The process of solidification is already described in details in a huge number of handbooks. Please see, for example:

1. Kurz W., Fisher D.J. Fundamentals of solidification. Third Edition. 1992, Trans Tech Publications, Aedermannsdorf, Switzerland. 305 p.

2. Easterling K. Introduction to the physical metallurgy of welding. Second edition. 1992 Butterworth-Heinemann, Oxford. 270 p.

3. Welding handbook. Ninth Edition. Volume 1. Welding Science And Technology. Eds.: Cynthia L. Jenney, Annette O’Brien. American Welding Society. Miami, 2001. 872 p.

4. Kou S. Welding metallurgy. Second Edition. 2003 by John Wiley & Sons, Inc. Hoboken, New Jersey. 461 p.

5. Lippold J.C. Welding Metallurgy and Weldability. 2015 by John Wiley & Sons, Inc. Hoboken, New Jersey. 400 p. i�

It can be understood on solidification of mixtures of iron and carbide atoms. During solidification, the preferentially certain phase is created, it sucks atoms of carbide from surrounding liquid. Once the surrounding liquid does not provide enough carbide, this phase ceases to be created and another phase requiring less carbide starts to be created.

This after a while leads to an increase of carbide in the remaining liquid, what again trigger the creation of the original phase requiring high carbide content.

All this process can go in several waves. It typically creates dendritic strictures with refining size of its branches.

Diffusion mechanism is solidification is influenced by the relative contents of its constitutive components/atom types.

One similar example to the topic is osmosis, where the difference of the partial pressures of the component on both sides of the membrane causes a flux of the component from high concentration into the lower one.

https://www.sciencedirect.com/search?qs=diffusion%20mechanism%20in%20solidification&show=25&sortBy=relevance

In general I agree with Mr. J. Kroc, but if you mean the further growth of the nucleus, it is possible due to the diffusion separation of the liquid ahead of the front of the growing colony of two solid phases. And this, in turn, is due to different equilibrium concentrations of the liquid in contact with these phases. It is also known that solidification is possible only under conditions of supercooling of a liquid, and in this case the equilibrium concentrations are described by the intersection points of the liquidus line continuations with the isotherm corresponding to a given degree of supercooling, so that the larger it is, the stronger the difference of equilibrium concentrations. Thus, separation diffusion develops in the liquid, leading to equilibrium concentrations at all interphase boundaries ahead of the front of the growing colony.

Thanks to Vera Berzovskaya for a detailed explanation. She motivated me to add this more detailed information.

More precise explanation follows in simple words. You can expect that there is a mixture of atoms of one element constituting the lattice, and some other element that is interstitial, it is not incorporated in the lattice itself.

This interstitial element makes extra deformation, pressure, in the lattice or pressure in the liquid. When there is nearby some lattice region without this substituting element, this element has a tendency to go in this depleted region.

The gradient of distribution of the substitution element is defining the driving force for its diffusion. Greater difference creates greater diffusion. The system has a tendency to achieve equilibrium.

During solidification, the equilibrium is not achieved because of the speed of cooling, finite diffusion speed of interstitial elements, face & grain boundaries, impurities influencing behavior if GB & face boundaries, etc. are constantly changing.

Solidification is a tight interplay between differences of atom concentrations (constituting driving force) of liquid and solid phases, possibility to create only one phase for given atomic concentrations, diffusion rates, and fluctuations.

The occurrence of nuclei can be explained as statistical fluctuations, which origin lays in quantum mechanical properties of atoms. The probability to create a growing nucleus increase with increasing driving force for its growth.

Actually, there already exist software based on cellular automata describing the solidification process and creation of dendrites. Such simulations are either purely cellular automata based it mixed with finite element methods describing the diffusion of atoms through the melted phase.

So what is the decisive factor in the solidification phase, the degree of lattice distortion (stresses) or concentration gradient?

I guess lattice distortion is a kinetic factor and the concentration gradient is thermodynamic.Mam would be happy if you correct me and explain the correct reason if I am wrong with detail.I cannot even justify why that is true if it all it is.

Distortion and concentration gradients are different for different combinations of matrix & added atoms, temperature, pressure, electric and magnetic fields, and acoustic fields play its role as well.

There must be at least some highly nonlinear regions in existence, which can lead to exotic and unexpected properties. Just to mention one of them, superalloys.

Systematic studies of those combinations can reveal even more exotic regions of unexpected mechanical properties. Theoretical simulations seem to be a promising tool for such a task but it will be a huge project.

I agree, this is very interesting, and now there are plenty of possibilities for mathematical modeling, but it seems to me that you still need to be based on the theory of the equilibrium state and describe how certain deviations from them (pressure or magnetic fields, etc.) affect solidification parameters (number of nuclei, their growth rate and, finally, concentration gradient).

I am using phase-field method to study Al-Cu solidification. You can take a look to the paper I added.

I add this book to the list cited above :

Solidification, J. A. DAntzig and M. Rappaz, EPFL Press.

If you want more details about phase-field modelling, please see

A. Karma and J. W. Rappel, phys. rev. E (1998)

and

B. Echebarria, R. Folch, A. Karma and M. Plapp, phys. rev. E (2004).