will tend to change its phase). Regarding and considering spherical particles, its mathematical expression is given by: where is the surface tension we need to break to create a nucleus. The value of the is never negative as it always takes energy to create an interface. The total Gibbs free energy is therefore: The critical radius is found by optimization, setting the derivative of equal to zero. yielding , where is the surface tension and 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. Interpretation When the Gibbs free energy change is positive, the nucleation process will not be prosperous. The nanoparticle radius is small, the surface term prevails the volume term . Contrary, if the variation rate is negative, it will be thermodynamically stable. The size of the cluster surpasses the critical radius. In this occasion, the volum term overcomes the superficial term . 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. Methods for reducing the critical radius Supercooling In order to decrease the value of the critical radius 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. Let be the temperature difference, where is the phase transition temperature. Let be the volume Gibbs free energy, enthalpy and entropy respectively. When , the system has null Gibbs free energy, so: In general, the following approximations can be done: and Consequently: So: Substituting this result on the expressions for and , the following equations are obtained: Notice that and diminish with an increasing supercooling. Analogously, a mathematical derivation for the superheating can be done. Supersaturation Supersaturation is a phenomenon where the concentration of a solute exceeds the value of the equilibrium concentration. From the definition of chemical potential ,where is the Boltzmann constant, is the solute concentration and is the equilibrium concentration. For a stoichiometric compound and considering and , where is the atomic volume: Defining the supersaturation as this can be rewritten as Finally, the critical radius and the Gibbs free energy of nuclear formation can be obtained