Abstract
The focus is on the development and application of kinetic models of largely mesoscopical character. The modular transformation model, that has the degree of transformation as output, is composed by an integration of nucleation, growth and impingement representations, which are separately modeled. Various nucleation modes (as initial site saturation, continuous nucleation and mixed nucleation), growth modes (as interface-controlled and diffusion-controlled) and impingement modes (as pertaining to isotropic growth, anisotropic growth, and random and nonrandom product–particle distributions) are dealt. The well-known Johnson–Mehl–Avrami (-Kolmogorov) equation (JMA (or JMAK) equation), for isothermal transformations, is obtained by the modular approach; it is shown that severe boundary conditions must hold for its validity. This equation can be generalized, leading to a class of JMA-like equations, which also pertain to nonisothermal transformations. Yet, it is shown that the (even) more general application of the modular approach has to be preferred. For example, only thus the genuinely constant values of the activation energies for nucleation and growth can be determined separately, by fitting of the model to, simultaneously, a set of transformation curves, either obtained isothermally, for a number of temperatures, or obtained nonisothermally, for a set of heating rates. Along the way a series of misconcepts and confusing statements in the literature are indicated. In a closing section, notes on the hierarchy of kinetic models are given. Mean field models, i.e., as dealt with in this chapter, at least currently, provide the best combination of correctness and practicability for the description of transformation kinetics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In case of a eutectoid reaction, γ → α + β, a “nose” in the TTT diagram occurs as well, which can be explained more or less similarly, as follows. The interlamellar, α/β, spacing becomes smaller for increasing undercooling (cf. Sect. 9.4.2), thereby initially more than counteracting the decrease of diffusional mobility upon increasing undercooling (i.e. decreasing temperature): the elemental redistribution at the transformation front requires less distance coverage by diffusion for smaller interlamellar spacing. Continued increase of the undercooling is associated with such pronounced decrease of the diffusional mobility that the transformation rate decreases pronouncedly. Hence, at intermediate undercooling an optimal combination of interlamellar spacing and diffusional mobility occurs that leads to the highest transformation rate.
- 2.
Note that, as a consequence of the cooling rate depending on the location in the specimen/workpiece, a phase transformation is not induced at the same time in all parts of the specimen/workpiece.
- 3.
- 4.
A genuine DSC apparatus records directly the difference of the amounts of heat absorbed/produced by a sample pan (containing the specimen to be investigated) and a reference pan. A DTA apparatus records the temperature difference of a sample pan (containing the specimen to be investigated) and a reference pan. By means of calibration with standard specimens of which heats of transformation (often pertaining to melting) are known, the output signal of a DTA apparatus can be presented as a heat produced/absorbed (by the specimen under investigation) rate, i.e. as holds for a genuine DSC apparatus. Commercial apparatus sold as DSC apparatus often actually are DTA apparatus, in the sense discussed here. Hence, for the discussion in this section DSC and DTA used as DSC are treated in the same way simultaneously (as DSC).
- 5.
Normally isochronal annealing, i.e. with constant heating rate Φ ≡ dT/dt, is applied, and thus dp/dT = (dp/dt)/Φ.
- 6.
Note the typographical errors in the corresponding Eq. (44) in Liu et al. (2007).
- 7.
- 8.
References
General
J.W. Christian, The Theory of Transformations in Metals and Alloys, Part I, Equilibrium and General Kinetic Theory, 2nd edn. (Pergamon Press, Oxford, 1975).
