# Monte Carlo Approach to the Evaluation of Nanoparticles Size Distribution from the Analysis of UV-Vis-NIR Spectra

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Section

`Quanta-Ray PRO-Series Nd:YAG`with wavelength $\lambda $ = 1064 nm, pulse length = 12 ns, mean power = 5 W, and repetition rate = 10 Hz) with the methodology described in Ref. [20]. A lens (focal length of 10 cm) focused the laser beam on a copper target at the bottom of a Teflon vessel, filled with 8 mL of liquid (acetone, methanol, and ethanol). The ablated mass was measured with a

`Sartorius M5`microbalance (sensitivity 0.01 mg) by weighting the target before and after the ablation, resulting, respectively, in 0.07 mg, 0.13 mg, and 0.70 mg for acetone, methanol, and ethanol with an accuracy of 0.02 mg.

`PerkinElmer LAMBDA 1050+`UV-Vis-NIR Spectrophotometer, measuring the absorbance from 200 nm to 1100 nm. A baseline correction was performed using the measured absorbance of the relative solvent for each solution.

#### 2.2. Computational Section

`Dataset_creation.nb`, described in Appendix C), uses the results of the Mie scattering theory to create a dataset containing the spectrum of spherical particles having a different size. The starting point for these simulations is the material’s and medium’s refractive index, which can be easily found in an online database [22] or in the Palik Handbook [23]. Unfortunately, more than ten refractive indexes for copper are available in the literature, in the visible range, with slight differences among them. So the copper refractive index was experimentally evaluated (see Appendix A). This code should work also in the UV region, where the refractive index of the solvents cannot be considered a constant, so their formulas are taken from the online database [22] and reported in Appendix B. A database is computed for each solvent for various radii and wavelength ranges.

`Mono_Fitting.nb`) and the other is for polydispersed ones (

`Poly_Fitting.nb`algorithm presented in Figure 2). Firstly, both the experimental data and cross section from the dataset are acquired. The experimental data are usually reported in arbitrary units, and the computed cross sections present values of the order of magnitude less than ${10}^{-13}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$. Working with these values is computationally inconvenient; for this reason, they will be rescaled. The total absorption spectrum of a given particle distribution $f\left(r\right)$ is obtained by integrating over the full particle radius in the range, but when computed, the integral must be discretized:

`Files reading`: Experimental data are acquired, sorted, and normalized. The dataset is acquired at the same wavelengths as the experimental points, and it is also rescaled. This automatically leads to the use of the wavelength range in which both the experimental data and the computed dataset are defined.`Assign starting point parameters`: choosing the starting point parameters for a function of three or six parameters is crucial. Starting with some random parameters can lead the gradient to descend toward a local minimum without specific physical significance. It is known that “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk—E. Fermi” [25]. To pursue this aim, two strategies are followed:`Monodisperse NPs`: The $ERROR$ is evaluated between the experimental data and every spectrum in the dataset. The spectrum that produces the minimum $ERROR$ gives the starting point for the distribution centroid ${\mu}_{1}$ and the scale parameter ${a}_{1}$. This evaluation is performed in a small range (a convenient one can be $400\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}\le \lambda \le 700\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ because gold and copper have their plasmonic peak within this range).`Polydisperse NPs`: The $ERROR$ is evaluated between the experimental data and every spectrum in the dataset in two different ranges. Small particles strongly contribute in the UV, so ${a}_{1}$ and ${\mu}_{1}$ (lognormal distribution) are assigned by finding the minimum $ERROR$ among the computed spectra for $\lambda \le 350\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$. Bigger particles and aggregates strongly contribute in the IR, so ${a}_{2}$ and ${\mu}_{2}$ (Gaussian distribution) are assigned by finding the minimum $ERROR$ among the computed spectra for $\lambda \ge 700\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$.

These edge values for $\lambda $ are purely indicative and can easily be changed in the code to find the optimal starting point for each sample. The initial values of ${w}_{1}=0.5$ and ${w}_{2}=3$ are assigned arbitrarily.`Monte Carlo step`: A cycle where a new set of parameters $\theta $ is randomly generated each time within a range of the initial parameter. Whenever the $ERROR$ obtained with the new set of parameters is lower than the initial $ERROR$, the parameters are updated, and the process is repeated for a fixed number of iterations, but new parameters can now vary in a smaller range than the previous one:$${\theta}_{j}:={\theta}_{j}(1+Range\xb7Random[-1,1])$$

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

`GoogleDrive`at the following link: https://drive.google.com/drive/folders/1oIPeAoi8S0_q63F3alHogbICgwl--KxU?usp=sharing accessed on 27 October 2023.

