*2.4. Effective Activation Energy of the Transformation Start*

Since ferrite nucleation is a thermally activated process, it can be described using the concept of activation energy [31–34]. The observed transformation start includes the nucleation of microscopic nuclei as well as the growth of the nuclei to a size at which the transformation can be observed by dilatometric measurement. The transformed fraction, *χ*, at a constant temperature can be described by an Avrami type Equation (13) [31–33]:

$$\chi = 1 - \exp[-(kt)^n] \approx (kt)^n \tag{13}$$

where the coefficient *k* is related to the nucleation and growth rates, and *n* is related to the shapes of the growing nuclei, and the approximation is valid for a small value of (*kt*)*<sup>n</sup>* (i.e., for the start of the transformation) [35]. We emphasize that during the intial stages of the transformation, before it has proceeded to the extent that it can be measured (e.g., 1% transformation), the values for *k* and *n* can be different than those produced later in the transformation [16]. This is because the energetically most favourable nucleation sites are consumed early in the process, and the transformed and diffusion regions do not overlap for the first ferrite regions that form on the austenite grain boundaries, whilst the mean carbon content of the austenite increases as the transformation proceeds.

Since both nucleation and growth are thermally activated processes, coefficient *k* can be described by the Arrhenius type Equation (14) [31–34]:

$$k = A \exp\left(\frac{-E\_{\Lambda}(T)}{RT}\right) \tag{14}$$

where *E*A(*T*) is the effective activation energy, which depends on the temperature due to effects of undercooling (i.e., thermodynamic driving force) and atomic diffusion; *R* is the gas constant; *A* = *Cω* is a prefactor, which relates to the attempt frequency, *ω*, and the concentration of heterogenous nucleation sites, *C*. The effective activation energy is a linear combination of the activation energies of nucleation and growth [31–34].

By substituting Equation (14) in to Equation (13) and solving for *t*, we can obtain an expression for the time, *τ*, required to produce a measurable transformation, *χ*m, at a given temperature, *T*, through Equation (15):

$$\tau = \text{Kexp}\left(\frac{E\_\text{A}(T)}{RT}\right) \tag{15}$$

where *K* = *χ*<sup>m</sup> 1 *<sup>n</sup>* /*A*. Since the effective activation energy depends on the temperature, it is not straightforward to obtain its value. However, it is possible to show how different factors affect the transformation start from the differences in transformation start times, as described in Section 3.
