Critical radius

Critical radius is the minimum particle size from which an aggregate is thermodynamically stable.

In other words, it is the lowest radius formed by atoms or molecules clustering together (in a gas, liquid or solid matrix) before a new phase inclusion (a bubble, a droplet or a solid particle) is viable and begins to grow.

Formation of such stable nuclei is called nucleation.

At the beginning of the nucleation process, the system finds itself in an initial phase.

Afterwards, the formation of aggregates or clusters from the new phase occurs gradually and randomly at the nanoscale.

Subsequently, if the process is feasible, the nucleus is formed.

Notice that the formation of aggregates is conceivable under specific conditions.

When these conditions are not satisfied, a rapid creation-annihilation of aggregates takes place and the nucleation and posterior crystal growth process does not happen.

Sometimes precipitation is rate-limited by the nucleation process.

An example would be when someone takes a cup of superheated water from a microwave and, when jiggling it with a spoon or against the wall of the cup, heterogeneous nucleation occurs and most of water particles convert into steam.

If the change in phase forms a crystalline solid in a liquid matrix, the atoms might then form a dendrite.

The crystal growth continues in three dimensions, the atoms attaching themselves in certain preferred directions, usually along the axes of a crystal, forming a characteristic tree-like structure of a dendrite.

The critical radius of a system can be determined from its Gibbs free energy.

The first one describes how probable it is to have a phase change and the second one is the amount of energy needed to create an interface.

is the Gibbs free energy per volume and obeys

For a low temperature, far from the fusion point, this energy is big (it is more difficult to change the phase) and for a temperature close to the fusion point it is small (the system will tend to change its phase).

and considering spherical particles, its mathematical expression is given by:

is the surface tension we need to break to create a nucleus.

is never negative as it always takes energy to create an interface.

is the absolute value of the Gibbs free energy per volume.

The Gibbs free energy of nuclear formation is found replacing the critical radius expression in the general formula.

When the Gibbs free energy change is positive, the nucleation process will not be prosperous.

Contrary, if the variation rate is negative, it will be thermodynamically stable.

The size of the cluster surpasses the critical radius.

From the expression of the critical radius, as the Gibbs volume energy increases, the critical radius will decrease and hence, it will be easier achieving the formation of nuclei and begin the crystallization process.

and promote nucleation, a supercooling or superheating process may be used.

Supercooling is a phenomenon in which the system's temperature is lowered under the phase transition temperature without the creation of the new phase.

be the volume Gibbs free energy, enthalpy and entropy respectively.

, the system has null Gibbs free energy, so:

and the Gibbs free energy of nuclear formation

Free energy change versus the nanoparticle radius. Below the critical radius, the clusters are not big enough to start the nucleation process. The Gibbs free energy change is positive and the process is not prosperous. This critical radius corresponds to the minimum size at which a particle can survive in solution without being redissolved. Above the critical radius, the particles will form and grow as it is thermodynamically favourable.
The line in blue represents the dependence in the case of a liquid and in green the case of a solid. It can be noted that when the concentration of the solute increases, the ΔG v increases, reducing the ΔG c and the critical radius, thus increasing the stability of the system.