## Conflicts of Interest

## Abbreviations

NPs | Nanoparticles |

PLAL | Pulsed Laser Ablation in Liquid |

MSE | Mean Squared Error |

SEM | Scanning Electron Microscopy |

TEM | Transmission Electron Microscopy |

## Appendix A. Copper Refractive Index

**Figure A1.**(

**a**) Copper reflectivity measured and fitted. (

**b**) Copper refractive index from the literature with the addition of the one in this work.

## Appendix B. Solvent Refractive Indexes

## Appendix C. Mie Scattering cross Section Simulation

`Dataset_creation.nb`is structured as follows.

- The scattering cross section: ${\sigma}_{sca}=\frac{2\pi}{k{\left(\lambda \right)}^{2}}\sum _{L=1}^{{L}_{max}}\left(\right|{a}_{L}{|}^{2}+|{b}_{L}{|}^{2})$.
- The extinction cross section: ${\sigma}_{ext}=\frac{2\pi}{k{\left(\lambda \right)}^{2}}\sum _{L=1}^{{L}_{max}}(2L+1)Re({a}_{L}+{b}_{L})$.
- The absorption cross section: ${\sigma}_{abs}={\sigma}_{ext}-{\sigma}_{sca}$.

**Figure A2.**Computed extinction cross section for particle of 150 nm and 200 nm radius truncated at different multipole order (${L}_{max}$).

## Appendix D. Results of Fitting with Different Refractive Indexes

**Table A1.**Best-fitting parameters and $MSE$ copper NPs produced in methanol, dataset computed with various refractive indexes and various radius ranges.

Particle Refractive Index | Dataset Range [nm] (Start:Step:Stop) | Parameter Error | $\mathbf{MSE}$ | $\mathit{a}1$ | $\mathit{\mu}$1 [nm] | $\mathit{w}1$ | $\mathit{a}2$ | $\mathit{\mu}$2 [nm] | $\mathit{w}2$ [nm] |
---|---|---|---|---|---|---|---|---|---|

This work | 0.5:0.5:250 | $3.3\%$ | $0.0004$ | $3.6\xb7{10}^{3}$ | $3.9$ | $0.289$ | $0.90$ | 84 | $4.5$ |

[22,26] | 0.5:0.5:250 | $3.3\%$ | $0.0002$ | $4.3\xb7{10}^{3}$ | $2.90$ | $0.37$ | $1.06$ | 83 | $8.9$ |

[23] | 0.5:0.5:250 | $3.3\%$ | $0.0002$ | $8.8\xb7{10}^{3}$ | $2.20$ | $0.43$ | $0.99$ | 88 | $2.36$ |

[22,27] | 0.5:0.5:250 | $3.3\%$ | $0.0002$ | $7.2\xb7{10}^{3}$ | $2.48$ | $0.39$ | $1.05$ | 83 | $5.6$ |

This work | 0.1:0.1:30 | $5\%$ | $0.01$ | $1.25\xb7{10}^{4}$ | $0.50$ | $0.0.25$ | |||

[22,26] | 0.1:0.1:30 | $5\%$ | $0.01$ | $1.12\xb7{10}^{3}$ | $1.13$ | $0.150$ | |||

TEM distribution from Ref. [20] | $2.1$ | $0.62$ |

**Table A2.**Best-fitting parameters and $MSE$ copper NPs produced in ethanol, dataset computed with various refractive indexes and various radius ranges.

Particle Refractive Index | Dataset Range [nm] (Start:Step:Stop) | Parameter Error | $\mathbf{MSE}$ | $\mathit{a}1$ | $\mathit{\mu}$1 [nm] | $\mathit{w}1$ | $\mathit{a}2$ | $\mathit{\mu}$2 [nm] | $\mathit{w}2$ [nm] |
---|---|---|---|---|---|---|---|---|---|

This work | 0.5:0.5:250 | $3.3\%$ | $0.0002$ | $5.9\xb7{10}^{4}$ | $1.37$ | $0.37$ | $0.53$ | 108 | $4.2$ |

[22,26] | 0.5:0.5:250 | $3.3\%$ | $0.0002$ | $4.2\xb7{10}^{4}$ | $1.63$ | $0.31$ | $0.53$ | 108 | $2.43$ |

[23] | 0.5:0.5:250 | $3.3\%$ | $0.0002$ | $8.8\xb7{10}^{3}$ | $2.20$ | $0.43$ | $0.99$ | 88 | $2.36$ |

[22,27] | 0.5:0.5:250 | $3.3\%$ | $0.0002$ | $8.7\xb7{10}^{3}$ | $2.60$ | $0.087$ | $0.55$ | 107 | $0.96$ |

This work | 0.1:0.1:30 | $5\%$ | $0.01$ | 310 | $1.50$ | $0.22$ | |||

[22,26] | 0.1:0.1:30 | $5\%$ | $0.01$ | $1.19\xb7{10}^{3}$ | $0.99$ | $0.20$ | |||

TEM distribution from Ref. [20] | $3.3$ | $0.52$ |

**Table A3.**Best-fitting parameters and $MSE$ copper NPs produced in acetone, dataset computed with various refractive indexes and various radius ranges.

