Monte Carlo Approach to the Evaluation of Nanoparticles Size Distribution from the Analysis of UV-Vis-NIR Spectra
Abstract
:1. Introduction
2. Materials and Methods
2.1. Experimental Section
2.2. Computational Section
- 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 is evaluated between the experimental data and every spectrum in the dataset. The spectrum that produces the minimum gives the starting point for the distribution centroid and the scale parameter . This evaluation is performed in a small range (a convenient one can be because gold and copper have their plasmonic peak within this range).
- Polydisperse NPs: The is evaluated between the experimental data and every spectrum in the dataset in two different ranges. Small particles strongly contribute in the UV, so and (lognormal distribution) are assigned by finding the minimum among the computed spectra for . Bigger particles and aggregates strongly contribute in the IR, so and (Gaussian distribution) are assigned by finding the minimum among the computed spectra for .
These edge values for are purely indicative and can easily be changed in the code to find the optimal starting point for each sample. The initial values of and are assigned arbitrarily. - Monte Carlo step: A cycle where a new set of parameters is randomly generated each time within a range of the initial parameter. Whenever the obtained with the new set of parameters is lower than the initial , 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:
3. Results
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
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
Appendix B. Solvent Refractive Indexes
Appendix C. Mie Scattering cross Section Simulation
- The scattering cross section: .
- The extinction cross section: .
- The absorption cross section: .
Appendix D. Results of Fitting with Different Refractive Indexes
Particle Refractive Index | Dataset Range [nm] (Start:Step:Stop) | Parameter Error | 1 [nm] | 2 [nm] | [nm] | ||||
---|---|---|---|---|---|---|---|---|---|
This work | 0.5:0.5:250 | 84 | |||||||
[22,26] | 0.5:0.5:250 | 83 | |||||||
[23] | 0.5:0.5:250 | 88 | |||||||
[22,27] | 0.5:0.5:250 | 83 | |||||||
This work | 0.1:0.1:30 | ||||||||
[22,26] | 0.1:0.1:30 | ||||||||
TEM distribution from Ref. [20] |
Particle Refractive Index | Dataset Range [nm] (Start:Step:Stop) | Parameter Error | 1 [nm] | 2 [nm] | [nm] | ||||
---|---|---|---|---|---|---|---|---|---|
This work | 0.5:0.5:250 | 108 | |||||||
[22,26] | 0.5:0.5:250 | 108 | |||||||
[23] | 0.5:0.5:250 | 88 | |||||||
[22,27] | 0.5:0.5:250 | 107 | |||||||
This work | 0.1:0.1:30 | 310 | |||||||
[22,26] | 0.1:0.1:30 | ||||||||
TEM distribution from Ref. [20] |
Particle Refractive Index | Dataset Range [nm] (Start:Step:Stop) | Parameter Error | 1 [nm] | |||
---|---|---|---|---|---|---|
This work | 0.1:0.1:30 | 128 | ||||
[22,26] | 0.1:0.1:30 | 172 | ||||
[23] | 0.1:0.1:30 | 216 | ||||
[22,27] | 0.1:0.1:30 | 294 | ||||
TEM distribution from Ref. [31] |
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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
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 StylePò, 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