To a first approximation, you can express the free energy of formation of a particle of some newly growing phase as the sum of the bulk free energy of formation and the interfacial energy. The bulk free energy contribution is negative and is proportional to R^3, and the interfacial energy is positive (in most cases) and is proportional to R^2, where R is the radius of the particle. In many systems, the interfacial term dominates for very small values of R, while the bulk energy term dominates for large values of R. This means that if you were to plot the total free energy of formation (i. e., the sum of the interfacial and bulk terms) as a function of R, it would initially increase with increasing values of R, up to some critical radius R*, and then it would decrease with increasing values of R. A nucleus that is smaller than the critical radius is a subcritical nucleus and it will experience a thermodynamic driving force to dissolve, because it can lower it's free energy by decreasing its radius R. A nucleus that is larger than the critical radius is a supercritical nucleus and it will experience a driving force to grow larger, because it can lower it's free energy by increasing its radius. When a particle has exactly the critical radius R*, it is a critical nucleus that is in unstable equilibrium, because dG/dR = 0 at R*. It has no thermodynamic driving force to shrink or to grow, but of course due to fluctuations it will fall to one side of the free energy barrier or the other and become either subcritical or supercritical.

We often use the critical radius to distinguish between distinct particles of the newly growing phase and clusters resulting from fluctuations in local composition. When determining particle density, only particles larger than the critical radius are considered.

Thanks for you help. But some simulations found that the density, composition and structures are different among precritical, critical and post critical nucleus. And we want to research these further challenges rather than limiting to the CNT.

An article which could be of interest for you:

"The microscopic pathway to crystallization in supercooled liquids"

Thanks for you help. This article is great and the following article is interesting to this research:

"Numerical Evidence for bcc Ordering at the Surface of a Critical fcc Nucleus"

@Leland: Excellent summary, just a semantical remark. Try to refrain from using the tautological term "critical nucleus": by definition the nucleus will be critical, hence its use here is redundant.

Hi,

I think this question can be addressed in a more general way without having to use any specific theory. In fact, the use of CNT to define critical and sub-critical clusters can lead to mistakes in certain limits because it just considers one variable, namely the cluster size. That said, a more general definition can be given just considering the thermodynamic force. A sub-critical cluster is a density fluctuation which is thermodynamically driven to shrink because of its physical properties (e.g. size, interior density, etc). On the other hand, a super-critical cluster concerns the diametrically opposed situation, i.e. the cluster is thermodynamically driven to grow. Hence, the critical cluster is just the border between the above situations, i.e. where the free energy does not push the cluster neither to grow nor to shrink but, ultimately this will be decided by the thermal noise.

A couple of articles which could be of interest for you:

1) A dynamical theory of nucleation for colloids and macromolecules.

(http://scitation.aip.org/content/aip/journal/jcp/136/3/10.1063/1.3677191)

2) Classical nucleation theory from a dynamical approach to nucleation.

(http://scitation.aip.org/content/aip/journal/jcp/138/24/10.1063/1.4811490)

I hope my answer can help you to better understand these concepts.

Best regards,

Year, the differences you mentioned above is important and helpful. Besides the theory on energy and thermodynamic, the variations in structure should be also considered, for example, the transition from bcc to fcc before and after nucleation.

Of course, the structure of clusters should be considered in more detailed descriptions of clusters. That is why I was suggesting the thermodynamic definition, since the structure can be defined as a physical property. Thus, depending on its structure and on the specific case you have, the cluster will be thermodynamically pushed to shrink or grow. The structure will play an important role in the polymorphic selection but this still remains being a challenging problem from a theoretical point of view.