DYNAMICS OF A POLYMERIZATION MODEL ON A GRAPH

This work is concerned with the dynamics of a polymerization process coupled with mass transfer and monomers injection, modeled by means of an infinite-dimensional system of Smoluchowski's equations in a finite graph. Under suitable assumptions on the system's aggregation coefficients, we show that, as a consequence of the injection mechanism, a sizable depletion of the pool of available reacting substances occurs at some finite time, that can be estimated in terms of the parameters of the problem. By analogy with well-known results in chemical engineering, we interpret that result as the onset of a sol-gel phase transition. According to the chemical engineering terminology, a sol-gel transition is characterized by the appearance of a "gel" defined as a fraction of the total chemical species which is not able to add to the polymerization process anymore. A gel just removes reacting species from the available pool but does not contributes back to the ongoing reaction. We suggest that this property might have some interest in the mathematical modeling of neurodegenerative processes, where the polymerization of some soluble proteins and their eventual aggregation into insoluble plaques play a remarkable role, which is not well understood as yet.