Particle Refractive Index | Dataset Range [nm] (Start:Step:Stop) | Parameter Error | $\mathbf{MSE}$ | $\mathit{a}1$ | $\mathit{\mu}$1 [nm] | $\mathit{w}1$ |
---|---|---|---|---|---|---|

This work | 0.1:0.1:30 | $3.3\%$ | $0.01$ | 128 | $2.32$ | $0.225$ |

[22,26] | 0.1:0.1:30 | $3.3\%$ | $0.006$ | 172 | $2.15$ | $0.288$ |

[23] | 0.1:0.1:30 | $3.3\%$ | $0.006$ | 216 | $2.00$ | $0.291$ |

[22,27] | 0.1:0.1:30 | $3.3\%$ | $0.007$ | 294 | $1.96$ | $0.227$ |

TEM distribution from Ref. [31] | $2.6$ | $0.14$ |

**Table A4.**Best-fitting parameters and $MSE$ gold NPs produced in Water (data adapted from Ref. [13]), dataset computed with various refractive indexes and various radius ranges.

Particle Refractive Index | Dataset Range [nm] (Start:Step:Stop) | Parameter Error | $\mathbf{MSE}$ | $\mathit{a}1$ | $\mathit{\mu}$1 [nm] | $\mathit{w}1$ |
---|---|---|---|---|---|---|

This work | 0.1:0.1:30 | $5\%$ | $0.004$ | 90 | $3.0$ | $0.20$ |

[22,28] | 0.1:0.1:30 | $5\%$ | $0.001$ | 72 | $3.4$ | $0.196$ |

[22,29] | 0.1:0.1:30 | $5\%$ | $0.004$ | 119 | $2.8$ | $0.32$ |

[22,27] | 0.1:0.1:30 | $5\%$ | $0.001$ | 41 | $3.2$ | $0.41$ |

Fitted with code adapted from Ref. [13] | $3.5$ | $0.05$ |

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**Figure 2.**Schematic description of the fitting algorithm with example graphics of polydisperse copper NPs produced in ethanol with PLAL.

**Figure 3.**Example of the same Monte Carlo gradient descent implemented on a two-variable function for a schematic visualization. On the left, the numbered boxes indicate the parameter values and the range in which random number generators work. On the right, the same numbered points descend toward the minimum of an example function.

**Figure 4.**Computed $MSE$ through the various Monte Carlo iteration for copper NPs solution (this work) and gold NPs solution (adapted from Ref. [13]). The points indicate the iteration in which there was a parameter update. The label in the legend indicates both if it refers to a monodispersion or a polydispersion and the used refractive index: “This Work”—Appendix A; “Hagemann”—Refs. [22,26], “Palik”—Ref. [23]; “JohnosnChristy”—Refs. [22,27]; “Olmon”—Refs. [22,28]; “BabarWeaver”—Refs. [22,29]. The fitting routine is iterated 400 times on gold NPs and 900 times on copper NPs. All the fitting results are presented in Appendix D.

**Figure 5.**Best-fitting extinction of copper NPs produced in (

**a**) methanol, (

**b**) ethanol, (

**c**) acetone, and (

**d**) gold NPs produced in water. In each graph, the blue dots represent the experimental cross section, and the yellow line represents the best fit. In (

**a**–

**c**), the experimental data are obtained using the methodology described in Section 2.1. In (

**d**), the data are adapted from Ref. [13]. The inset reports the fitting $MSE$ and distribution parameters.

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## Share and Cite

**MDPI and ACS Style**

Pò, C.L.; Iacono, V.; Boscarino, S.; Grimaldi, M.G.; Ruffino, F.
Monte Carlo Approach to the Evaluation of Nanoparticles Size Distribution from the Analysis of UV-Vis-NIR Spectra. *Micromachines* **2023**, *14*, 2208.
https://doi.org/10.3390/mi14122208

**AMA Style**

Pò CL, Iacono V, Boscarino S, Grimaldi MG, Ruffino F.
Monte Carlo Approach to the Evaluation of Nanoparticles Size Distribution from the Analysis of UV-Vis-NIR Spectra. *Micromachines*. 2023; 14(12):2208.
https://doi.org/10.3390/mi14122208

**Chicago/Turabian Style**

Pò, Cristiano Lo, Valentina Iacono, Stefano Boscarino, Maria Grazia Grimaldi, and Francesco Ruffino.
2023. "Monte Carlo Approach to the Evaluation of Nanoparticles Size Distribution from the Analysis of UV-Vis-NIR Spectra" *Micromachines* 14, no. 12: 2208.
https://doi.org/10.3390/mi14122208