# Simulating Polymerization by Boltzmann Inversion Force Field Approach and Dynamical Nonequilibrium Reactive Molecular Dynamics

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## Abstract

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## 1. Introduction

- (1)
- Initiation:$$\begin{array}{c}\hfill R+h\nu ={R}^{\u2022}\\ \hfill {R}^{\u2022}+M=R{M}^{\u2022}\end{array}$$
- (2)
- Propagation:$$\begin{array}{c}\hfill R{M}_{n}^{\u2022}+M=R{M}_{n+1}^{\u2022}\end{array}$$
- (3)
- Termination:$$\begin{array}{c}\hfill R{M}_{n}^{\u2022}+R{M}_{m}^{\u2022}=R{M}_{n+m}R\\ \hfill R{M}_{n}^{\u2022}+R{M}_{m}^{\u2022}=R{M}_{n}+R{M}_{m}\end{array}$$

## 2. Models and Methods

#### 2.1. Coarse-Grained Potential

#### 2.2. Coarse-Grained Reaction Modeling

#### 2.3. Dynamical Approach to Nonequilibrium Molecular Dynamics

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Structure of the 1,6-hexanediol dimethacrylate monomer. With yellow circles, the carbon atoms, of type A and B, used in the CG procedure as described in the text are marked.

**Figure 2.**Long-range radial distributions obtained in all-atoms simulations, compared to the coarse-grained ones (distance in Å). In the left panels, the coarse-grained distributions in the first step of the IBI procedure are reported, and the right panels report the distributions in the final iteration.

**Figure 3.**Short-range distributions obtained in AA simulation compared to CG ones (distance in Å). In the left panels, the coarse-grained distributions in the first step of the IBI procedure are reported, and in the right panels, the distributions in the final iteration.

**Figure 4.**Illustration of the chain propagation reaction in the CG model. A new bond is created between the atoms A1, A2 if a reaction occurs (see text) and two angle potentials, B1-A1-A2 and A1-A2-B2, are added. In red, the reactive atoms; in white, the potentially reactive atoms; and in black, the inactive ones are drawn.

**Figure 5.**Sketch representing the NEMD paradigm. The continuous line represents the trajectory sampling the initial PDF $f\left(\mathrm{\Gamma},{t}_{0}\right)$, at the initial time, ${t}_{0}$. The pointed lines represent the independent trajectories evolved in time up to t sampling $f\left(\mathrm{\Gamma},t\right)$ from the initial configurations belonging to $f\left(\mathrm{\Gamma},{t}_{0}\right)$.

**Figure 6.**Snapshot of the polymeric structure with an initial radical concentration of $3\%$ and a $40\%$ double bonds conversion without the pressure correction term. Note that the bonds crossing the periodic boundaries are not drawn in the figure.

**Figure 7.**Comparison of the original coarse-grained potential in Kcal/mol obtained with the IBI method and the pressure corrected one for the inter-particle term AA as a function of their mutual distance in Å.

**Figure 8.**(

**a**) Double bond conversion as a function of simulation time averaged by D-NEMD for the full-atomistic (AA) and the coarse-grained (CG) force fields at different initial radical concentrations. (

**b**) Largest cluster size in function of double bond conversion. The curves were used to calculate the gel points.

**Figure 9.**Cycles size distribution for the AA (from Ref. [10]) and CG systems at different stages of the polymerization process in double bonds conversion values and different initial radical concentrations. The distribution was computed as the mean over a set of 20 independent trajectories. The standard deviation (not reported in figure) was found to be always less than 5% of the corresponding mean values.

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**MDPI and ACS Style**

Monteferrante, M.; Succi, S.; Pisignano, D.; Lauricella, M.
Simulating Polymerization by Boltzmann Inversion Force Field Approach and Dynamical Nonequilibrium Reactive Molecular Dynamics. *Polymers* **2022**, *14*, 4529.
https://doi.org/10.3390/polym14214529

**AMA Style**

Monteferrante M, Succi S, Pisignano D, Lauricella M.
Simulating Polymerization by Boltzmann Inversion Force Field Approach and Dynamical Nonequilibrium Reactive Molecular Dynamics. *Polymers*. 2022; 14(21):4529.
https://doi.org/10.3390/polym14214529

**Chicago/Turabian Style**

Monteferrante, Michele, Sauro Succi, Dario Pisignano, and Marco Lauricella.
2022. "Simulating Polymerization by Boltzmann Inversion Force Field Approach and Dynamical Nonequilibrium Reactive Molecular Dynamics" *Polymers* 14, no. 21: 4529.
https://doi.org/10.3390/polym14214529