If you look at Jim Zoval's data for the growth of platinum nanoparticles, you will notice that significant size dispersion is present even though we assert that the nucleation of platinum nanoparticles occurs instantaneously in these experiments, and that growth occurs at diffusion control. If every platinum nanoparticle nucleates simultaneously, and grows at diffusion control, an obvious question is: What is the origin of particle size dispersion in these growth experiments?

We've used the Brownian Dynamics (BD) simulation method to study the growth of nanoparticle ensembles in an attempt to elucidate the factors which are responsible for size dispersion in the case where particles nucleate instantaneously, grow at a diffusion controlled rate from solution precursors, and are confined during growth to a flat surface.

Both random and hexagonal 2D ensembles were considered with coverages ranging from 5 x 10⁹ to 1 x 10¹¹ cm^-2 were modeled, and you can read the paper for details of these simulations. The main points of this paper can be summarized with reference to the figure shown below:

In this Figure, the reaction rate (expressed as a deposition current, blue trace) and standard deviation of the particle radius (_R, red trace) are plotted as a function of time for an ensemble of metal nanoparticles (10, 50, or 200, as indicated) that nucleated RANDOMLY on the surface. Three temporal regimes can be distinguished in the growth of these nanoparticle ensembles: At short times (less than 20 ns), the standard deviation of the particle radius, _R, rapidly increases to 0.05 to 0.1 nm. At intermediate times in the interval from 20 ns to 200 ns, _R peaks and begins to decline, nearly to zero in some simulations. This "convergent" growth segment continues until overlap of the diffusion layers of adjacent nanoparticles on the surface is nearly complete. Schematically, the situation is as shown in the cartoon shown below at t=2.

We've derived an analytical expression for _R in this time regime which is based on the stochastic nature of the deposition process, and excellent agreement with the simulation data was obtained. Seriously. Finally, at yet longer times (at which diffusion to the surface is planar), the behavior of random and hexagonal ensembles diverge: Random ensembles again transition into a divergent growth regime in which _R increases monotonically with time; the size dispersion of hexagonal arrays, however, continues to decrease with deposition time. In this time regime, the data support the conclusion that size dispersion is caused by an inhomogeneous distribution of interparticle distances which translates into an inhomogeneity in the diffusion limited flux at each particle.