Local orientational mobility in regular hyperbranched polymers

We study the dynamics of local bond orientation in regular hyperbranched polymers modeled by Vicsek fractals. The local dynamics is investigated through the temporal autocorrelation functions of single bonds and the corresponding relaxation forms of the complex dielectric susceptibility. We show that the dynamic behavior of single segments depends on their remoteness from the periphery rather than on the size of the whole macromolecule. Remarkably, the dynamics of the core segments (which are most remote from the periphery) shows a scaling behavior that differs from the dynamics obtained after structural average. We analyze the most relevant processes of single segment motion and provide an analytic approximation for the corresponding relaxation times. Furthermore, we describe an iterative method to calculate the orientational dynamics in the case of very large macromolecular sizes.

Figure 1

Structure of VF of $F=4$ and generation $G=3$. Different shells $m$ are color-coded: $m=0$ (red), $m=1$ (blue), and $m=2$ (black). The droplike shapes indicate examples of branches ${\mathcal{B}}^{\left(i\right)}$. Their root segments are color-coded by green.

Figure 2

(a) $M_1^a\left(t\right)$ functions for the root segments of different VF shells $m$. The structures are of different size $G$ and of $F=3$. (b) The contributions [Eq. (8)] of the relaxation decays in $M_1^a\left(t\right)$ of (a) for all relaxation times ${\tau }_{\lambda }={\tau }_0{\lambda }^{-1}$.

Figure 3

The relaxation times of the symmetric modes of (b) and (d) coincide with the relaxation times of sets (a) and (c), respectively.

Figure 4

The $M_1\left(t\right)$ functions for VF of generation $G=4$ and different $F$. The results are presented for root segments belonging to different shells indicated by $m$.

Figure 5

The real part $\mathrm{\Delta }{\epsilon }_a^{\prime }\left(\omega \right)$ and the imaginary part $\mathrm{\Delta }{\epsilon }_a^{\prime \prime }\left(\omega \right)$ of the complex dielectric susceptibility for single segments of VF of different size $G$ and of $F=3$. The choice of the segments is the same as in Fig. 2.

Figure 6

Plots (a)–(c): The imaginary part $\mathrm{\Delta }{\epsilon }_a^{\prime \prime }\left(\omega \right)$ of the complex dielectric susceptibility for the core segments of VF for different parameters $F$ and $G$. Plot (d): Contributions $C_{\lambda }^a$ for $\mathrm{\Delta }{\epsilon }_a^{\prime \prime }\left(\omega \right)$ of plots (a)–(c), the size of VF is $G=6$.

Figure 7

Imaginary part of the dielectric susceptibility averaged over segments of the same type. The VF is of generation $G=5$ and $F=3$.

Figure 8

Iterative construction of VF structures. Here, the structure with both the cubes and the spheres form a VF of generation $G=3$. In the decimated structure, where we keep only the branching nodes (cubes), ones obtains a VF of former generation $G=2$. All nodes with a spherical shape appeared at the iteration from $G=2$ to $G=3$.

Figure 9

The imaginary part $\mathrm{\Delta }{\epsilon }_a^{\prime \prime }\left(\omega \right)$ and the contributions to it coming from the relaxation times $\left\{{\tau }_k^{\text{br}}\right\}$ of the complex dielectric susceptibility for the core segments of VF for $F=3$ and 4 and $G=6$.