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. In precipitation models, nucleation is generally a prelude to models of the crystal growth process. 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. Mathematical derivation The critical radius of a system can be determined from its Gibbs free energy. It has two components, the volume energy and the surface 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. The mathematical expression of , considering spherical particles, is given by: where is the Gibbs free energy per volume and obeys . It is defined as the energy difference between one system at a certain temperature and the same system at the fusion temperature and it depends on pressure, the number of particles and temperature: . 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