# Relaxation Dynamics of Semiflexible Fractal Macromolecules

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

- The mean squared length of the bonds is fixed $\langle {\mathbf{d}}_{a}\xb7{\mathbf{d}}_{b}\rangle ={\ell}^{2}$.
- For adjacent bonds a and b, directly connected over a bead i, $\langle {\mathbf{d}}_{a}\xb7{\mathbf{d}}_{b}\rangle =\pm {\ell}^{2}{q}_{i}$ holds. The sign is determined by the relative orientation of the bonds. The positive sign describes the case of head-to-tail orientation of a and b; otherwise, the minus sign is obtained. The common stiffness parameter related to a and b is denoted by ${q}_{i}$.
- Due to the freely-rotating condition imposed on non-adjacent bonds a and c (connected over the unique path $({b}_{1},...,{b}_{k})$), one obtains $\langle {\mathbf{d}}_{a}\xb7{\mathbf{d}}_{c}\rangle =\langle {\mathbf{d}}_{a}\xb7{\mathbf{d}}_{{b}_{1}}\rangle \langle {\mathbf{d}}_{{b}_{1}}\xb7{\mathbf{d}}_{{b}_{2}}\rangle \cdots \langle {\mathbf{d}}_{{b}_{k}}\xb7{\mathbf{d}}_{c}\rangle {\ell}^{-2k}$. For linear chains, this restriction under the continuous chain limit $\ell \to 0$ and ${q}_{i}\to 1$ leads to the definition of the persistence length ${L}_{p}$; see Equation (3.15) of [44]. However, for branched structures, no smooth curve description due to the branching points is possible.

## 3. Results and Discussion

#### 3.1. Dynamical Matrix of T-fractals

- Terminal beads with functionality (i.e., number of NN) $f=1$ have exactly one internal NN with $f=3$. Hence, one obtains the matrix element:$${\mu}_{1}=1+\frac{2{q}^{2}}{1-q-2{q}^{2}}$$
- For T-fractals of generation $G=1$, the single internal bead is directly connected to three terminal beads. The corresponding matrix element is given by:$${\mu}_{2}=\frac{3}{1-2q}$$
- An internal bead with two terminal and one internal NN is described by the diagonal element:$${\mu}_{3}=\frac{3}{1-2q}+\frac{2{q}^{2}}{1-q-2{q}^{2}}$$
- The diagonal element ${\mu}_{4}$ corresponds to internal beads with three internal NN:$${\mu}_{4}=\frac{3}{1-2q}+3\frac{2{q}^{2}}{1-q-2{q}^{2}}$$
- If an internal bead has two internal and one terminal NN, it is described by the diagonal element:$$\begin{array}{c}\hfill {\mu}_{5}=\frac{3}{1-2q}+2\frac{2{q}^{2}}{1-q-2{q}^{2}}\end{array}$$

- If one of the two considered beads is a terminal bead, we obtain the NN element:$${\nu}_{1}=-\frac{1}{1-2q}$$
- Otherwise, two internal beads in NN position result in the matrix element:$${\nu}_{2}=-\frac{1+2q}{1-2q}$$

#### 3.2. Hierarchical Eigenmodes of T-fractals

#### 3.3. Reduced Matrices

#### 3.4. Eigenvalue Spectra of ${\mathrm{A}}^{\mathrm{STP}}$

#### 3.5. Mechanical Relaxation

## 4. Conclusions

## Supplementary Materials

**a**) first, (

**b**) second, (

**c**) third and (

**d**) fourth groups. The gluing beads colored by black are immobile.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

GGS | generalized Gaussian structures |

STP | semiflexible treelike polymers |

NN | nearest neighbor |

NNN | next-nearest neighbor |

## Appendix A. Iterative Procedure for the Calculation of the Matrix A_{n+1} from A_{n}

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**Figure 1.**(

**a**) Iterative construction of a T-fractal up to generation $G=3$. The beads added in the second and in the third iteration steps are colored by blue and red, respectively; (

**b**) A two-dimensional representation of a T-fractal of generation $G=3$, and (

**c**) its random configuration in three-dimensional space.

**Figure 2.**Schematic representation of the non-vanishing elements of matrix ${\mathbf{A}}^{\mathrm{STP}}$. For diagonal elements (${\mu}_{i}$), the corresponding beads are highlighted by color. For off-diagonal elements (${\nu}_{i}$ and ρ), the corresponding interactions are indicated through wavy lines.

**Figure 3.**Branches ${\mathcal{Z}}^{({G}_{\mathcal{Z}})}$ of different branch generation ${G}_{\mathcal{Z}}=1,2,3$ for a T-fractal of generation $G=3$.

**Figure 4.**Examples for eigenmodes of the first (

**a**), second (

**b**) and third (

**c**) group. The beads of a branch that move with the same amplitude have the same color. The numbers placed in the beads correspond to the variables $\{{x}_{i}\}$ used in Equations (23), (25) and (26). The beads k colored by black are the gluing beads.

**Figure 5.**Iterative construction of a ${\mathcal{Z}}^{(n)}$ branch consisting of three smaller branches; see the text for details.

**Figure 6.**Composition of the number of independent variables $F(5)$ for the fifth group; see the text for details.

**Figure 7.**Eigenvalue spectra of $G=9$ T-fractals plotted in semi-logarithmic (

**a**) and in double logarithmic (

**b**) scales for different values of the stiffness parameter q.

**Figure 8.**Schematic representation of the eigenmode corresponding to the largest eigenvalue of a $G=4$ T-fractal.

**Figure 9.**Gyration radius of T-fractals for different values of the stiffness parameter q plotted in (

**a**) as a function of generation G and in (

**b**) as a function of number of beads N.

**Figure 10.**(

**a**) Reduced storage moduli $[{G}^{\prime}(\omega )]$ of $G=9$ T-fractals and (

**b**) the corresponding local slopes of the $[{G}^{\prime}(\omega )]$ curves for different values of stiffness parameter q.

**Figure 11.**(

**a**) Reduced loss moduli $[{G}^{\u2033}(\omega )]$ of $G=9$ T-fractals and (

**b**) the corresponding local slopes of the $[{G}^{\u2033}(\omega )]$ curves for different values of stiffness parameter q.

G | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

N | 4 | 10 | 28 | 82 | 244 | 730 | 2188 | 6562 | 19,684 | 59,050 |

$F(G+1)$ | 2 | 3 | 6 | 14 | 35 | 90 | 234 | 611 | 1598 | 4182 |

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Mielke, J.; Dolgushev, M. Relaxation Dynamics of Semiflexible Fractal Macromolecules. *Polymers* **2016**, *8*, 263.
https://doi.org/10.3390/polym8070263

**AMA Style**

Mielke J, Dolgushev M. Relaxation Dynamics of Semiflexible Fractal Macromolecules. *Polymers*. 2016; 8(7):263.
https://doi.org/10.3390/polym8070263

**Chicago/Turabian Style**

Mielke, Jonas, and Maxim Dolgushev. 2016. "Relaxation Dynamics of Semiflexible Fractal Macromolecules" *Polymers* 8, no. 7: 263.
https://doi.org/10.3390/polym8070263