#
Designing High-Refractive Index Polymers Using Materials Informatics^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Polymer Properties

#### 2.2. Machine Learning

#### 2.3. Computational Details

## 3. Results and Discussion

#### 3.1. Analysis of Regression Models

#### 3.2. Molecular Evolution Analysis

#### 3.3. Comparison with DFT

#### 3.4. Analysis of Selected Monomers

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

QC | Quantum Chemistry |

QSPR | Quantitative Structure Property Relationship |

ML | Machine Learning |

DFT | Density Functional Theory |

TD-DFT | Time-Dependent Density Functional Theory |

## References

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**Figure 1.**Schematic shows an outline of the design approach. The “A” on the fragments indicate points of attachment. The crossover operations (indicated by blue arrows) involve random selection of fragments (building blocks highlighted by the circles) in the two parent structures and swaps them, thus producing typically two offspring. In a given structure, the mutation operator (red arrow) may either replace or delete a randomly selected fragment.

**Figure 2.**The building block scaffolds used in this study. The attachment points are indicated by the letter “A”. The fragment recombination proceeds according to a set of fragment compatibility rules [29].

**Figure 3.**Variable importance plots for the (

**A**) Partial Least Squares Regression (PLSR) model for n, and Random Forest (RF) models for (

**B**) ${T}_{g}$ and (

**C**) ${T}_{d}$. In all cases, only the top 10 most important variables are shown.

**Figure 4.**Scores plot for the first two latent variables shows the spread of the predicted refractive indices for the designed monomers. The dashed arrow in the centre of the plot shows the direction of the increasing refractive indices as indicated by the PLSR model. Structures of selected monomers along this line reflect the chemical diversity in the population.

**Figure 5.**The box plot shows the range of refractive index values with respect to each scaffold (see Figure 2).

**Figure 7.**Plot shows the observed vs. the Density Functional Theory (DFT)-predicted refractive indices. The density in Equation (1) is calculated using a QSPR model. An overall correlation of 0.81 was obtained. See Table S13 in the Supplementary Materials for additional details.

**Table 1.**Summary of the experimental data available for refractive index (n), density ($\rho $), glass transition temperatures (${T}_{g}$) and decomposition temperatures (${T}_{d}$ for 10% weight loss). ${N}_{obs}$ is the number of available samples, while ${N}_{cal}$ and ${N}_{test}$ are the respective numbers in the calibration and test sets (based on a random 50:50 split of the data).

Property | ${\mathit{N}}_{\mathit{obs}}$ | Range | ${\mathit{N}}_{\mathit{cal}}$ | ${\mathit{N}}_{\mathit{test}}$ |
---|---|---|---|---|

n | 237 | 1.34–1.71 | 120 | 117 |

$\rho $ | 195 | 0.84–2.1 | 99 | 96 |

${T}_{g}$ (${}^{\circ}$C) | 601 | −143–399 | 304 | 297 |

${T}_{d}$ (${}^{\circ}$C) | 175 | 125–563 | 90 | 85 |

**Table 2.**Summary of the regression model performances for the refractive index (n), glass transition temperatures (${T}_{g}$) and decomposition temperatures (${T}_{d}$). Here, $MAE$ is the mean absolute error, $RMSE$ is the root mean squared error and ${R}^{2}$ the squared correlation between the observed and predicted values.

Model | Property | Calibration | Testing | ||
---|---|---|---|---|---|

${\mathit{R}}_{\mathit{cv}}^{\mathbf{2}}$ | $\mathit{RMSE}\phantom{\rule{4pt}{0ex}}\mathbf{\left(}\mathit{MAE}\mathbf{\right)}$ | ${\mathit{R}}^{\mathbf{2}}$ | $\mathit{RMSE}\phantom{\rule{4pt}{0ex}}\mathbf{\left(}\mathit{MAE}\mathbf{\right)}$ | ||

PLSR | n | 0.79 | 0.04 (0.03) | 0.79 | 0.04 (0.03) |

${T}_{g}$ (${}^{\circ}$C) | 0.81 | 52 (34) | 0.83 | 49 (38) | |

${T}_{d}$ (${}^{\circ}$C) | 0.61 | 49 (24) | 0.62 | 51 (41) | |

RF | n | 0.83 | 0.03 (0.01) | 0.88 | 0.03 (0.02) |

${T}_{g}$ (${}^{\circ}$C) | 0.86 | 44 (14) | 0.88 | 40 (30) | |

${T}_{d}$ (${}^{\circ}$C) | 0.80 | 35 (12) | 0.72 | 45 (30) | |

$\rho $ | 0.64 | 0.13 (0.04) | 0.66 | 0.14 (0.08) |

**Table 3.**Summary of the random forest classification performances for the polymer solubilities in different solvents. See Tables S7–S12 in the Supplementary Materials for performances with respect to each solvent. The 10-fold cross-validated ${\kappa}_{Cal}$ and ${\kappa}_{Test}$ values are reported for each solvent. Here, S, soluble, PS, partially soluble/swelling/soluble on heating and I, insoluble.

