Free Energies of Formation of Metal Clusters and Nanoparticles from Molecular Simulations: Aln with n ) 2-60

Efficient simulation methods are presented for determining the standard Gibbs free energy changes for the reactions, M + Mn-1 T Mn (R1), involved in the formation of atomic clusters and nanoparticles (also called particles) in the vapor phase. The standard Gibbs free energy of formation (¢fG°) of a particle is obtained from these Gibbs free energy changes (¢G°) by a recursion relationship using the experimental ¢fG° of the monomer. In the present study, this method has been applied to reactions involving Aln particles with n ) 2-60. This method has been validated for n ) 2, where the experimental thermodynamic properties of Al2 have been recompiled using the latest available experimental or highly accurate theoretical data. For n ) 2-4, two completely different approaches, a Monte Carlo configuration integral (MCCI) integration of partition functions and a Monte Carlo direct simulation of the equilibrium constants (MCEC), employing four wellvalidated potential energy functions have been used to calculate ¢G° of R1. Excellent agreement is observed for these two methods. Although different potential energy functions give different stage-1 results for n e 10, three high-level correction (HLC) terms, namely, a correction for the potential energy difference of the global minima, another for the electronic excitation contribution, and a third based on calculating isomeric rovibrational contribution, have been applied to mitigate deficiencies in the potential energy functions. For n ) 2, good agreement has been found between the corrected simulation results and experimental data. For larger n, the more efficient MCEC method has been used. Finally, accurate ¢G° of R1 and thus ¢fG° of Aln particles with n ) 2-60 have been determined. This is the first example of the determination of nanoparticle free energies of formation.