Empirical and Semi-Empirical Kinetics

Abstract

The term $C_{{i}}$ indicates concentration, and each term within () brackets indicates a rate of change of concentration. Each of the terms within () brackets is an index of reaction rate, and these are interrelated.

Appendices

Appendix: Kinetics of Dynamic Recrystallization

The Johnson–Mehl equation, Eq. 2.38, has been widely used to describe the kinetics of static recrystallization process (Ye et al. 2002). For this application, $\alpha = X$, fraction recrystallized. However, during dynamic recrystallization (DRX) in rapid forming processes like hammer forging, the whole operation finishes within a fraction of second and it is not possible to measure time $t$ to requisite precision, without very sophisticated experimental techniques. Therefore, for DRX studies on solution annealed (grain size $\sim200\;\upmu{\text{m}}$) alloy D9 (a version of austenitic stainless steel, chosen for in-core application in fast nuclear reactors), (Mandal 2004) replaced $t$ in Eq. 2.38 by true strain $\varepsilon$, to write a revised version of Johnson–Mehl equation in the form:

$$ X = 1 - \exp ( - k\varepsilon^{n} )\quad{\text{i}} . {\text{e}} .,\ln \,( - \ln \,(1 - X)) = \ln k + n\;\ln \;\varepsilon $$

(2.55)

This revised equation can be readily rationalized with Eq. 2.28, by considering an average constant true strain rate $\dot{\varepsilon }_{0}$ during the forging process so that $t = {\varepsilon \mathord{\left/ {\vphantom {\varepsilon {\dot{\varepsilon }_{0} }}} \right. \kern-0pt} {\dot{\varepsilon }_{0} }}$. Compared to Eq. 2.28, the exponent $n$ remains unchanged, but the $t$ term now depends upon $\dot{\varepsilon }_{0}$ as well, and as such is expected vary for example from hammer forging to the slower process of hydraulic press forging. The results did support the expected linear variation of $\ln \,( - \ln \,(1 - X))$ with $\ln \varepsilon$, for both hammer forging and the slower press forging operations, Fig. 2.16. The temperatures mentioned in these figures are the starting temperatures for forging, and the lines correspond to least square fit of the data. The values for the exponent $n$ are also indicated in these figures.

Fig. 2.16

Modified Johnson–Mehl plot of recrystallization kinetics at various temperatures in a forge hammer, b hydraulic press operation (Mandal 2004)

After hot working, the grain size in the product microstructure was smaller than the initial; grain size. This fact plus the small values for in the range of 1.07–1.41 led to the conclusion that alloy D-9 shows a growth control DRX. Large numbers of growing nuclei present mutually inhibit grain boundary mobility and restrict the grain growth. From the low $n$ values, and also form metallographic studies, it was concluded that nucleation of new strain-free grain takes place in the interfacial area of grain and twin boundaries. For the hammer forging operation, $n$ increased with increasing initial temperature, this was attributed to availability of higher thermal activation energy with increasing temperature. For the hydraulic press forging operation on the other hand, $n$ values tended to decrease, particularly for the two high temperatures. This was attributed to the lower strain rate, which results in significant temperature drop during the forging operation.

Two-slope Behaviour with Johnson–Mehl Equation

It has been mentioned that of the many rate equations proposed for describing different reactions, most heterogeneous reactions of first order that are encountered in metallic alloys are best described by the Johnson–Mehl equation, Eq. 2.28. Occasionally, however, a two-slope behaviour in a plot of $\ln \,( - \ln \,(1 - \alpha ))$ against $\ln t$ is observed (Vandermeer and Gordon 1963; Bergmann et al. 1983), as shown schematically in Fig. 2.17.

Fig. 2.17

Schematic of two-slope behaviour with Johnson–Mehl equation

One such example is the transformation of $\updelta$-ferrite phase in austenitic welds on exposure to high temperatures is considered here. During high-temperature exposure as encountered in service, the ferrite transforms to a variety of secondary phases such as M₂₃C₆ carbide, $\sigma ,\chi ,\alpha^{{\prime }}$ which degrade the mechanical and corrosion properties of the weld metal. The formation of these phases is influenced by a number of factors such as weld metal composition, ferrite content and its morphology, in addition to temperature and time of transformation. Johnson–Mehl equation has been employed by many investigators to describe the ferrite transformation in stainless steel weld metals. For example, Gill et al. (1992) determined the kinetics of transformation of $\updelta$-ferrite at $873$, $923$, $973$ and $1023$ K. In low carbon weld metal, the transformation could be described by the Johnson–Mehl equation; using microstructural evidences, it was concluded that the dissolution of ferrite takes place predominantly by its replacements byσ phase. In contrast, in the high carbon weld, transformation of $\updelta$-ferrite was represented by two slopes, indicating that $\updelta$-ferrite transforms rapidly during initial stages of ageing, which is followed by a region of sluggish transformation. The value of $\alpha$ where the change of slope occurs was found to increase with ageing temperature. Using microstructural evidences, Gill et al. (1992) attributed the two-slope behaviours to two competing reactions: replacement by carbides and austenite at initial ageing times which leaves the ferrite particles depleted of ${\text{Cr}}$ and ${\text{Mo}}$, followed by transformation to σ and/or austenite at longer ageing times, which is sluggish. These authors systematically documented the variation in the value of the exponent(s) with weld chemistry and temperatures. Writing the Johnson–Mehl equation as $\ln \,( - \ln \,(1 - \alpha )) = \ln b + n\,\ln t$, and they showed that the temperature dependence of $b$ could be expressed in Arrhenius form, whence they determined values for apparent activation energy. The apparent activation energies are useful for data correlation, but as indicated above, bereft of fundamental significance.