Dynamics of randomly branched polymers: configuration averages and solvable models

Treating the relaxation dynamics of an ensemble of random hyperbranched macromolecules in dilute solution represents a challenge even in the framework of Rouse-type approaches, which focus on generalized Gaussian structures (GGSs). The problem is that one has to average over a large class of realizations of molecular structures, and that each molecule undergoes its own dynamics. We show that a replica formalism allows to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum. Interestingly, for a specific probability distribution of the spring strengths of the GGSs, the integral equation takes a particularly simple form. Given that several dynamical observables, such as the mechanical moduli G'(omega) and G"(omega), as well as the averaged monomer displacement are relatively simple functions of the eigenvalues, we can use the obtained spectra to compute the corresponding averaged dynamical forms. Comparing the results obtained from this approach and from extensive diagonalizations of hyperbranched GGSs we find a very good agreement.