Avrami-Kolmogorov AK Phase-Change Model and Its Two-Phase Modes

Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 29 where J S = J χ , t ln ρ χ ,t ρ eq χ is the entropy flux, and σ e = −k B Z J χ , t ∂ ∂χ ln ρ χ , t ρ eq χ d χ , 27 is the entropy production which is expressed in terms of currents and conjugated thermo- dynamic forces defined in the χ –space. We will now assume a linear meaning: small departure from the equilibrium depen- dence between fluxes and forces and establish a linear relationship between them [33] J χ , t = −k B L [ ρ χ , t] ∂ ∂χ ln ρ χ , t ρ eq χ , 28 where L ≡ L[ ρ χ , t] is an Onsager’s coefficient, which in general depends on the state variable ρ χ , t, in particular on the reaction coordinate χ [33]. To derive this expression, locality in χ −space has to be assured, for which only fluxes and forces with the same tensorial characteristics become mutually coupled [37]. Then, the resulting kinetic equation follows by inserting Eq. 28 to the continuity equation 25 ∂ρ χ ,t ∂ t = ∂ ∂χ D χ , t ρ eq χ ∂ ∂χ ρ χ , t ρ eq χ , 29 where we have defined the diffusion coefficient as D χ , t ≡ k B L ρ χ ,t ρ χ ,t . This equation, which because of Eq. 22 applied together with Eq. 23 can also be written as ∂ρ χ ,t ∂ t = ∂ ∂χ D χ , t ∂ρ χ , t ∂χ + D χ , t k B T ∂∆ W ∂χ ρ χ , t , 30 is the FPS type equation [36] accounting readily for the evolution of the probability den- sity ρ in our χ -space. This implies that the spherulitic formation of interest is given the FPS dynamics [33, 37], where the dynamics are realized as drifted diffusion in the phase space of the mesoscopic reaction coordinate, χ , which is the volume of a single spherulite: A ’real’ volume in the space of d E = 3 and an area in d E = 2, where d E - the Euclidean dimension of the space.

2.1.8. Avrami-Kolmogorov AK Phase-Change Model and Its Two-Phase Modes

AK phase-change description continues to remain the most popular method for obtaining crystallization kinetics information [51, 52]. The conceptual foundation of this description is based on the famous combinatorial raindrop problem leading to Poisson statistics. For spherulites it can be reformulated by quantitatively determining the probability of a point being passed over by exactly M evolving spherulites, ρ sph M, being in the original combi- natorial description termed as the number of the wave-fronts. ρ sph M takes on the standard Poissonian form ρ sph M = e −M M M F M , 31 30 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. where M stands for the average number of the evolving spherulites passing through a point; F M denotes the factorial of M. The probability of any point not being passed over by a spherulite is given by value of ρ sph M = 0, eventually resulting in ρ sph M = 0 = e −M , 32 since F M = 0 = 1. As a consequence of the above, ρ sph M = 0 also represents the points being still amor- phous, i.e. not been passed over by the spherulites, and thus, it is equal to amorphous fractional completion, θ , which, in turn, results in having 1 − θ as the crystalline fractional completion. Comparing then 1 − θ with ρ sph M = 0 yields at once θ = 1 − e −M , 33 which provides the formula for the crystallized fraction. The problem finally reduces to determine M as a function of geometric assumptions on the nuclei forms as well as the time t after which the children spherulitic phase is born from the parent phase of apparently amorphous overall character. It then leads to the general solution θ ≡ θ t = 1 − e − ω t ν , 34 where M = ω t ν , resulting from a direct comparison of the two last formulae, involves two parameters ω and ν . ω is recognized to be dependent on the shape of the growing crystalline entities as well as on the amount and type of nucleation. ν depends upon the nucleation type and growth geometry but not upon the amount of nucleation [52]. For example, in biopolymers the so-called transcrystallization is a process in the course of which nucle- ation does not prevail to be a rate-determining factor. One of most solid observations on the spherulitic formation, as often as possible re- ported in the literature, is that the formation occurs asymptotically in a constant-tempo kinetic-thermodynamic regime [44, 40, 41, 39, 42, 53, 54, 55]. This is due to the fact that the droplets-involving amorphization kinetics, usually pronounced in a vigorous manner, is successfully balanced in a dynamic way during the phase change by a counter-effect which appears to be the fibrillization. It then leads in a common way to the typically undesired formation of the the polycrystals termed spherulites, see Fig. 1 [51]. The effect is fully manifested in the late-time zone, and because the fibrillization leads ultimately to a surface-clustering sub-effect caused by the pieces rodes of the crystalline skeleton, it somehow acts in its final stage as the surface tension in case of equilibrium systems, i.e. as if the system was, at the moment, in equilibrium with its outer thermostat phase. Yet, the system unavoidably departs from this local quasi-equilibrium if there is still an ample place, left for the pieces to enter the as-yet unoccupied space within the growing object. If there is actually no ample place left, the overall evolution terminates, showing up a characteristic cessation-to-growth stage, frequently reported by experimenters [56].

2.1.9. Amorphization vs Polycrystallization: