at all times. The theory allows for the description of a variety of RNG crystallization processes in thin films and bulk materials. Expressions useful for experimental data analysis are presented for the grain size distribution and the moments in terms of fundamental and measurable parameters of the model.
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We determine the non-equilibrium grain size distribution during the crystallization of a solid in $d$ dimensions at fixed thermodynamic conditions, for the random nucleation and growth model, and in absence of grain coalescence. Two distinct generalizations of the theory established earlier are considered. A closed analytic expression of the grain size distribution 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 non-equilibrium grain size distribution is robust against anisotropic growth of grains and fluctuations of nuclei sizes.
We study the avalanche statistics observed in a minimal random growth model. The growth is governed by a reproduction rate obeying a probability distribution with finite mean a and variance va. These two control parameters determine if the avalanche size tends to a stationary distribution, (Finite Scale statistics with finite mean and variance or Power-Law tailed statistics with exponent in (1, 3]), or instead to a non-stationary regime with Log-Normal statistics. Numerical results and their statistical analysis are presented for a uniformly distributed growth rate, which are corroborated and generalized by analytical results. The latter show that the numerically observed avalanche regimes exist for a wide family of growth rate distributions and provide a precise definition of the boundaries between the three regimes.
The space subdivision in cells resulting from a process of random nucleation and growth is a subject of interest in many scientific fields. In this paper, we deduce the expected value and variance of these distributions while assuming that the space subdivision process is in accordance with the premises of the Kolmogorov-Johnson-Mehl-Avrami model. We have not imposed restrictions on the time dependency of nucleation and growth rates. We have also developed an approximate analytical cell size probability density function. Finally, we have applied our approach to the distributions resulting from solid phase crystallization under isochronal heating conditions.
The logarithmic-normal (lognormal) distribution is one of the most frequently observed distributions in nature and describes a large number of physical, biological and even sociological phenomena. The origin of this distribution is therefore of broad interest but a general derivation from basic principles is still lacking. Using random nucleation and growth to describe crystallization processes we derive the time development of grain size distributions. Our derivation provides, for the first time, an analytical expression of the size distribution in the form of a lognormal type distribution. We apply our results to the grain size distribution of solid phase crystallized Si-films.
A simple numerical model which calculates the kinetics of crystallization involving randomly distributed nucleation and isotropic growth is presented. The model can be applied to different thermal histories and no restrictions are imposed on the time and the temperature dependencies of the nucleation and growth rates. We also develop an algorithm which evaluates the corresponding emerging grain size distribution. The algorithm is easy to implement and particularly flexible making it possible to simulate several experimental conditions. Its simplicity and minimal computer requirements allow high accuracy for two- and three-dimensional growth simulations. The algorithm is applied to explore the grain morphology development during isothermal treatments for several nucleation regimes. In particular, thermal nucleation, pre-existing nuclei and the combination of both nucleation mechanisms are analyzed. For the first two cases, the universal grain size distribution is obtained. The high accuracy of the model is stated from its comparison to analytical predictions. Finally, the validity of the Kolmogorov-Johnson-Mehl-Avrami model is verified for all the cases studied.
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