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Nonequilibrium grain size distribution with generalized growth and nucleation rates

Published online by Cambridge University Press:  04 June 2013

Kimberly S. Lokovic ,
Ralf B. Bergmann  and
Andreas Bill
Kimberly S. Lokovic
Affiliation:
Department of Physics & Astronomy, California State University Long Beach, Long Beach, California 90840
Ralf B. Bergmann
Affiliation:
Institute for Applied Beam Technology (BIAS), 28359 Bremen, Germany
Andreas Bill*
Affiliation:
Department of Physics & Astronomy, California State University Long Beach, Long Beach, California 90840
*
a)Address all correspondence to this author. e-mail: abill@csulb.edu
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Abstract

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.

Type
Articles
Information
Journal of Materials Research , Volume 28 , Issue 11 , 14 June 2013 , pp. 1407 - 1412
Copyright
Copyright © Materials Research Society 2013 

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References

REFERENCES

Wagner, S.: Amorphous silicon: Vehicle and test bed for large-area electronics. Phys. Status Solidi A 207, 501–509 (2010).CrossRefGoogle Scholar
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).Google Scholar
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).CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
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, A05–A03 (2009).CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Sekimoto, K.: Kinetics of magnetization switching in a 1-d system - size distribution of unswitched domains. Physica A 125, 261 (1984).CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
Axe, J.D. and Yamada, Y.: Scaling relations for grain autocorrelation functions during nucleation and growth. Phys. Rev. B 34, 1599 (1986).CrossRefGoogle ScholarPubMed
Ben-Naim, E. and Krapivsky, P.L.: Nucleation and growth in one dimension. Phys. Rev. E 54, 3562 (1996).CrossRefGoogle ScholarPubMed
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).Google Scholar
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).CrossRefGoogle Scholar
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).Google Scholar
Rios, P.R.: Comparison between a computer simulated and an analytical grain size distribution. Scr. Mater. 40, 665 (1999).CrossRefGoogle Scholar
Wang, C. and Liu, G.: Grain size distribution obtained from monte carlo simulation and the analytical mean field model. ISIJ Int. 43, 774 (2003).CrossRefGoogle Scholar
Crespo, D. and Pradell, T.: Evaluation of time-dependent grain-size populations for nucleation and growth kinetics. Phys. Rev. B 54, 3101 (1996).CrossRefGoogle ScholarPubMed
Pineda, E. and Crespo, D.: Microstructure development in kolmogorov, johnson-mehl, and avrami nucleation and growth kinetics. Phys. Rev. B 60, 3104 (1999).CrossRefGoogle Scholar
Bruna, P., Pineda, E., and Crespo, D.: Cell size distribution in random tessellations of space. Phys. Rev. E 70, 066119 (2004).Google Scholar
Crespo, D., Bruna, P., and Gonzalez-Cinca, R.: On the validity of avrami formalism in primary crystallization. J. Appl. Phys. 100, 054907 (2006).Google Scholar
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).Google Scholar
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).CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
Kakinuma, H.: Comprehensive interpretation of the preferred orientation of vapor-phase grown polycrystalline silicon films. J. Vac. Sci. Technol., A 13(5), 2310–2317 (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).CrossRefGoogle Scholar
Kolmogorov, A.N.: A statistical theory for the recrystallization of metals. Izv. Akad. Nauk SSSR, Ser. Mat. 1, 355 (1937).Google Scholar
Johnson, W.N. and Mehl, R.F.: Reaction kinetics in processes of nucleation and growth. Trans AIME 135, 416 (1939).Google Scholar
Avrami, M.: Kinetics of phase change. I: General theory. J. Chem. Phys. 7, 1103 (1939).CrossRefGoogle Scholar
Avrami, M.: Kinetics of phase change. II: Transformation-time relations for random distribution of nuclei. J. Chem. Phys. 8, 1940 (1940).CrossRefGoogle Scholar
Avrami, M.: Kinetics of phase change. III: Granulation, phase change and microstructure. J. Chem. Phys. 9, 177 (1941).CrossRefGoogle Scholar