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Nonequilibrium grain size distribution with generalized growth and nucleation rates

  • Kimberly S. Lokovic (a1), Ralf B. Bergmann (a2) and Andreas Bill (a3)

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.

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1.Wagner, S.: Amorphous silicon: Vehicle and test bed for large-area electronics. Phys. Status Solidi A 207, 501509 (2010).
2.Aberle, A.G. and Widenborg, P.I., “Crystalline Silicon Thin-film Solar Cells via High-temperature and Intermediate-temperature Approaches,” in Handbook of Photovoltaic Science and Engineering, Vol. 11, 2nd ed, edited by Luque, A. and Hedegus, S. (John Wiley and Sons, Ltd., Chichester, UK, 2011).
3.Bergmann, R.B. and Bill, A.: On the origin of logarithmic-normal distributions: An analytical derivation, and its application to nucleation and growth processes. J. Cryst. Growth 310, 3135 (2008).
4.Teran, A.V., Bergmann, R.B., and Bill, A.: Time-evolution of grain size distributions in random nucleation and growth crystallization processes. Phys. Rev. B 81, 075319 (2010).
5.Bill, A., Teran, A.V., and Bergmann, R.B.: Modeling the grain size distribution during solid phase crystallization of silicon. Mater. Res. Soc. Symp. Proc. 1153, A05A03 (2009).
6.Gelbard, F.M. and Seinfeld, J.H.: Exact solution of the general dynamic equation for aerosol growth by condensation. J. Colloid Interface Sci. 68, 173 (1979).
7.Shi, G. and Seinfeld, J.H.: Transient kinetics of nucleation and crystallization: Part I. nucle-ation. J. Mater. Res. 6, 2091 (1991). “Transient kinetics of nucleation and crystallization: Part I. Nucleation,” 6, 2097(1991); Frank G. Shi and John H. Seinfeld, “Nucleation in the pre-coalescence stages: universal kinetic laws, ” Materials Chemistry and Physics, 37, 1–15 (1994), ISSN 0254–0584.
8.Sekimoto, K.: Kinetics of magnetization switching in a 1-d system - size distribution of unswitched domains. Physica A 125, 261 (1984).
9.Sekimoto, K.: Evolution of the domain structure during the nucleation and growth process with non-conserved order parameter. Int. J. Mod. Phys. B 5, 1843 (1991).
10.Axe, J.D. and Yamada, Y.: Scaling relations for grain autocorrelation functions during nucleation and growth. Phys. Rev. B 34, 1599 (1986).
11.Ben-Naim, E. and Krapivsky, P.L.: Nucleation and growth in one dimension. Phys. Rev. E 54, 3562 (1996).
12.Zhang, H., Jun, S., and Bechhoefer, J.: Nucleation and growth in one dimension - 1-the generalized kolmogorov-johson-mehl-avrami model. Phys. Rev. E 71, 011908 (2005).
13.Brown, W.K. and Wohletz, K.H.: Derivation of the weibull distribution based on physical principles and its connection to the rosin-rammler and lognormal distributions. J. Appl. Phys. 78, 2758 (1995).
14.Thompson, C.V., Fayad, W., and Frost, H.J.: Steady-state grain-size distributions resulting from grain growth in two dimensions. Scr. Mater. 40, 1199 (1999).
15.Rios, P.R.: Comparison between a computer simulated and an analytical grain size distribution. Scr. Mater. 40, 665 (1999).
16.Wang, C. and Liu, G.: Grain size distribution obtained from monte carlo simulation and the analytical mean field model. ISIJ Int. 43, 774 (2003).
17.Crespo, D. and Pradell, T.: Evaluation of time-dependent grain-size populations for nucleation and growth kinetics. Phys. Rev. B 54, 3101 (1996).
18.Pineda, E. and Crespo, D.: Microstructure development in kolmogorov, johnson-mehl, and avrami nucleation and growth kinetics. Phys. Rev. B 60, 3104 (1999).
19.Bruna, P., Pineda, E., and Crespo, D.: Cell size distribution in random tessellations of space. Phys. Rev. E 70, 066119 (2004).
20.Crespo, D., Bruna, P., and Gonzalez-Cinca, R.: On the validity of avrami formalism in primary crystallization. J. Appl. Phys. 100, 054907 (2006).
21.Niklasson, G.A., Söderlund, J., Kiss, L.B., and Granqvist, C.G.: Lognormal size distributions in particle growth processes without coagulation. Phys. Rev. Lett. 80, 2386 (1998).
22.Farjas, J. and Roura, P.: Numerical model of solid phase transformations governed by nucleation and growth: Microstructure development during isothermal crystallization. Phys. Rev. B 75, 184112 (2007).
23.Farjas, J. and Roura, P.: Cell size distribution in a random tessellation of space governed by the Kolmogorov-Johnson-Mehl-Avrami model: Grain size distribution in crystallization. Phys. Rev. B 78, 144101 (2008).
24.Kakinuma, H.: Comprehensive interpretation of the preferred orientation of vapor-phase grown polycrystalline silicon films. J. Vac. Sci. Technol., A 13(5), 23102317 (1995). ISSN 07342101; P.Hartman, ed., Crystal Growth: An Introduction (North Holland Publishing Company, 1973) ch. 14; K.P. Gentry, T. Gredig, and I.K. Schuller, “Asymmeric Grain Distribution in Phthalocyanine Thin Flims,” Phys. Rev. B, 80, 174118(2009).
25.Kolmogorov, A.N.: A statistical theory for the recrystallization of metals. Izv. Akad. Nauk SSSR, Ser. Mat. 1, 355 (1937).
26.Johnson, W.N. and Mehl, R.F.: Reaction kinetics in processes of nucleation and growth. Trans AIME 135, 416 (1939).
27.Avrami, M.: Kinetics of phase change. I: General theory. J. Chem. Phys. 7, 1103 (1939).
28.Avrami, M.: Kinetics of phase change. II: Transformation-time relations for random distribution of nuclei. J. Chem. Phys. 8, 1940 (1940).
29.Avrami, M.: Kinetics of phase change. III: Granulation, phase change and microstructure. J. Chem. Phys. 9, 177 (1941).
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Journal of Materials Research
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