Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-09T06:56:44.525Z Has data issue: false hasContentIssue false
  • English
  • Français

The Time Cone method for Nucleation and Growth Kinetics on a Finite Domain

Published online by Cambridge University Press:  21 February 2011

John W. Cahn
John W. Cahn*
Affiliation:
Materials Sci. and Eng. Lab., NIST, Gaithersburg, MD 20899
Get access

Abstract

The Kolmogorov-Johnson-Mehl-Avrami theory is an exact statistical solution for the expected fraction transformed in a nucleation and growth reaction in an infinite specimen, when nucleation is random in the untransformed volume and the radial growth rate after nucleation is constant until impingement. Many of these restrictive assumptions are introduced to facilitate the use of statistics. The introduction of “phantom nuclei” and “extended volumes” are constructs that permit exact estimates of the fraction transformed. An alternative, the time cone method, is presented that does not make use of either of these constructs. The method permits obtaining exact closed form solutions for any specimen that is convex in time and space, and for nucleation rates and growth rates that are both time and position dependent. Certain types of growth anisotropies can be included. The expected fraction transformed is position and time dependent. Expressions for transformation kinetics in simple specimen geometries such as plates and growing films are given, and are shown to reduce to expected formulas in certain limits.

Type
Research Article
Information
MRS Online Proceedings Library (OPL) , Volume 398: Symposium C – Thermodynamics and Kinetics of Phase Transformations , 1995 , 425
Copyright
Copyright © Materials Research Society 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1 Kolmogorov, A. N., Izv. Akad. Nauk SSSR, Ser. Matem No. 3, 355–359 (1937).Google Scholar
2 Johnson, W. and Mehl, R. F., TAIME 135, 416–442, Discussion 442–458 (1939); Un-published details of the derivations in Appendixes A-E deposited as ADI number 1182 with the American Society for Information Science (ISIS), 8720 Georgia Avenue, Silver Spring, MD 20910–3603.Google Scholar
3 Avrami, Melvin, JCP 7, 1103–1112 (1939).Google Scholar
4 Avrami, Meivin, JCP 8, 212–224 (1940).Google Scholar
5 Avrami, Meivin, JCP 9, 177–184 (1941).Google Scholar
6 Meijering, J. L., Philips Res. Rpts. 8, 270–290 (1953).Google Scholar
7 Miles, R. E., in Proceedings of the Symposium on Statistical and Probalistic Problems in Metallurgy, edited by Nicholson, W. L., Suppl. Adv. Appl. Prob. 243–266 (1972)Google Scholar
8 Gilbert, E. N., Ann. Math. Statist. 33 958–972 (1962).Google Scholar
9 Gilbert, E. N., in Applications of Undergraduate Mathematics in Engineering, edited by Noble, B. (Macmillan, NY 1967), Chapt. 16.Google Scholar
10 Belen'ky, V. Z., Geometric Probability Models of Crystallization (Nauka, Moscow, 1980), in Russian.Google Scholar
11 Christian, J. W., The Theory of Transformations in Metals and Alloys, 2nd ed. (Pergamon, 1975) Part I, Chapt. 12.Google Scholar
12 Jackson, J. L., Science 183, 446–447 (1974).Google Scholar
13 Hull, F. C., Colton, R. A., and Mehl, R. F., Trans AIME 150, 185–207 (1942).Google Scholar
14 V. A. Shneidman and M. C. Weinberg, Time-dependent nucleation and growth of (a) crystalline phase during a rapid quench into a glassy state, this volume.Google Scholar
15 John W. Cahn, unpublished results.Google Scholar
16 Appendix D of Johnson and Mehl. [2]Google Scholar
17 Cahn, John W., Acta Met. 4, 449–459 (1956).Google Scholar
18 Levine, L. E., Narayan, K. L., Kelton, K. F., Finite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equations, preprint.Google Scholar
19 Kelton, K. F., Lakshmi, K., Levine, L. E., Cull, T. C. and Ray, C. S., Computer modelling of nonisothermal crystallization, preprint.Google Scholar
20 Stoyan, D. and Kendall, W. S., Stochastic Geometry and its Applications, (J. Wiley, NY, 1995).Google Scholar
21 Frank, F. C., in Growth and Perfection of Crystals, edited by Doremus, R. H., Roberts, B. W., and Turnbull, D. (John Wiley, New York, 1958), 411–419.Google Scholar
22 Cahn, J. W., Taylor, J. E., and Handwerker, C. A., in Sir Charles Frank, QBE, FRS, An eightieth birthday tribute, edited by Chambers, R. G., Enderby, J. E., Keller, A., Lang, A. R. and Steeds, J. W. (Adam Hilger, New York, 1991), p 88–118.Google Scholar
23 Taylor, J. E., Handwerker, C. A., and Cahn, J. W., Acta Met Mat 40, 1443–1474 (1992).Google Scholar
24 Erukhimovitch, V. and Baram, J., Rev. B50, 5854–5856 (1994); B 51 6221–6230 (1995).Google Scholar
25 Baram, J. and Erukhimovitch, V., Innovations in Mater. Res. 1, 109–118 (1996).Google Scholar
26 Chernov, A. A., Modern Crystallography, HI Crystal Growth (Springer, Berlin, 1984), Sect. 7.1.Google Scholar
27 Mora, M. T., this volume.Google Scholar
28 Ham, F. S., J. Appl. Phys., 30, 1518–1525 (1959).Google Scholar
29 John W. Cahn, manuscript in preparation.Google Scholar