Can Smoluchowski equation account for gelation transition?

Abstract

We revisit the scaling theory of the Smoluchowski equation with special emphasis on the dimensional analysis to derive the scaling ansatz and to give an insightful foundation to it. It has long been argued that the homogeneity exponent $λ$ of the aggregation kernel divides the aggregation process into two regimes (i) $λ\leq 1$ nongelling and (ii) $λ>1$ gelling. However, our findings contradict with this result. In particular, we find that the Smoluchowski equation is valid if and only if $λ<1$. We show that beyond this limit i.e. at $λ\geq 1$, it breaks down and hence it fails to describe a gelation transition. This also happens to be accompanied by violation of scaling.