  The time-dependent kinetics of formation and evolution of nano-size atomic
clusters is investigated and illustrated with the nucleation dynamics of
ion-seed Ar$_n$H$^+$ particles. The rates of growth and degradation of
Ar-atomic shells around the seed ion are inferred from Molecular Dynamics (MD)
simulations. Simulations of cluster formation have been performed with accurate
quantum-mechanical binary interaction potentials. Both the nonequilibrium and
equilibrium growth of Ar$_n$H$^+$ are investigated at different temperature and
densities of the atomic gas and seed ions. Formation of Ar$_{n\leq 40}$ shells
is the main mechanism which regulates the kinetics of nano-cluster growth and
the diffusive fluctuations of the cluster size distribution. The time-evolution
of the cluster intrinsic energy and cluster size distributions are analyzed at
the non-thermal, quasi-equilibrium, and thermal equilibrium stages of
Ar$_n$H$^+$ formation. We've determined the self-consistent model parameters
for the temporal fluctuations of the cluster size and found coefficients of the
diffusive growth mechanism describing the equilibrium distribution of
nano-clusters. Nucleation of haze and nano-dust particles in astrophysical and
atmospheric ionized gases are discussed.
