Two universality classes for random hyperbranched polymers.

We grow AB2 random hyperbranched polymer structures in different ways and using different simulation methods. In particular we use a method of ad hoc construction of the connectivity matrix and the bond fluctuation model on a 3D lattice. We show that hyperbranched polymers split into two universality classes depending on the growth process. For a "slow growth" (SG) process where monomers are added sequentially to an existing molecule which strictly avoids cluster-cluster aggregation the resulting structures share all characteristic features with regular dendrimers. For a "quick growth" (QG) process which allows for cluster-cluster aggregation we obtain structures which can be identified as random fractals. Without excluded volume interactions the SG model displays a logarithmic growth of the radius of gyration with respect to the degree of polymerization while the QG model displays a power law behavior with an exponent of 1/4. By analyzing the spectral properties of the connectivity matrix we confirm the behavior of dendritic structures for the SG model and the corresponding fractal properties in the QG case. A mean field model is developed which explains the extension of the hyperbranched polymers in an athermal solvent for both cases. While the radius of gyration of the QG model shows a power-law behavior with the exponent value close to 4/5, the corresponding result for the SG model is a mixed logarithmic-power-law behavior. These different behaviors are confirmed by simulations using the bond fluctuation model. Our studies indicate that random sequential growth according to our SG model can be an alternative to the synthesis of perfect dendrimers.

Relaxation dynamics of a polymer network modeled by a multihierarchical structure.

We numerically analyze the scaling behavior of experimentally accessible dynamical relaxation forms for polymer networks modeled by a finite multihierarchical structure. In the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix, we determine the averaged monomer displacement under local external forces as well as the mechanical relaxation quantities (storage and loss moduli). Hence we generalize the known analysis for both classes of fractals to the case of multihierarchical structure, for which even though we have a mixed growth algorithm, the above cited observables still give information about the two different underlying topologies. For very large lattices, reached via an algebraic procedure that avoids the numerical diagonalizations of the corresponding connectivity matrices, we depict the scaling of both component fractals in the intermediate time (frequency) domain, which manifests two different slopes.

Dynamics of Vicsek fractals, models for hyperbranched polymers.

We consider the dynamics of Vicsek fractals of arbitrary connectivity, models for hyperbranched polymers. Their basic dynamical properties depend on their eigenvalue spectra, which can be determined iteratively. This paves the way for theoretical studies to very high precision for regular, finite, arbitrarily large hyperbranched structures.