# Characterization of Monte Carlo Dynamic/Kinetic Properties of Local Structure in Bond Fluctuation Model of Polymer System

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Bond Fluctuation Model and Simulations of Physical Phenomena in Polymer Systems

#### 1.2. Local Free Volume—Static Characterization of Mosaic-Like States

#### 1.3. Heterogeneous Dynamics and Complexity

#### 1.4. Objective

## 2. Materials and Methods

#### 2.1. Monte Carlo Bond Fluctuation Model

#### 2.2. Local Void Parameter $V(\overrightarrow{r},t)$

#### 2.3. Local Mobility Parameter $C(\overrightarrow{r},t)$

#### 2.4. Parameter $D(\overrightarrow{r},t)$

## 3. Results

#### 3.1. Evolution in Time of Local Free Volume: Parameter $V(\overrightarrow{r},t)$

#### 3.2. Evolution in Time of Local Mobility Parameter $C(\overrightarrow{r},t)$

#### 3.3. Parameter $D(\overrightarrow{r},t)$

#### 3.4. Localization of Glass Transition

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Instantaneous 2-dimensional maps of parameter V: $T=0.1$ (

**left**), $T=0.25$ (close to glass transition, (

**middle**)) and $T=0.5$ (

**right**).

**Figure 2.**(

**a**) Parameter V—a scheme of calculations. (

**b**) Snapshot of the simulation: configuration of polymer system. $10\%$ of all polymer chains are shown.

**Figure 3.**Time-dependence of local free volume V for three representative cells at $T=0.15$ (

**a**), 0.25 (

**b**) and 0.5 (

**c**), see text for more details.

**Figure 4.**Double logarithmic plot of non-normalized probability distribution $\rho ({t}_{L},V)$ for $V=6$ (

**a**) and $V=0$ (

**b**). $T=0.1$ (black line), $T=0.25$ (red line) and $T=0.5$ (blue line). Green dashed line represents power law.

**Figure 5.**Empirical cumulative functions $F(V,t)$ for $V=6$ (

**left**) and $V=0$ (

**right**). $T=0.25$ (blue line) and $T=0.5$ (red line). Horizontal lines mark the values 0.95 and 0.99.

**Figure 6.**Normalized probability density $\rho (\stackrel{}{V\left(\right)open="("\; close=")">\overrightarrow{r}}\xaf$ for $T=0.1$ (

**a**), $T=0.25$ (

**b**) and $T=0.5$ (

**c**). Solid line ($T=0.5$) represents gaussian fit.

**Figure 7.**Instantaneous 2-dimensional maps of parameter C: $T=0.1$ (

**a**), $T=0.25$ (

**b**) and $T=0.5$ (

**c**).

**Figure 8.**Empirical probability distributions $\rho \left(C\right)$: $T=0.1$ (

**a**), $T=0.25$ (

**b**) and $T=0.5$ (

**c**).

**Figure 9.**Normalized probability density $\rho (\stackrel{}{C\left(\right)open="("\; close=")">\overrightarrow{r}}\xaf$ for $T=0.1$ (

**a**), $T=0.25$ (

**b**) and $T=0.5$ (

**c**). Solid lines represent gaussian fits (see text).

**Figure 10.**Time-dependence of parameter D in three representative cells for $T=0.15$ (

**a**), 0.25 (

**b**) and 0.5 (

**c**).

**Figure 11.**Instantaneous 2-dimensional maps of parameter D: $T=0.1$ (

**a**), $T=0.25$ (

**b**) and $T=0.5$ (

**c**).

**Figure 12.**Empirical probability distributions $\rho \left(D\right)$: $T=0.1$ (

**a**), $T=0.25$ (

**b**) and $T=0.5$ (

**c**).

**Figure 13.**Normalized empirical probability distributions $\rho (\stackrel{}{D\left(\right)open="("\; close=")">\overrightarrow{r}}\xaf$ for $T=0.1$ (

**a**), $T=0.25$ (

**b**) and $T=0.5$ (

**c**).

**Figure 15.**Plots of the amplitudes of the peaks for $\stackrel{}{V\left(\right)open="("\; close=")">\overrightarrow{r}}\xaf$ (black circles) and $\stackrel{}{D\left(\right)open="("\; close=")">\overrightarrow{r}}\xaf$ (blue triangles) against temperature. Red solid and dashed lines represent Boltzmann fits, Equation (9).

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Radosz, W.; Pawlik, G.; Mituś, A.C.
Characterization of Monte Carlo Dynamic/Kinetic Properties of Local Structure in Bond Fluctuation Model of Polymer System. *Materials* **2021**, *14*, 4962.
https://doi.org/10.3390/ma14174962

**AMA Style**

Radosz W, Pawlik G, Mituś AC.
Characterization of Monte Carlo Dynamic/Kinetic Properties of Local Structure in Bond Fluctuation Model of Polymer System. *Materials*. 2021; 14(17):4962.
https://doi.org/10.3390/ma14174962

**Chicago/Turabian Style**

Radosz, Wojciech, Grzegorz Pawlik, and Antoni C. Mituś.
2021. "Characterization of Monte Carlo Dynamic/Kinetic Properties of Local Structure in Bond Fluctuation Model of Polymer System" *Materials* 14, no. 17: 4962.
https://doi.org/10.3390/ma14174962