# The Dynamic Behavior of a Single Semiflexible Ring Chain in a Linear Polymer Matrix

^{*}

## Abstract

**:**

## 1. Introduction

_{b-linear}increases. This is because linear chains tend to be rod-like with a parallel arrangement as their bending energy increases. Additionally, the hollow part of the ring chain is filled with bundles of rod-like semiflexible linear chains.

## 2. Model and Method

_{ring}= 30, and the length of the linear chain is L

_{linear}= 30. Each chain’s potential energy is found by the following equation:

_{0}is the distance between two adjacent monomers of a polymer chain. K = 30 k

_{B}T/σ

^{2}is the spring coefficient, and the finite ductility correlation parameter R

_{0}= 1.5σ is used to avoid chain crossing.

_{bending}is the bending energy. The unit of K

_{bending}is k

_{B}T, and all simulations are carried out at k

_{B}T = 1.0 in our system. K

_{b-ring}represents the bending energy of the ring chain, while K

_{b-linear}represents the bending energy of the linear chain. Additionally, θ

_{0}is the equilibrium value of the angle. For rings, θ

_{0}= [π × (L

_{ring}− 2)]/L

_{ring}, θ

_{0}= [π × (30 − 2)]/30 = 0.93π. In the coarse-grained model developed in this study, θ

_{0}= π for linear chains. When K

_{bending}is larger, the chain is more difficult to bend, that is, the chain is rigid. In this paper, the bending energy of the ring is fixed as K

_{b-ring}= 50, and the bending energy of the linear chain ranges from K

_{b-linear}= 0 to K

_{b-linear}= 50.

_{c}is the truncation radius r

_{c}= 2

^{1/6}σ. It reaches zero at the minimum distance r = r

_{c}= 2

^{1/6}σ of the respective Lennard-Jones potential, and is set to zero after that distance. That is, ring–ring, linear–linear, and ring–linear interactions are all pure repulsive forces, and ε = 1.0 k

_{B}T is the interaction strength.

_{ring}× N

_{ring}+ L

_{linear}× N

_{linear})/L

^{3}= 0.5, where N

_{ring}and N

_{linear}are the number of ring chains and the number of linear chains, respectively, and L

_{ring}and L

_{linear}are the length of the ring chain and the length of the linear chain. In this study, L

_{ring}= 30, N

_{ring}= 1. The simulation parameters covered in this paper are given in Table 1. As the rigidity of the linear chain increases, it becomes more difficult for the system to be pressed to the target size. In order to ensure that the density of the system is about ρ = 0.5, the side length is around L ≈ 50 for different K

_{b-linear}. The stronger the rigidity, the greater the pressure required. For example, for K

_{b-linear}= 0, the pressure is 0.381, and for K

_{b-linear}= 50, the pressure is 0.451. Additionally, L

_{x}, L

_{y}, and L

_{z}for all the systems is provided in Table 1. For our system, the pressure in the three directions is the same, all three directions are scaled in the same proportions, and the final target side length is the same: L

_{x}= L

_{y}= L

_{z}.

^{7}steps, where the first 10

^{7}steps are used to ensure balance. After Δt

_{1}, data are collected at intervals of 10

^{4}steps. We use a Nosé–Hoover thermostat; the reduced temperature is T* = 1.0 in ε/k

_{B}, the initial temperature T

_{start}= 1.0, the end temperature T

_{stop}= 1.0, the damping factor T

_{damp}= 0.5, and the thermostat mass m = 1.0. The velocity Verlet algorithm is used to integrate Newtonian equations of motion with a time step of Δt = 0.005τ

_{0}, where τ

_{0}= (mσ

^{2}/k

_{B}T)

^{1/2}is the inherent MD unit of time. The units of reduction σ = 1, m = 1, and τ

_{0}= (mσ

^{2}/k

_{B}T)

^{1/2}= 1 are the units of length, mass, and time, respectively. All simulations were carried out using the free and open-source LAMMPS molecular dynamics software package (Sandia National Lab, Albuquerque, NM, USA) [38].

