Abstract
Using a Markov rate-process model, exact expressions are found for the steady growth rate of an edge of a two-dimensional crystal in terms of the numberM of particles along the edge, the height difference (or number of permanent steps)K along the edge, the nucleation rate α, and the speed α+ν of movement of steps. The familiar growth regimes can be identified with asymptotic regimes for the parametersK, (v/α)1/2, andM. From a mathematical viewpoint, there are seven basic regimes, of which the known physical regimes are special cases.
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References
C. H. Bennett, M. Bütticker, R. Landauer, and H. Thomas,J. Stat. Phys. 24: 419 (1981).
Y. S. Chow and H. Teicher,Probability Theory (Springer, New York, 1978).
E. T. Copson,An Introduction to the Theory of Functions of a Complex Variable (Clarendon, Oxford, 1962).
E. T. Copson,Asymptotic Expansions (Cambridge University Press, Cambridge, 1965).
F. C. Frank,Disc. Faraday Soc. 5:48 (1949).
F. C. Frank,J. Crystal Growth 22:233 (1974).
D. J. Gates and M. Westcott,Proc. R. Soc. Lond. A 416:443 (1988).
D. J. Gates and M. Westcott,Proc. R. Soc. Lond. A 416:463 (1988).
D. J. Gates and M. Westcott,J. Stat. Phys. 59:73 (1990).
D. J. Gates and M. Westcott, Markov models of steady crystal growth.Ann. Appl. Prob. 3:339 (1993).
D. J. Gates,J. Stat. Phys. 52:245 (1988).
N. Goldenfeld,J. Phys. A 17:2807 (1984).
I. S. Grashteyn and I. M. Ryzhik,Tables of Integrals, Series and Products (Academic Press, New York, 1980).
J. D. Hoffman, G. T. Davis, and J. I. Lauritzen, Jr., InTreatise on Solid-State Chemistry. Vol. 3.Crystalline and Non-Crystalline Solids, N. B. Hanay, ed. (Plenum Press, New York, 1976), p. 335.
J. D. Hoffman,Polymer 24:3 (1983).
F. Khoury and E. Passaglia, InTreatise on Solid-State Chemistry. Vol. 3.Crystalline and Non-Crystalline Solids, N. B. Hanay, ed. (Plenum Press, New York, 1976), p. 497.
J. I. Lauritzen, Jr.,J. Appl. Phys. 44:4353 (1973).
D. M. Sadler,Polymer 28:14540 (1987).
J. D. Watson,The Double Helix (Weidenfield and Nicolson, London, 1968).
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Gates, D.J., Westcott, M. Seven basic regimes of steady crystal growth in two dimensions. J Stat Phys 77, 199–215 (1994). https://doi.org/10.1007/BF02186839
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DOI: https://doi.org/10.1007/BF02186839