# Crystallisation in Melts of Short, Semi-Flexible Hard-Sphere Polymer Chains: The Role of the Non-Bonded Interaction Range

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Structural Properties of the Non-Perturbed System

#### 3.2. The Role of Non-Bonded Interactions

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FCC | Face-centred cubic |

HCP | Hexagonal close packed |

BCC | Body-centred cubic |

DOS | Density of states |

SAMC | stochastic approximation Monte Carlo |

1D | one-dimensional |

2D | two-dimensional |

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**Figure 1.**(

**a**) micro-canonical entropy for the system of tangent hard-sphere flexible polymers with purely repulsive non-bonded interaction. Entropy shifted according to its maximal value. (

**b**) Temperature dependence of the heat capacity per chain calculated with the entropy shown in the left panel.

**Figure 2.**Canonical probability distribution for the potential width $\delta =0.15d$ at two temperatures: (

**a**–

**c**) $T=0.31\epsilon $ and (

**d**–

**f**) $T=0.35\epsilon $. Columns from left to right correspond to the cases of no additional interactions, square-well attraction ${\epsilon}_{\mathrm{c}}=0.01\epsilon $ and square-shoulder repulsion ${\epsilon}_{\mathrm{c}}=-0.01\epsilon $. Probabilities less than than 1% of the maximal value are not shown.

**Figure 3.**Canonical probability distribution for the potential width $\delta =0.35d$ at two temperatures: (

**a**–

**c**) $T=0.31\epsilon $ and (

**d**–

**f**) $T=0.35\epsilon $. Columns from left to right correspond to the cases of no additional interactions, square-well attraction ${\epsilon}_{\mathrm{c}}=0.01\epsilon $ and square-shoulder repulsion ${\epsilon}_{\mathrm{c}}=-0.01\epsilon $. Probabilities less than than 1% of the maximal value are not shown.

**Figure 4.**The radial distribution function calculated for two temperatures: $T=0.31\epsilon $ (black line) and $T=0.35\epsilon $ (red line).

**Figure 5.**Heat capacity maps calculated for temperatures around the phase transition. The corresponding interaction ranges are shown on the maps.

**Figure 6.**The phase transition temperature ‘shift-rate’ (12). The line represents a fit of central points by the cubic polynomial $f\left(x\right)=-0.0647-7.247x+{45.662}^{2}-61.310{x}^{3}$, with $x=\delta /d$. The neutral interaction width estimated from the equation $f\left(x\right)=0$ is ${\delta}_{c}=0.245d$.

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**MDPI and ACS Style**

Shakirov, T. Crystallisation in Melts of Short, Semi-Flexible Hard-Sphere Polymer Chains: The Role of the Non-Bonded Interaction Range. *Entropy* **2019**, *21*, 856.
https://doi.org/10.3390/e21090856

**AMA Style**

Shakirov T. Crystallisation in Melts of Short, Semi-Flexible Hard-Sphere Polymer Chains: The Role of the Non-Bonded Interaction Range. *Entropy*. 2019; 21(9):856.
https://doi.org/10.3390/e21090856

**Chicago/Turabian Style**

Shakirov, Timur. 2019. "Crystallisation in Melts of Short, Semi-Flexible Hard-Sphere Polymer Chains: The Role of the Non-Bonded Interaction Range" *Entropy* 21, no. 9: 856.
https://doi.org/10.3390/e21090856