# Relaxation Dynamics of Semiflexible Fractal Macromolecules

^{1}

^{2}

^{*}

## 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}

## References

- Gao, C.; Yan, D. Hyperbranched polymers: From synthesis to applications. Prog. Polym. Sci.
**2004**, 29, 183–275. [Google Scholar] [CrossRef] - Voit, B.I.; Lederer, A. Hyperbranched and highly branched polymer architectures: Synthetic strategies and major characterization aspects. Chem. Rev.
**2009**, 109, 5924–5973. [Google Scholar] [CrossRef] [PubMed] - Yan, D.; Gao, C.; Frey, H. Hyperbranched Polymers: Synthesis, Properties, and Applications; John Wiley & Sons: Hoboken, NJ, USA, 2011; Volume 8. [Google Scholar]
- Lederer, A.; Burchard, W. Hyperbranched Polymers: Macromolecules in between Deterministic Linear Chains and Dendrimer Structures; Royal Society of Chemistry: Cambridge, UK, 2015; Volume 16. [Google Scholar]
- Fischer, M.; Vögtle, F. Dendrimers: From design to application—A progress report. Angew. Chem. Int. Ed.
**1999**, 38, 884–905. [Google Scholar] [CrossRef] - Ballauff, M.; Likos, C.N. Dendrimers in solution: Insight from theory and simulation. Angew. Chem. Int. Ed.
**2004**, 43, 2998–3020. [Google Scholar] [CrossRef] [PubMed] - Sowinska, M.; Urbanczyk-Lipkowska, Z. Advances in the chemistry of dendrimers. New J. Chem.
**2014**, 38, 2147–2708. [Google Scholar] [CrossRef] - Lederer, A.; Burchard, W.; Khalyavina, A.; Lindner, P.; Schweins, R. Is the universal law valid for branched polymers? Angew. Chem. Int. Ed.
**2013**, 52, 4659–4663. [Google Scholar] [CrossRef] [PubMed] - Hölter, D.; Burgath, A.; Frey, H. Degree of branching in hyperbranched polymers. Acta Polym.
**1997**, 48, 30–35. [Google Scholar] [CrossRef] - Lyulin, A.V.; Adolf, D.B.; Davies, G.R. Computer simulations of hyperbranched polymers in shear flows. Macromolecules
**2001**, 34, 3783–3789. [Google Scholar] [CrossRef] - Sheridan, P.F.; Adolf, D.B.; Lyulin, A.V.; Neelov, I.; Davies, G.R. Computer simulations of hyperbranched polymers: The influence of the Wiener index on the intrinsic viscosity and radius of gyration. J. Chem. Phys.
**2002**, 117, 7802–7812. [Google Scholar] [CrossRef] - Polińska, P.; Gillig, C.; Wittmer, J.P.; Baschnagel, J. Hyperbranched polymer stars with Gaussian chain statistics revisited. Eur. Phys. J. E
**2014**, 37, 12. [Google Scholar] [CrossRef] [PubMed] - Jurjiu, A.; Dockhorn, R.; Mironova, O.; Sommer, J.U. Two universality classes for random hyperbranched polymers. Soft Matter
**2014**, 10, 4935–4946. [Google Scholar] [CrossRef] [PubMed] - Wawrzyńska, E.; Sikorski, A.; Zifferer, G. Monte Carlo simulation studies of regular and irregular dendritic polymers. Macromol. Theory Simul.
**2015**, 24, 477–489. [Google Scholar] [CrossRef] - Lederer, A.; Burchard, W.; Hartmann, T.; Haataja, J.S.; Houbenov, N.; Janke, A.; Friedel, P.; Schweins, R.; Lindner, P. Dendronized hyperbranched macromolecules: Soft matter with a novel type of segmental distribution. Angew. Chem. Int. Ed.
**2015**, 54, 12578–12583. [Google Scholar] [CrossRef] [PubMed] - Kahng, B.; Redner, S. Scaling of the first-passage time and the survival probability on exact and quasi-exact self-similar structures. J. Phys. A Math. Gen.
**1989**, 22, 887. [Google Scholar] [CrossRef] - Matan, O.; Havlin, S. Mean first-passage time on loopless aggregates. Phys. Rev. A
**1989**, 40, 6573. [Google Scholar] [CrossRef] - Maritan, A.; Sartoni, G.; Stella, A.L. Singular dynamical renormalization group and biased diffusion on fractals. Phys. Rev. Lett.