Solvent | #Samples | I | PS | S | ${\mathit{\kappa}}_{\mathit{Cal}}$ | ${\mathit{\kappa}}_{\mathit{Test}}$ |
---|---|---|---|---|---|---|

CHCl${}_{3}$ | 136 | 53 | 34 | 48 | 0.56 | 0.50 |

NMP | 145 | 10 | 42 | 93 | 0.62 | 0.36 |

DMAc | 105 | 8 | 41 | 56 | 0.52 | 0.48 |

DMSO | 154 | 19 | 56 | 79 | 0.53 | 0.58 |

THF | 120 | 15 | 59 | 46 | 0.49 | 0.62 |

**Table 4.**Summary of the calculated properties for selected monomers. For the refractive index ${n}_{pred}$, ${\rho}_{pred}$, ${T}_{g}$ and ${T}_{d}$, the prediction uncertainties are also provided. ${n}_{DFT}$ is the refractive index calculated according to Equation (1) with polarizabilities obtained from DFT. The Abbe number ${v}_{d}$ is calculated according to Equation (2) and makes use of the DFT-calculated polarizabilities and QSPR based density estimation. Absorption maxima ${\lambda}_{max}$ (in chloroform solvent) are calculated using Time-dependent Density Functional Theory (TD-DFT). MW, molecular weight.

Structure | MW | ${\mathit{n}}_{\mathit{pred}}$ | ${\mathit{T}}_{\mathit{g}}$ | ${\mathit{T}}_{\mathit{d}}$ | ${\mathit{\rho}}_{\mathit{pred}}$ | ${\mathit{n}}_{\mathit{DFT}}$ | ${\mathit{v}}_{\mathit{d}}$ | $\mathbf{\Delta}\mathit{n}$ | ${\mathit{\lambda}}_{\mathit{max}}$ |
---|---|---|---|---|---|---|---|---|---|

M0001 | 927 | 1.98 ± 0.11 | 256 ± 27 | 438 ± 65 | 1.35 ± 0.23 | 1.79 | 7.79 | 0.09 | 367 |

M0002 | 570 | 1.75 ± 0.05 | 226 ± 62 | 456 ± 56 | 1.37 ± 0.32 | 1.72 | 22.85 | 0.07 | 356 |

M0003 | 571 | 1.74 ± 0.15 | 210 ± 51 | 398 ± 84 | 1.29 ± 0.16 | 1.67 | 5.85 | 0.09 | 420 |

M0004 | 663 | 1.79 ± 0.10 | 242 ± 50 | 408 ± 65 | 1.36 ± 0.31 | 1.65 | 7.45 | 0.38 | 411 |

M0005 | 801 | 1.80 ± 0.04 | 222 ± 47 | 466 ± 50 | 1.36 ± 0.22 | 1.98 | 1.98 | 0.05 | 429 |

M0006 | 716 | 1.84 ± 0.06 | 206 ± 41 | 396 ± 74 | 1.27 ± 0.16 | 1.80 | 10.88 | −0.12 | 299 |

M0007 | 637 | 1.78 ± 0.14 | 223 ± 42 | 439 ± 81 | 1.37 ± 0.25 | 1.76 | 3.49 | −0.05 | 455 |

M0008 | 596 | 1.78 ± 0.09 | 180 ± 64 | 370 ± 87 | 1.33 ± 0.32 | 1.70 | 13.13 | −0.03 | 347 |

M0009 | 649 | 1.72 ± 0.04 | 198 ± 85 | 387 ± 79 | 1.63 ± 0.44 | 1.90 | 33.36 | 0.03 | 257 |

M0010 | 935 | 1.77 ± 0.11 | 226 ± 38 | 428 ± 60 | 1.44 ± 0.35 | 1.73 | 24.80 | −0.13 | 305 |

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**MDPI and ACS Style**

Venkatraman, V.; Alsberg, B.K.
Designing High-Refractive Index Polymers Using Materials Informatics. *Polymers* **2018**, *10*, 103.
https://doi.org/10.3390/polym10010103

**AMA Style**

Venkatraman V, Alsberg BK.
Designing High-Refractive Index Polymers Using Materials Informatics. *Polymers*. 2018; 10(1):103.
https://doi.org/10.3390/polym10010103

**Chicago/Turabian Style**

Venkatraman, Vishwesh, and Bjørn Kåre Alsberg.
2018. "Designing High-Refractive Index Polymers Using Materials Informatics" *Polymers* 10, no. 1: 103.
https://doi.org/10.3390/polym10010103