# Polymorph Stability and Free Energy of Crystallization of Freely-Jointed Polymers of Hard Spheres

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Free Energy Difference between FCC and HCP Polymorphs

#### 2.2. Monte Carlo Simulations

**Figure 1.**System snapshots along the MC simulation. From left to right: $1\times {10}^{8}$ (Region I), $4\times {10}^{11}$ (Region II), $8\times {10}^{11}$ (Region III) and $1.4\times {10}^{12}$ (Region IV, end of simulation) MC steps. From top to bottom: (

**A**) Sites are colored according to their structural similarity as quantified through the CCE norm descriptor [93] with blue, red and green corresponding to sites with HCP, FCC and FIV character, respectively. Amorphous (AMO) sites are shown in yellow and with reduced dimensions for visual clarity. The stable FCC crystal (fourth, rightmost snapshot) is obtained in the steady state (up to fluctuations) MC production phase, after approximately $9\times {10}^{12}$ MC steps; (

**B**) Sites are colored according to their parent chain and are shown with wrapped coordinates, subjected to periodic boundary conditions; (

**C**) Sites are colored according to their parent chain and are shown with coordinates fully unwrapped in space; (

**D**) Two randomly selected chains are shown in red and blue with sphere coordinates fully unwrapped in space. Image panels created with the VMD software [95]. Details on the MC simulation and the corresponding trajectory can be found in [78].

## 3. Decorrelation of Translational and Conformational Degrees of Freedom

## 4. Free Energy of Crystallization

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Left y-axis (black color): Distance $\left|\mathbf{R}\right|$ between a monomer and the centroid of its Voronoi cell, versus one of the torsion angles $\varphi $ which belong to the same monomer, averaged over all frames in Region IV. Symbol size is proportional to the frequency of occurrence of the corresponding torsion angle $\varphi $. Right y-axis (red color): the probability distribution of torsion angles. Error bars are fluctuation amplitudes, angles have been grouped in 20 bins.

**Figure 3.**Left y-axis (black color): Distance $\left|\mathbf{R}\right|$ between a monomer and the centroid of its Voronoi cell, versus the bending angle $\theta $ of the monomer, averaged over Region IV. Symbol size is proportional to the frequency of occurrence of the corresponding bending angle $\theta $. Right y-axis (red color): the probability distribution of bending angles. Error bars are fluctuation amplitudes, angles have been grouped in 20 bins.

**Figure 4.**(Main panel) Characteristic ratio, ${C}_{n}$, and (inset) ratio of the mean square end-to-end distance divided by six times the mean square radius of gyration, $\frac{\langle {R}_{ee}^{2}\rangle}{6\langle {R}_{g}^{2}\rangle}$, as a function of chain length, l, in the disordered solid (Region I) and in the almost perfect FCC crystal (Region IV).

**Figure 5.**Probability distribution of the modulus of the end-to-end vector, $P\left(\right|{\mathbf{R}}_{ee}\left|\right)$, for chains in the length interval $l\in [970,1030]$ in the disordered solid (Region I) and in the almost perfect FCC crystal (Region IV). A small interval of l instead of single values of l has been used to obtain better statistics.

**Figure 6.**Definition of bending angles ${\theta}_{1},{\theta}_{2}$ and torsion angle $\varphi $. These angles are also used in the joint orientational functions of bending and torsion angles ${f}_{bt}({\theta}_{1},\varphi ,{\theta}_{2})$ (see Section 4). The angle $\varphi $ gives the rotation around the line defined by monomers 2-3 and is measured with respect to the plane defined by the three successive bonds 1-2-3.

**Figure 7.**Isosurface representation of the integrands in Equation (13) in the amorphous ${f}_{bt}^{am}({\theta}_{1},\varphi ,{\theta}_{2})$ (left), and in the crystal ${f}_{bt}^{cr}({\theta}_{1},\varphi ,{\theta}_{2})$ (right). Isosurface coloring corresponds to $1.5\times {10}^{-5}$ (transparent pink), $3.0\times {10}^{-5}$ (transparent yellow), $6.0\times {10}^{-5}$ (transparent green), $9.{0}^{-5}$ (solid blue).

**Figure 8.**Sections of the joint orientational functions of bending and torsional angles in the initial amorphous state ${f}_{bt}^{am}({\theta}_{1},\varphi ,{\theta}_{2})$ for four values of the torsion angle $\varphi $.

**Figure 9.**Sections of the joint orientational functions of bending and torsional angles in the stable FCC polymorph ${f}_{bt}^{cr}({\theta}_{1},\varphi ,{\theta}_{2})$ for four values of the torsion angle $\varphi $.

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Herranz, M.; Benito, J.; Foteinopoulou, K.; Karayiannis, N.C.; Laso, M.
Polymorph Stability and Free Energy of Crystallization of Freely-Jointed Polymers of Hard Spheres. *Polymers* **2023**, *15*, 1335.
https://doi.org/10.3390/polym15061335

**AMA Style**

Herranz M, Benito J, Foteinopoulou K, Karayiannis NC, Laso M.
Polymorph Stability and Free Energy of Crystallization of Freely-Jointed Polymers of Hard Spheres. *Polymers*. 2023; 15(6):1335.
https://doi.org/10.3390/polym15061335

**Chicago/Turabian Style**

Herranz, Miguel, Javier Benito, Katerina Foteinopoulou, Nikos Ch. Karayiannis, and Manuel Laso.
2023. "Polymorph Stability and Free Energy of Crystallization of Freely-Jointed Polymers of Hard Spheres" *Polymers* 15, no. 6: 1335.
https://doi.org/10.3390/polym15061335