**1993**, 71, 1027. [Google Scholar] [CrossRef] [PubMed] - Burioni, R.; Cassi, D.; Regina, S. Cutting-decimation renormalization for diffusive and vibrational dynamics on fractals. Phys. A
**1999**, 265, 323–332. [Google Scholar] [CrossRef] - Burioni, R.; Cassi, D.; Corberi, F.; Vezzani, A. Phase-ordering kinetics on graphs. Phys. Rev. E
**2007**, 75, 011113. [Google Scholar] [CrossRef] [PubMed] - Agliari, E. Exact mean first-passage time on the T-graph. Phys. Rev. E
**2008**, 77, 011128. [Google Scholar] [CrossRef] [PubMed] - Haynes, C.P.; Roberts, A.P. Global first-passage times of fractal lattices. Phys. Rev. E
**2008**, 78, 041111. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.; Lin, Y.; Zhou, S.; Wu, B.; Guan, J. Mean first-passage time for random walks on the T-graph. New J. Phys.
**2009**, 11, 103043. [Google Scholar] [CrossRef] - Lin, Y.; Wu, B.; Zhang, Z. Determining mean first-passage time on a class of treelike regular fractals. Phys. Rev. E
**2010**, 82, 031140. [Google Scholar] [CrossRef] [PubMed] - Agliari, E.; Blumen, A.; Mülken, O. Quantum-walk approach to searching on fractal structures. Phys. Rev. A
**2010**, 82, 012305. [Google Scholar] [CrossRef] - Dolgushev, M.; Guérin, T.; Blumen, A.; Bénichou, O.; Voituriez, R. Contact kinetics in fractal macromolecules. Phys. Rev. Lett.
**2015**, 115, 208301. [Google Scholar] [CrossRef] [PubMed] - Gurtovenko, A.; Blumen, A. Generalized Gaussian Structures: Models for polymer systems with complex topologies. In Polymer Analysis Polymer Theory; Springer Berlin Heidelberg: Berlin, Germany, 2005; Volume 182, pp. 171–282. [Google Scholar]
- Rouse, P.E. A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys.
**1953**, 21, 1272–1280. [Google Scholar] [CrossRef] - Markelov, D.A.; Dolgushev, M.; Gotlib, Y.Y.; Blumen, A. NMR relaxation of the orientation of single segments in semiflexible dendrimers. J. Chem. Phys.
**2014**, 140, 244904. [Google Scholar] [CrossRef] [PubMed] - Markelov, D.A.; Falkovich, S.G.; Neelov, I.M.; Ilyash, M.Y.; Matveev, V.V.; Lähderanta, E.; Ingman, P.; Darinskii, A.A. Molecular dynamics simulation of spin–lattice NMR relaxation in poly-l-lysine dendrimers: Manifestation of the semiflexibility effect. Phys. Chem. Chem. Phys.
**2015**, 17, 3214–3226. [Google Scholar] [CrossRef] [PubMed] - Bixon, M.; Zwanzig, R. Optimized Rouse–Zimm theory for stiff polymers. J.Chem. Phys.
**1978**, 68, 1896–1902. [Google Scholar] [CrossRef] - Guenza, M.; Perico, A. A reduced description of the local dynamics of star polymers. Macromolecules
**1992**, 25, 5942–5949. [Google Scholar] [CrossRef] - La Ferla, R. Conformations and dynamics of dendrimers and cascade macromolecules. J. Chem. Phys.
**1997**, 106, 688–700. [Google Scholar] [CrossRef] - Von Ferber, C.; Blumen, A. Dynamics of dendrimers and of randomly built branched polymers. J. Chem. Phys.
**2002**, 116, 8616–8624. [Google Scholar] [CrossRef] - Dolgushev, M.; Blumen, A. Dynamics of semiflexible treelike polymeric networks. J. Chem. Phys.