## 3. Result and Discussion

_{b-linear}= 0 in Figure 1a. As shown in Figure 1, the semiflexible ring chain presents a hollow two-dimensional flat disc shape, when the linear chain is flexible, linear chains are a spherical shape, and several linear chains are distributed in the middle of the circular chain, as shown in Figure 1a. As the bending energy of the linear chain increases, as shown in Figure 1b, the linear chain stretches and has a certain length with a radius of gyration R

_{g-linear}= 5.60 for K

_{b-linear}= 10, distributed around the ring chain. The bending energy is moderate for K

_{b-linear}= 10, and linear chains are between a spherical shape and rod shape. As a result, the linear chains are bent and distributed across the hollow part of the ring chain, as shown in Figure 1b for a typical linear chain with K

_{b-linear}= 10 highlighted in yellow. In both above cases, the diffusion in all directions of the ring chain is almost the same. As the bending energy of the linear chain increases to K

_{b-linear}= 50, as shown in Figure 1c, the linear chain tends to be arranged in parallel, and penetrates the ring. This is mainly caused by the fact that there are ordered nematic phases in a semiflexible linear chain system at high monomer concentrations. All linear chains in our simulation are made up of 30 monomers, as the size of the linear chains depends on the chain stiffness; the root-mean square radius of gyration R

_{g-linear}ranges from to 2.98 to 7.02 for linear chains when the bending energy of the chains increases from K

_{b-linear}= 0 to 50, and R

_{g-linear}= 5.60 for K

_{b-linear}= 10 (see Figure S1 in the Supplementary Information). The hollow part of the ring chain is filled by bundles of rod-like semiflexible linear chains. For this condition, the diffusion of the ring chain parallel to the plane is limited, and the diffusion slows down significantly.

_{b-linear}= 50, the motion of the semiflexible ring is hindered for a period of time. As shown in Figure 1c, the linear chains pass through the center of the ring, the ring is trapped, and the movement slows down. The ring chain becomes subdiffused, and Γ has a distinct peak, as shown in the red line in Figure 2, which indicates the presence of kinetic heterogeneity. In fact, the diffusion of single particles in crystalline, glass, and granular fluids also satisfies non-Gaussian dynamics due to the escape “cage” process.

^{2}θ − 1)/2>

_{b-linear}= 0), the value of S remains nearly constant at 0, and linear chains are always oriented randomly. However, for rod-like linear polymers with K

_{b-linear}= 50, Figure 3 shows that S has a maximum of 0.74, indicating that thy are almost parallel with respect to each other for linear chains.

_{cm}

^{2}in Figure 4 represents the “diffusion trajectory” of the semiflexible ring chain, i.e., the square displacement of the centroid of the ring from the corresponding simulated sample at a certain moment relative to the initial position and zero time. For the ring diffused in the flexible linear chain matrix, such as the black curve in Figure 4, random walk diffusion behavior can always be found. However, for the semiflexible linear chain system of Figure K

_{b-linear}= 50, the red curve shows a shorter plateau, as shown in Figure 4, a plain that is temporarily stuck due to the ring being bound by the linear chain through for stage S01. After this chain is temporarily removed at S01, linear chains can be retracted or “slipped” from the ring relatively easily. From the corresponding molecular conformation, it is found that the rings are still connected along the trajectory segment between stage S01 and stage S02. The ring is soon crossed by another chain, corresponding to stage S02, as shown in the illustration. The plateau becomes more visible in the figure. After stage S02, the chain temporarily escapes the caged state, and the ring begins to return to normal diffusion.

_{u}(t), defined by

_{normal}, was estimated by fitting an unweighted least squares line to the linear, long-time region of a semilog plot of ρ

_{u}(t) vs. time (see Figure 5). The inverse of the relaxation time is the negative of the slope of the line.