G. Kostorz (ed.), Phase Transformations in Materials (Wiley-VCH, Weinheim, Germany, 2001)
F. Liu, F. Sommer, C. Bos, E.J. Mittemeijer, Analysis of solid state phase transformation kinetics: models and recipes. Int. Mater. Rev. 52, 193–212 (2007)
E.J. Mittemeijer, Analysis of the kinetics of phase transformations. J. Mater. Sci. 27, 3977–3987 (1992)
Specific
M. Avrami, Kinetics of phase change. I. General theory. J. Chem. Phys. 7, 1103–1112 (1939)
M. Avrami, Kinetics of phase change. II. Transformation-time relations for random distribution of nuclei. J. Chem. Phys. 8, 212–224 (1940)
M. Avrami, Granulation, phase change, and microstructure. Kinetics of phase change. III. J. Chem. Phys. 9, 177–184 (1941)
R. Bauer, B. Rheingans, E.J. Mittemeijer, The kinetics of the precipitation of Co from supersaturated Cu-Co alloy. Metall. Mater. Trans. A 42A, 1750–1759 (2011)
C. Bos, F. Sommer, E.J. Mittemeijer, An atomistic analysis of the interface mobility in a massive transformation. Acta Mater. 53, 5333–5341 (2005)
M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations; Their Thermodynamic Basis, 2nd edn. (Cambridge University Press, 2012)
E.A. Jägle, E.J. Mittemeijer, Interplay of kinetics and microstructure in the recrystallization of pure copper: comparing mesoscopic simulations and experiments. Metall. Mater. Trans. A 43A, 2534–2551 (2012)
W.A. Johnson, R.F. Mehl, Reaction kinetics in processes of nucleation and growth. Trans. Am. Inst. Min. Metall. Eng. 135, 416–458 (1939)
R. Kampmann, R. Wagner, Kinetics of precipitation in metastable binary alloys—theory and application to Cu1.9at.%Ti and Ni14at.%Al, in Decomposition of Alloys: the Early Stages ed. by P. Haasen, V. Gerold, R. Wagner, M.F. Ashby (Pergamon Press, 1984), pp. 91–103
A.T.W. Kempen, F. Sommer, E.J. Mittemeijer, The kinetics of the austenite-ferrite phase transformation of Fe-Mn: differential thermal analysis during cooling. Acta Mater. 50, 3545–3555 (2002)
H.E. Kissinger, Reaction kinetics in differential thermal analysis. Anal. Chem. 29, 1702–1706 (1957)
A.N. Kolmogorov, On the statistics of crystallization in metals. Izv. Akad. Nauk SSSR Ser. Mat. 3, 355–359 (1937) (in Russian with abstract in German)
F. Liu, F. Sommer, E.J. Mittemeijer, Parameter determination of an analytical model for phase transformation kinetics: application to crystallization of amorphous Mg-Ni Alloys. J. Mater. Res. 19, 2586–2596 (2004a)
F. Liu, F. Sommer, E.J. Mittemeijer, Determination of nucleation and growth mechanisms of the crystallization of amorphous alloys. application to calorimetric data, Acta Mater. 52, 3207–3216 (2004b)
H. Nitsche, F. Sommer, E.J. Mittemeijer, The Al nano-crystallization process in amorphous Al85Ni8Y5Co2. J. Non-Crystall. Solids 351, 3760–3771 (2005)
D. Raabe, Computational Materials Science (Wiley-VCH, 1998)
B. Rheingans, E.J. Mittemeijer, Modelling precipitation kinetics: evaluation of the thermodynamics of nucleation and growth. CALPHAD: Comput. Coupling Phase Diag. Thermochem. 50, 49–58 (2015a)
B. Rheingans, E.J. Mittemeijer, Analysis of precipitation kinetics on the basis of particle-size distributions. Metall. Mater. Trans. A 46A, 3423–3439 (2015b)
J. Sietsma, S. van der Zwaag, A concise model for mixed-mode phase transformations in the solid state. Acta Mater. 52, 4143–4152 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Mittemeijer, E.J. (2021). Phase Transformations: Kinetics. In: Fundamentals of Materials Science. Springer, Cham. https://doi.org/10.1007/978-3-030-60056-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-60056-3_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-60055-6
Online ISBN: 978-3-030-60056-3
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)