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An analysis of phase transition models*

Published online by Cambridge University Press:  26 September 2008

A. Fasano
Affiliation:
Dipartimento di Matematica U. Dini, Viale Morgagni 67/A, 50134 Firenze, Italy
M. Primicerio
Affiliation:
Dipartimento di Matematica U. Dini, Viale Morgagni 67/A, 50134 Firenze, Italy

Abstract

We consider phase transition processes in which the thermodynamic variables are the temperature and an order parameter. Various classes are identified and many specific examples are illustrated. In this framework the question of the range of applicability of the so-called ‘additivity rules’ is investigated, showing that they apply only to a very special type of processes.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

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References

[1]Caginalp, G. & Jones, J. (1995). A derivation and analysis of phase field models of thermal alloys. Ann. Phys. 237, 66107.CrossRefGoogle Scholar
[2]Cahn, J. (1956). Transformation kinetics during continuous cooling. Acta Metall. 4, 572575.CrossRefGoogle Scholar
[3]Fried, E. & Gurtin, M. (1993). Continuum theory of thermally induced phase transition on an order parameter. Physica 68, 326343.Google Scholar
[4]Hayes, W. (1985). Mathematical Models in Material Sciences. MSc thesis, Oxford.Google Scholar
[5]Mazzullo, S. (1987). Crystallization of polymers processing and the associated free and moving boundary problems. Himont Italia.Google Scholar
[6]Penrose, O. & Fife, P. (1990). Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43, 4462.CrossRefGoogle Scholar
[7]Rabesiaka, J. & Kovacs, A. (1961). Isothermal crystallization kinetics of polyethylene. J. Appl. Phys. 32, 23142320.CrossRefGoogle Scholar
[8]Schneider, W., Köppl, A. & Berger, J. (1988). Non-isothermal Crystallization of Polymers. Intern. Polym. Processing 11 (314), 151154.CrossRefGoogle Scholar
[9]Visintin, A. (1987). Mathematical models of solid–solid phase transitions in steel. IMA J. Appl. Math. 39, 143157.CrossRefGoogle Scholar
[10]Sprekels, J. & Zheng, S. (1993). Global smooth solutions to a thermodynamically constant model of phase-field type in higher space dimension. J. Math. Anal. Appl. 176, 200223.CrossRefGoogle Scholar
[11]Berger, J., Köppl, A. & Schneider, W. (1988). Non-isothermal crystallization. Crystallization of polymers. System of rate equations. Intern. Polym. Processing 2, 151154.Google Scholar
[12]Mandelkern, L. (1964). Crystallization of Polymers. McGraw-Hill, New York.Google Scholar
[13]Malkin, A. Ya., Beghisev, V. P., Keapin, I. A. & Bolgov, S. A. (1984). General treatment of polymer crystallization kinetics. Parts 1 and 2. Polym. Eng. Sci., Polym. Phys. Ed. 24, 13961401, 14021408.Google Scholar
[14]Avrami, M. (1939, 1940, 1941). Kinetics of phase change. I. J. Chem. Phys. 7, 11031112; II. J. Chem. Phys. 8, 212–224CrossRefGoogle Scholar
III. J. Chem. Phys. 9, 177184.Google Scholar
[15]Barenblatt, G.I. & Prostokishin, V. M. (1993). A mathematical model of damage accumulation taking into account microstructural effects. Euro. J. Appl. Math. 4, 225240.CrossRefGoogle Scholar
[16]Mannucci, P. (1993). Studio degli effetti termici in un problema di polimerizzazione. PhD Tesis, Firenze.Google Scholar
[17]Mazzullo, S., Paolini, M. & Verdi, C. (1988). Polymer crystallization and processing: free boundary problems and their numerical approximation. ECMI 88, Glasgow.Google Scholar
[18]O'Neill, K. & Miller, R. (1982). Numerical solutions for a rigid-ice model of secondary frost heave. CRREL Rep. 82–13, Hanover, NH.Google Scholar
[19]Fasano, A. & Primicerio, M. (1985). Heat and mass transfer in quasi-steady ground freezingprocesses. In Proc. ESMI I (Hazewinkel, M. and Matheij, R., eds.), Teubner-Kluwer, Stuttgart.Google Scholar
[20]Fasano, A. & Primicerio, M. (1988). A phase-change model with a zone of coexistence of phases. IMAJ. Appl. Math. 41, 3146.CrossRefGoogle Scholar
[21]Visintin, A. (1985). On hysteresis in phase transition. Control and Cyb. 14, 297307.Google Scholar
[22]Christian, J. W. (1965). The Theory of Transformations in Metals and Alloys. Pergamon Press.Google Scholar
[23]Kolmogorov, A. (1937). Statistical theory of crystallization of metals. Bull. Acad. Sci. USSR Mat. Sci. 1, 355359.Google Scholar
[24]Andreucci, D., Fasano, A., Paolini, M., Primicerio, M. & Verdi, C. (1994). Numerical simulation of polymer crystallization. Math. Models Meth. Appl. Sci. 4, 135145.CrossRefGoogle Scholar
[25]Andreucci, D., Fasano, A. & Primicerio, M. (1991). Polymer crystallization kinetics: the effect of impingement. In ECMI IV, 316 (Wacker, H. J. and Zulehner, W., eds.), Teubner-Kluwer, Stuttgart.Google Scholar
[26]Green, W. & Ampt, G. (1911). Studies on soil physics. The flow of air and water through soils. J. Agric. Sci. 4, 124.Google Scholar
[27]Scheil, E. (1935). Anlaufzest den Austenitumwandung. Arch, für Eisenhüttenwesen 8, 565579.CrossRefGoogle Scholar
[28]Agarwal, P. K. & Brimacombe, J. K. (1981). Mathematical model of heat flow and austenite-pearlite transformation in eutectoid carbon steel rods for wire. Metallurgical Trans. B 12B, 121.CrossRefGoogle Scholar
[29]Andreucci, D. & Verdi, C. Existence, uniqueness, and error estimates for a model of polymer crystallization. Preprint.Google Scholar