_{R′}(t), is defined by

_{orientation}, was estimated by fitting an unweighted least squares line to the linear, long-time region of a semilog plot of ρ

_{R’}(t) vs. time (see Figure 5). The inverse of the relaxation time is the negative of the slope of the line. As K

_{b-linear}increases, the ring chain is threaded by linear chains, the central part of the semiflexible ring is filled, the left and right swing movement in the normal direction is significantly limited, and the relaxation time in the normal direction increases significantly, from τ

_{normal}= 113,378 for K

_{b-linear}= 0 to τ

_{normal}= 590,000 for K

_{b-linear}= 50, as shown in Table 2. The diffusion is dominated by the motion along the orientation direction of the ring. Our simulation time for balancing is significantly longer than the slowest of the four relaxation times.

_{b-linear}increases, the gap between τ

_{normal}and τ

_{orientation}gradually becomes larger, which is consistent with the screenshot shown in Figure 1c. Additionally, the normal relaxation time of the semiflexible ring is much longer than the relaxation time of the parallel direction τ

_{normal}> τ

_{orientation}, that is, the semiflexible ring spreads along the extension direction of the rod-like linear chain through its center. However, it is temporarily confined to the cage in the direction of the plane parallel. Our results help us to understand the special topological conformation and dynamic behavior of the semiflexible ring in ring–linear polymer blend systems.

## 4. Conclusions

_{b-linear}value, and the relaxation time in the normal direction grows dramatically. Our research can be used to better comprehend the microphysical motion processes of semiflexible ring chains in linear polymers.

## Supplementary Materials

_{g-linear}as a function of K

_{b-linear}.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Snapshots of a single semiflexible ring chain in a linear polymer matrix with different bending energies: K

_{b-linear}= 0 (

**a**), K

_{b-linear}= 10 (

**b**), K

_{b-linear}= 50 (

**c**).

**Figure 2.**Non-Gaussian parameters Γ of the semiflexible ring chain in linear chains with different bending energies K

_{b-linear}.

**Figure 4.**Representative trajectories of a semiflexible ring chain in a linear chain with different bending energies K

_{b-linear}.

**Figure 5.**lnρ

_{n}(t) or lnρ

_{R’}(t) as a function of t for a semiflexible ring chain for K

_{b-linear}= 0 (

**a**), K

_{b-linear}= 10 (

**b**), K

_{b-linear}= 50 (

**c**).

L_{ring} | N_{ring} | L_{linear} | N_{linear} | K_{b-ring} | K_{b-linear} | Pressure | Side Length of Simulation Box L _{x}/L_{y/}L_{z} | r_{c} |
---|---|---|---|---|---|---|---|---|

30 | 1 | 30 | 2082 | 50 | 0 | 0.381 | 50.23 | 2^{1/6} |

10 | 0.418 | 51.57 | ||||||

20 | 0.426 | 50.82 | ||||||

30 | 0.435 | 51.43 | ||||||

40 | 0.441 | 50.56 | ||||||

50 | 0.451 | 50.90 |

**Table 2.**The relaxation times τ

_{normal}and τ

_{orientation}for a semiflexible ring chain in linear chains with different bending energies K

_{b-linear}.

K_{b-linear} | τ_{orientation} | τ_{normal} |
---|---|---|

0 | 17,295 | 113,378 |

10 | 18,642 | 561,797 |

20 | 20,000 | 572,100 |

30 | 22,000 | 579,100 |

40 | 23,000 | 582,400 |

50 | 24,026 | 590,000 |

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**MDPI and ACS Style**

Zhou, X.; Qin, Y.
The Dynamic Behavior of a Single Semiflexible Ring Chain in a Linear Polymer Matrix. *Biophysica* **2023**, *3*, 476-484.
https://doi.org/10.3390/biophysica3030031

**AMA Style**

Zhou X, Qin Y.
The Dynamic Behavior of a Single Semiflexible Ring Chain in a Linear Polymer Matrix. *Biophysica*. 2023; 3(3):476-484.
https://doi.org/10.3390/biophysica3030031

**Chicago/Turabian Style**

Zhou, Xiaolin, and Yifan Qin.
2023. "The Dynamic Behavior of a Single Semiflexible Ring Chain in a Linear Polymer Matrix" *Biophysica* 3, no. 3: 476-484.
https://doi.org/10.3390/biophysica3030031