Dynamics of semiflexible regular hyperbranched polymers

Abstract

We study the dynamics of semiflexible Vicsek fractals (SVF) following the framework established by Dolgushev and Blumen [J. Chem. Phys. 131, 044905 (2009)], a scheme which allows to model semiflexible treelike polymers of arbitrary architecture. We show, extending the methods used in the treatment of semiflexible dendrimers by Fürstenberg et al. [J. Chem. Phys. 136, 154904 (2012)], that in this way the Langevin-dynamics of SVF can be treated to a large part analytically. For this we show for arbitrary Vicsek fractals (VF) how to construct complete sets of eigenvectors; these reduce considerably the diagonalization problem of the corresponding equations of motion. In fact, such eigenvector sets arise naturally from a hierarchical procedure which follows the iterative construction of the VF. We use the obtained eigenvalues to calculate the loss moduli G(")(ω) of SVF for different degrees of stiffness of the junctions. Finally, we compare the results for SVF to those found for semiflexible dendrimers.

Relaxation Dynamics of Semiflexible Fractal Macromolecules

“…Such a procedure was first introduced for fully flexible dendrimers of functionality f = 3 [43] and later extended to arbitrary functionalities [47,48]; see also recent general results [49] for flexible dendritic structures. The works in [39,40] illustrate that an extension of the procedure is also applicable to semiflexible dendrimers and semiflexible Vicsek fractals. Here, we find (G + 1) groups of eigenvectors for a T-fractal of generation G. Among them, the first G groups are based on the branches Z (1) to Z (G) , whereas the (G + 1)-th group involves the motion of all beads, including the central one (in the case of semiflexible dendrimers [39], the groups 1 to G represent the dynamics of dendrons of the generations 1 to G, and the (G + 1)-th group involves the motion of all dendrimer's beads).…”

“…In Equation (40), the F(n − 1) × F(n − 1) matrix A n−1 describes the two terminal Z (n−1) branches, whereas the internal Z (n−1) branch is described by the (F(n) − F(n − 1)) × (F(n) − F(n − 1)) matrix L n−1 . The blocks W 12 and W 21 reflect the interaction of the two external branches with the internal branch.…”

“…This work employs the framework of semiflexible treelike polymers (STP) [35], which allows one to study arbitrary treelike architectures and to obtain many results in closed form. In particular, the STP framework allows us to determine in this work a complete set of eigenmodes of semiflexible T-fractals, following the procedure put forward by Cai and Chen for fully-flexible dendrimers [43] that has been recently extended to semiflexible structures, namely dendrimers [39] and Vicsek fractals [40]. This procedure reduces the numerical effort and gives an intuitive sense to the structure of eigenmodes and of the corresponding relaxation spectra.…”

“…Semiflexibility was first introduced to the dynamics 37 of discrete chains by Bixon and Zwanzig [30], later it was included to the description of other 38 macromolecular architectures [31][32][33][34][35][36][37][38][39][40][41]. This work employs the framework of semiflexible treelike 39 polymers (STP) [34] which allows to study arbitrary treelike architectures and to obtain many results 40 in closed form. In particular, the STP framework allows us to determine in this work a complete 41 set of eigenmodes of semiflexible T-fractals, following the procedure put forward by Cai and Chen…”

“…An improved description of the polymer's dynamics is achieved by introducing local semiflexibility in the GGS model, which turns out to be very important for dendritic structures [29,30]. Semiflexibility was first introduced to the dynamics of discrete chains by Bixon and Zwanzig [31]; later, it was included to the description of other macromolecular architectures [32][33][34][35][36][37][38][39][40][41][42]. This work employs the framework of semiflexible treelike polymers (STP) [35], which allows one to study arbitrary treelike architectures and to obtain many results in closed form.…”