Phantom nuclei

In addressing the impingement problem Avrami introduced the concept of phantom nuclei and the extended volume of transformed material that results from such nuclei. Phantom nuclei are nuclei that are allowed to develop in the volume that has already been transformed. Thus their designation as phantom. The total number of nuclei, real and fictitious, that are generated in the time interval dr is given by... 

Vint may be modeled in many different ways. One of the extreme examples is the solvation model proposed years ago by Klopman (1967), which is quoted here to show the flexibility of this approach, and not to suggest its use (the limits of this model have been known since a long time). In this model each nucleus of M is provided with an extra phantom charge (the solvaton), which introduces, via Coulombic interactions, a modification of the solute electronic wavefunction and of the expectation value of the energy, Em, mimicking solvent effects. 

For the free surface of nucleation, the authors introduce the concept of phantom nuclei. They assume that at any time, all the surface of the grains is free of nucleation. But, if a nucleus is created on a point of the surface already transformed (called phantom nucleus), its growth would be such that it will always be contained in a nucleus bom earlier (Figure 10.10), and the easy way by the authors consists of not calculating the transformed fraction but the fraction of the not transformed powder, the fraction of which is by no means influenced by the hypothetical creation and the growth of phantom nuclei as a consequence of our preceding remark. 

With regard to nucleation, we will have to take into account the fact that as the volume of the initial phase decreased by the growth of the final solid phase, and thus the whole volume is not free any more for nucleation, the concept of phantom nucleus will solve this problem. 

In Fignre A.9.2, between the times t and f+dt, the change dF of the real volume is represented by blackened smfaces, whereas the change dFf of fictitious volume is represented by the whole of the smfaces ranging between all the concentric spheres. This fictitious volume includes enlargements of phantom nucleus , which would have been bom inside the transformed phase and which, insofar as the reaction of growth is limited by an interfacial step, would always be included inside small islands of the new phase and would not increase the siuface of the interface between the two solid phases. The fraction dF/dFf corresponds to the probabihty of the increase of the nuttiber of nucleus at the expense of the intact matter. If it is assumed that the nuclei are randomly distributed, this probability is equal to the not-yet transformed fraction of the solid at time t, that is, -a. Thus, by taking accoimt of equations [A.9.1] and [A.9.2] we can write ... 

To express the fictitious rate of the transformation, we again take equation [10.10] in which r -t) is the nucleation frequency (munber of nucleus produced per second in all the volume) and r (t, z) is the growth reaction speed at time / of a nucleus which was bom at z. But as we saw, this relation that does not take into account the blocking of the interfaces, and the ingestion of the phantom nucleus applies to the fictitious rate, it will have to be corrected by equation [A.9.3] to express the real. We will be able to thus write for the latter ... 

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