Nucleation in a liquid droplet

The droplet-based microreactors in microfluidic systems have been used to synthesize nanocrystals of a variety of metals and semiconductors, which involves the nucleation and growth processes. Considering the limited numbers of solvent atoms and solute atoms/particles in a stationary droplet, we derive analytical expressions of the changes of the Gibbs free energy and the Helmholtz free energy for the concurrent formation of multiple microclusters of the same size in the liquid droplet. Both the changes of the Gibbs free energy and the Helmholtz free energy are dependent on the ratio of the number of microclusters to the solvent atoms and the interface energy between the solution and the microclusters. Using the change of the free energy, which is an approximation of the Gibbs free energy and the Helmholtz free energy, we obtain the critical nucleation number of the solute atoms/particles in the microclusters for the concurrent nucleation of multiple nuclei of the same size. The critical nucleation number of the solute atoms/particles is dependent on the ratio of the number of nuclei in the droplet to the solvent atoms, and the maximum change of the free energy for the concurrent nucleation of multiple nuclei of the same size increases with the increase of the ratio of the number of the nuclei in the droplet to the number of the solvent atoms.