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An interference problem with application to crystal growth

Part of: Systems theory and control: General Applications to specific types of physical systems

Published online by Cambridge University Press:  01 July 2016

F. B. Knight  and
J. L. Steichen
F. B. Knight*
Affiliation:
University of Illinois at Urbana-Champaign
J. L. Steichen*
Affiliation:
MathX
*
∗ Postal address: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61810, USA.
∗∗ Postal address: MathX, 25 Lancaster Lane, Monsey, NY 10952, USA. Email address: steichen.mathx@astras.us
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Abstract

We model diffusion-controlled crystal growth as an interference problem. The crystal layers grow by nucleation (initiation of crystallization centers) followed by attachment of molecules to the nucleus. A forming crystal layer completes by either spreading across the length of the crystal or by colliding with another spreading crystal layer. This model differs from the classical Johnson-Mehl-Kolmogorov model in that nucleation happens only on boundaries of a ‘seed’ crystal as opposed to nucleation from random points in a given region. Our results also differ from the limiting results found for this classical model. We use the invariant measure of an embedded Markov process to find the growth rate of the crystal in terms of the nucleation rates. Ergodic theorems are then used to derive explicit formulae for some stationary probabilities.

Keywords

Markov processcrystal growthmodelingsimulation

MSC classification

Secondary: 82D25: Crystals 93A30: Mathematical modeling (models of systems, model-matching, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 725 - 746
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Arnol´d, V. I. (1973). Ordinary Differential Equations. MIT Press, Cambridge, MA.Google Scholar
[2] Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
[3] Chiu, S. N. (1995). Limit theorems for the time of completion of Johnson–Mehl tessellations. Adv. Appl. Prob. 27, 889–910.Google Scholar
[4] Cowan, R., Chiu, S. N. and Holst, L. (1995). A limit theorem for the replication time of a DNA molecule. J. Appl. Prob. 32, 296–303.Google Scholar
[5] Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
[6] Helms, L. L. (1974). Ergodic properties of several interacting Poisson particles. Adv. Math. 12, 32–57.Google Scholar
[7] Holst, L., Quine, M. P. and Robinson, J. (1996). A general stochastic model for nucleation and linear growth. Ann. Appl. Prob. 6, 903–921.Google Scholar
[8] Johnson, W. A. and Mehl, R. F. (1939). Reaction kinetics in processes of nucleation and growth. Trans. Amer. Inst. Min. Metal. Petro. Eng. 135, 416–458.Google Scholar
[9] Knight, C. A. and Rider, K. (2002). Free-growth forms of tetrahydrofuran clathrate hydrate crystals from the melt: plates and needles from a fast-growing vicinal cubic crystal. Philosoph. Mag. A 82, 1609–1632.Google Scholar
[10] Kolmogorov, A. N. (1937). On the statistical theory of metal crystallization. Izv. Akad. Nauk SSSR Ser. Mat. 3, 355–360.Google Scholar
[11] Laudise, R. A. (1970). The Growth of Single Crystals. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[12] Meijering, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.Google Scholar
[13] Orey, S. (1971). Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand, London.Google Scholar
[14] Roy, B. N. (1992). Crystal Growth from Melts: Applications to Growth of Groups 1 and 2 Crystals. John Wiley, Chichester.Google Scholar
[15] Tammann, G. (1925). The States of Aggregation. Van Nostrand, New York.Google Scholar
[16] Toschev, S. (1973). Homogeneous nucleation. In Crystal Growth: An Introduction, ed. Hartman, P., North-Holland, Amsterdam, pp. 1–49.Google Scholar
[17] Vanderbei, R. J. and Shepp, L. A. (1988). A probabilistic model for the time to unravel a strand of DNA. Commun. Statist. Stoch. Models 4, 299–314.Google Scholar