Aggregation, the coalescence of monomers into large, structured clusters, is found at the heart of many phenomena in science and industry. In a simplified model, cluster growth is a discrete stochastic process of adding or shedding monomers, one at a time. The Becker-Doring (BD) ordinary differential equations govern the growth of small clusters by surface reactions. Large clusters, growing by the diffusion of the surrounding monomers, are described by the Lifshitz-Slyozov equations. Small clusters nucleate and form large ones by a slow process caused by fluctuations over a free-energy barrier. The investigator, Joseph Farjoun, studies the following aspects of nucleation: (a) The Zeldovich nucleation formula disagrees with the experimental results. It is highly sensitive to the super-saturation's value, and hence on the distribution of cluster-sizes. The investigator seeks a more accurate formula by resolving the super-saturation to higher order. In addition, a simulation based on renormalized Monte-Carlo methods is providing more data on the nucleation rate. (b) The long-term distribution of clusters sizes, as determined by the Zeldovich nucleation rate and LS growth rate, is a classical problem. The investigator studies a connection to the smooth similarity-solution: A deeper analysis of the partial answer in his PhD thesis. (c) The analysis of 2-dimensional nucleation is a natural extension of the work in the investigator's PhD thesis. The logarithmic behavior of monomer concentration about nuclei in 2-dimensions forces the nucleation process to be spatially non-homogeneous. The investigator studies this by connecting to previous work on 2-dimensional diffusion. (d) The investigator studies the dynamic dependency of the aggregation process on other physical parameters. Specifically, he analyzes aggregation in a system with a varying temperature. The nucleation rate and the total amount of clusters formed are determined by a balance between the decreasing temperature and the depletion of monomers, which have competing effects on the super-saturation.
Phenomena that are modeled by aggregation -- solidification, precipitation, and the condensation of vapor into droplets -- appear in many biophysical and industrial contexts: Glass-to-crystal transformations, crystal nucleation in under-cooled liquids, and in polymers, colloidal crystallization, growth of spherical aggregates beyond the critical micelle concentration (CMC), and the segregation by coarsening of binary alloys quenched into the miscibility gap, to name but a few specific examples. The field of micro-chip fabrication stands to benefit greatly from a theory capable of predicting the short- and long-term behavior of 2-dimensional clusters. Resolving the long-standing disagreement between the Zeldovich formula (which predicts the rate at which clusters are created) and the experiments, finding the time-scales of the various phases of the aggregation process, and a theory for 2-dimensional aggregation (all of which are topics of research in this project) allows for more accurate predictions and better design. A more realistic model, which accounts for the varying temperature, can be used in the design of vapor-forming industries (e.g. power plants), and in weather prediction. The current state of the art in aggregation is that of extremes: the behavior at very short and very long times, the behavior of very small or very large clusters have been studied extensively. In this project, the work of the investigator, Joseph Farjoun, bridges these gaps.
