Equilibrium size distribution and phase separation of multivalent, molecular assemblies in dilute solution

Multivalent molecules can bind a limited number of multiple neighbors via specific interactions. In this paper, we investigate theoretically the self-assembly and phase separation of such molecules in dilute solution. We show that the equilibrium size (n) distributions of linear or branched assemblies qualitatively differ; the former decays exponentially with the relative size n/N[combining macron] (N[combining macron] = n), while the latter decays as a power law, with an exponential cutoff only for n ⪆ N[combining macron]2 ≫ N[combining macron]. In some cases, finite, branched assemblies are unstable and show a sol-gel transition at a critical concentration. In dilute solutions, non-specific interactions result in phase separation, whose critical point is described by an effective Flory Huggins theory that is sensitive to the nature of these distributions.