**2009**, 131, 044905. [Google Scholar] [CrossRef] [PubMed] - Dolgushev, M.; Blumen, A. Dynamics of semiflexible chains, stars, and dendrimers. Macromolecules
**2009**, 42, 5378–5387. [Google Scholar] [CrossRef] - Kumar, A.; Biswas, P. Dynamics of semiflexible dendrimers in dilute solutions. Macromolecules
**2010**, 43, 7378–7385. [Google Scholar] [CrossRef] - Kumar, A.; Rai, G.J.; Biswas, P. Conformation and intramolecular relaxation dynamics of semiflexible randomly hyperbranched polymers. J. Chem. Phys.
**2013**, 138, 104902. [Google Scholar] [CrossRef] [PubMed] - Fürstenberg, F.; Dolgushev, M.; Blumen, A. Analytical model for the dynamics of semiflexible dendritic polymers. J. Chem. Phys.
**2012**, 136, 154904. [Google Scholar] [CrossRef] [PubMed] - Fürstenberg, F.; Dolgushev, M.; Blumen, A. Dynamics of semiflexible regular hyperbranched polymers. J. Chem. Phys.
**2013**, 138, 034904. [Google Scholar] [CrossRef] [PubMed] - Qi, Y.; Dolgushev, M.; Zhang, Z. Dynamics of semiflexible recursive small-world polymer networks. Sci. Rep.
**2014**, 4, 7576. [Google Scholar] [CrossRef] [PubMed] - Galiceanu, M.; Reis, A.S.; Dolgushev, M. Dynamics of semiflexible scale-free polymer networks. J. Chem. Phys.
**2014**, 141, 144902. [Google Scholar] [CrossRef] [PubMed] - Cai, C.; Chen, Z.Y. Rouse dynamics of a dendrimer Model in the ϑ Condition. Macromolecules
**1997**, 30, 5104–5117. [Google Scholar] [CrossRef] - Winkler, R.G.; Reineker, P.; Harnau, L. Models and equilibrium properties of stiff molecular chains. J. Chem. Phys.
**1994**, 101, 8119–8129. [Google Scholar] [CrossRef] - Mansfield, M.L.; Stockmayer, W.H. Unperturbed dimensions of wormlike stars. Macromolecules
**1980**, 13, 1713–1715. [Google Scholar] [CrossRef] - Doi, M. Introduction to Polymer Physics; Oxford university press: Qxford, UK, 1996. [Google Scholar]
- Gurtovenko, A.A.; Gotlib, Y.Y.; Blumen, A. Rouse dynamics of polymer networks bearing dendritic wedges. Macromolecules
**2002**, 35, 7481–7491. [Google Scholar] [CrossRef] - Gurtovenko, A.A.; Markelov, D.A.; Gotlib, Y.Y.; Blumen, A. Dynamics of dendrimer-based polymer networks. J. Chem. Phys.
**2003**, 119, 7579–7590. [Google Scholar] [CrossRef] - Koda, S. Equivalence between a generalized dendritic network and a set of one-dimensional networks as a ground of linear dynamics. J. Chem. Phys.
**2015**, 142, 204112. [Google Scholar] [CrossRef] [PubMed] - Mason, J.C.; Handscomb, D.C. Chebyshev Polynomials; Chapman & Hall/CRC: Boca Raton, FL, USA, 2003. [Google Scholar]
- Alexander, S.; Orbach, R. Density of states on fractals: «fractons». J. Phys. Lett.
**1982**, 43, 625–631. [Google Scholar] [CrossRef] - Dolgushev, M.; Berezovska, G.; Blumen, A. Cospectral polymers: Differentiation via semiflexibility. J. Chem. Phys.
**2010**, 133, 154905. [Google Scholar] [CrossRef] [PubMed] - Sommer, J.-U.; Blumen, A. On the statistics of generalized Gaussian structures: Collapse and random external fields. J. Phys. A
**1995**, 28, 6669–6674. [Google Scholar] [CrossRef] - Friedrich, C. Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol. Acta
**1991**, 30, 151–158. [Google Scholar] [CrossRef] - Schiessel, H.; Blumen, A. Mesoscopic pictures of the sol-gel transition: Ladder models and fractal networks. Macromolecules
**1995**, 28, 4013–4019. [Google Scholar] [CrossRef] - Sokolov, I.M.; Klafter, J.; Blumen, A. Fractional kinetics. Phys. Today
**2002**, 55, 48–54. [Google Scholar] [CrossRef]

**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 |

© 2016 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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