# New Statistical Models for Copolymerization

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

#### 3.1. Bernoulli Model

#### 3.1.1. Chain Lengths

#### 3.1.2. Fingerprint Model

#### 3.1.3. Reactivity Ratios

#### 3.2. Geometric Model

#### 3.2.1. Chain Length

#### 3.2.2. Fingerprint Model

#### 3.2.3. Reactivity Ratios

#### 3.3. Single Chain Models

#### 3.4. Parameter Estimation

#### 3.5. Model Evaluation

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ODE | Ordinary differential equation |

MS | Mass spectrometry |

MALDI-TOF MS | Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry |

NRMSE | Normalized root mean square error |

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**Figure 1.**

**Left**: Comparison of the distribution of chain lengths computed by the Monte Carlo simulations with ${10}^{2}$ vs. ${10}^{6}$ chains at reactivity ratio ${r}_{A}=1.0$;

**Right**: Normalized root mean square error (NRSME) of the fingerprints computed by Monte Carlo simulations with different numbers of chains compared to the fingerprint computed from all chains produced by all Monte Carlo simulations at reactivity ratio ${r}_{A}=1.0$.

**Figure 2.**Comparison of the distribution of chain lengths computed by the Monte Carlo simulations with ${r}_{A}=0.01$ (

**left**) and ${r}_{A}=2.0$ (

**right**) vs. the length distributions computed by the Bernoulli and Geometric models.

**Figure 3.**

**Left**: Concentration of monomers $\left[\mathsf{A}\right]$ and $\left[\mathsf{B}\right]$ during the Monte Carlo simulation with ${r}_{A}=2.0$. We divided the time into discrete synthesis steps and determined the average concentrations $\tilde{\left[\mathsf{A}\right]}$ and $\tilde{\left[\mathsf{B}\right]}$;

**Right**: Monomer probabilities ${p}_{\mathsf{A}}$ and ${p}_{\mathsf{B}}$ for each synthesis step calculated from the average concentrations.

**Figure 4.**Copolymer fingerprint computed by the Monte Carlo simulation with ${r}_{A}=2.0$ (filled contours) compared to the fingerprints computed by the statistical models (solid and dashed contours).

**Left**: Bernoulli model with and without reactivity parameters (RP);

**Right**: Geometric model with and without reactivity parameters (RP).

**Figure 5.**

**Left**: Normalized root mean square error (NRMSE) of the copolymer fingerprints computed by Monte Carlo simulations compared to the fingerprints computed by the statistical models;

**Right**: Log likelihoods of the polymer chains produced by the Monte Carlo simulations under the Bernoulli and Geometric models with and without RP. Note that the minimal and maximal log likelihoods are so close to the means that the error bars are indiscernible.

**Figure 6.**Comparison of the running time (

**left**) and memory (

**right**) measurements of the Monte Carlo simulations using Gillespie’s algorithm with ${10}^{2}$ to ${10}^{6}$ chains and the Bernoulli (B) and Geometric (G) models with and without RP.

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**MDPI and ACS Style**

Engler, M.S.; Scheubert, K.; Schubert, U.S.; Böcker, S. New Statistical Models for Copolymerization. *Polymers* **2016**, *8*, 240.
https://doi.org/10.3390/polym8060240

**AMA Style**

Engler MS, Scheubert K, Schubert US, Böcker S. New Statistical Models for Copolymerization. *Polymers*. 2016; 8(6):240.
https://doi.org/10.3390/polym8060240

**Chicago/Turabian Style**

Engler, Martin S., Kerstin Scheubert, Ulrich S. Schubert, and Sebastian Böcker. 2016. "New Statistical Models for Copolymerization" *Polymers* 8, no. 6: 240.
https://doi.org/10.3390/polym8060240