Marginally compact fractal trees with semiflexibility

We study marginally compact macromolecular trees that are created by means of two different fractal generators. In doing so, we assume Gaussian statistics for the vectors connecting nodes of the trees. Moreover, we introduce bond-bond correlations that make the trees locally semiflexible. The symmetry of the structures allows an iterative construction of full sets of eigenmodes (notwithstanding the additional interactions that are present due to semiflexibility constraints), enabling us to get physical insights about the trees' behavior and to consider larger structures. Due to the local stiffness, the self-contact density gets drastically reduced.

Figure 1

Fractal trees ${\mathcal{T}}_1$ (a) and ${\mathcal{T}}_2$ (b) studied in this work. Both structures are at iteration $I=2$. Beads shown as open circles represent the trees of inital iteration $I=1$. The sketch is aimed only to present the topology of the fractal trees; their spatial conformations may appear in vastly different forms.

Figure 2

Schematic sketch of eigenmodes of ${\mathcal{T}}_1$ for $I=1$. Beads having the same amplitude are color coded. Black beads are immobile.

Figure 3

Schematic sketch of eigenmodes of ${\mathcal{T}}_2$ for $I=1$. Beads having the same amplitude are color coded. Black beads are immobile.

Figure 4

Double-logarithmic representation of the eigenvalues of the dynamical matrix of ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ at iteration $I=5$. The parameter $q=0$ is related to fully flexible trees and $q=0.9$ to the semiflexible ones. All spectra show for lower eigenvalues a scaling ${\lambda }_p\sim p^{5/3}$.

Figure 5

Double-logarithmic representation of the mean-square gyration radii $\langle R_g^2\rangle$ of trees ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ as function of total number of beads $N\left(I\right)=8^I+1$ for different values of the stiffness parameter $q$. Inset shows rescaled radii with the minimal nonvanishing eigenvalue ${\lambda }_1$.

Figure 6

Half-logarithmic representation of the asphericity $\langle A_d\rangle$ and prolateness $\langle P_d\rangle$ of trees ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ as function of total number of beads $N\left(I\right)=8^I+1$ for different values of the stiffness parameter $q$. As expected for large self-similar objects (here for $N>500$), both characteristics saturate to a plateau.

Figure 7

Half-logarithmic representation of the number of self-contacts per monomer for trees ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ as function of total number of beads $N\left(I\right)=8^I+1$ for different values of the stiffness parameter $q$.

Figure 8

Kratky representation $k^{2}F\left(k\right)$ of the form factor of trees ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ at iteration $I=4$ for different values of the stiffness parameter $q$. In the intermediate region of $Q=k\sqrt{\langle R_g^2\rangle }$, the data saturate on the scaling $Q^{-1}$.

Figure 9

Double-logarithmic representation of the monomeric MSD for trees ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ at iteration $I=5$ having different values of stiffness parameter $q$.

Figure 10

Top: Double-logarithmic representation of the shear-stress relaxation modulus $G\left(t\right)$ for trees ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ at iteration $I=5$ having different values of stiffness parameter $q$. Inset shows $G\left(t\right)$ with rescaled by factor ${\lambda }_1$ time. Bottom: Double-logarithmic representation of the storage $G^{\prime }\left(\omega \right)$ and loss $G^{\prime \prime }\left(\omega \right)$ moduli corresponding to $G\left(t\right)$ of the main top plot. For all curves $c{k}_{B}T=1$ is taken.

Figure 11

Schematic sketch of substructures (“leaves” $\mathcal{L}$) of tree ${\mathcal{T}}_1$.

Figure 12

Schematic sketch of substructures (“leaves” $\mathcal{L}$) of tree ${\mathcal{T}}_2$.