# The Mathematical Simulation for the Photocatalytic Fatigue of Polymer Nanocomposites Using the Monte Carlo Methods

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Photodegradation of Polymers and Metals Fatigue

## 3. Derivation of the Equation of the “Photocatalytic Fatigue” Curve

## 4. Calculation of the Experimental Parameters for the Equations of “Photocatalytic Fatigue” by the Monte Carlo Method

## 5. Equations of the “Photocatalytic Fatigue”

#### 5.1. Phenomenological Equation for Polypropylene

#### 5.2. Phenomenological Equation for Polyester

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Kalos, M.H.; Whitlock, P.A. Monte Carlo Methods. Second Revised and Enlarged Edition; WILEY-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2008; p. 204. [Google Scholar]
- Fishman, G.S. Monte Carlo: Concepts, Algorithms and Applications; Springer: New York, NY, USA, 1996; p. 698. [Google Scholar]
- Nocedal, J.; Wright, S. Numerical Optimization; Springer: New York, NY, USA, 2006; Volume XXII, p. 664. [Google Scholar]
- Vasiliev, F.P. Optimization Methods; Publishing house of the Moscow Center for Continuous Mathematical Education (MCCME): Moscow, Russia, 2011; p. 434. [Google Scholar]
- Asmussen, S.; Glynn, P.W. Stochastic Simulation: Algorithms and Analysis; Springer: New York, NY, USA, 2007; p. 476. [Google Scholar]
- Kroese, D.P.; Taimre, T.; Botev, Z.I. Handbook of Monte Carlo Methods; John Wiley & Sons: New York, NY, USA, 2011; p. 772. [Google Scholar]
- Ermakov, S.M. Monte-Carlo Method and Related Issues; Nauka: Moscow, Russia, 1975; p. 472. [Google Scholar]
- Ermakov, S.M.; Zhiglyavsky, A.A. Mathematical Theory of Optimal Experiment; Nauka: Moscow, Russia, 1987; p. 320. [Google Scholar]
- Zhiglyavsky, A.A. Mathematical Theory of Global Random Search; Publishing House of Leningrad State University: Leningrad, Russia, 1985; p. 293. [Google Scholar]
- Vladimirova, L.; Fatyanova, I. Construction of regression experiment optimal plan using parallel computing. In Proceedings of the 2015 International Conference “Stability and Control Processes” in Memory of V.I. Zubov (SCP), Saint-Petersburg, Russia, 5–9 October 2015; pp. 361–363. [Google Scholar]
- Hanemann, T.; Szabo, D.V. Polymer-Nanoparticle Composites: From Synthesis to Modern Applications. Materials
**2010**, 3, 3468–3517. [Google Scholar] [CrossRef] - Oladele, I.O.; Omotosho, T.F.; Adediran, A.A. Polymer-Based Composites: An Indispensable Material for Present and Future Applications. Int. J. Polym. Sci.
**2020**, 12, 8834518. [Google Scholar] [CrossRef] - Mao, Z.; Yanga, Z.; Zhang, J. SrTiO
_{3}as a new solar reflective pigment on the cooling property of PMMA-ceramic composites. Ceram. Int.**2019**, 45, 16078–16087. [Google Scholar] [CrossRef] - Shi, S.; Shen, D.; Xu, T.; Zhang, Y. Thermal, optical, interfacial and mechanical properties of titanium dioxide/shape memory polyurethane nanocomposites. Compos. Sci. Technol.
**2018**, 164, 17–23. [Google Scholar] [CrossRef] [Green Version] - Serpone, N.; Emeline, A.V. Semiconductor Photocatalysis—Past, Present, and Future Outlook. J. Phys. Chem. Lett.
**2012**, 3, 673–677. [Google Scholar] [CrossRef] - Chang, H.T.; Wu, N.M.; Zhu, F. A Kinetic Model for Photocatalytic Degradation of Organic Contaminants in a Thin Film TiO
_{2}Catalyst. Water Res.**2000**, 34, 407–416. [Google Scholar] [CrossRef] - Danish, M.S.S.; Estrella, L.L.; Alemaida, I.M.A.; Lisin, A.; Moiseev, N.; Ahmadi, M.; Senjyu, T. Photocatalytic applications of metal oxides for sustainable environmental remediation. Metals
**2021**, 11, 80. [Google Scholar] [CrossRef] - Sanongraj, W.; Chen, Y.; Crittenden, J.C.; Destaillats, H.; Hand, D.W.; Perram, D.L.; Taylor, R. Mathematical Model for Photocatalytic Destruction of Organic Contaminants in Air. J. Air Waste Manag. Assoc.
**2007**, 57, 1112–1122. [Google Scholar] [CrossRef] [Green Version] - Lee, Q.Y.; Li, H. Photocatalytic Degradation of Plastic Waste: A Mini Review. Micromachines
**2021**, 12, 907. [Google Scholar] [CrossRef] - Kamrannejad, M.M.; Hasanzadeha, A.; Nosoudi, N.; Mai, L.; Babaluo, A.A. Photocatalytic Degradation of Polypropylene/TiO
_{2}Nano-composites Materials. Mater. Res.**2014**, 17, 1039–1046. [Google Scholar] [CrossRef] [Green Version] - Mavric, Z.; Tomsic, B.; Simoncic, B. Recent advances in the ultraviolet protection finishing of textiles. Tekstilec
**2018**, 61, 201–220. [Google Scholar] [CrossRef] - Wiener, J.; Chladova, A.; Shahidi, S.; Peterova, L. Effect of UV Irradiation On Mechanical and Morphological Properties of Natural and Synthetic Fabric Before and After Nano-Tio
_{2}Padding. Autex Res. J.**2017**, 17, 370–378. [Google Scholar] [CrossRef] [Green Version] - Ranby, B.J.; Rabek, J.F. Photodegradation, Photo-Oxidation, and Photostabilization of Polymers; Wiley: London, UK, 1975. [Google Scholar]
- Klemchuk, P.P. Influence of pigments on the light stability of polymers: A critical review. Polym. Photochem.
**1983**, 3, 1–27. [Google Scholar] [CrossRef] - Emanual, N.M.; Buchachenko, A.L. Chemical Physics of Molecular Destruction and Stabilization of Polymers; Nauka: Moscow, Russia, 1988; p. 368. [Google Scholar]
- Egerton, G.S. Photosensitizing Properties of Dyes and White Pigments. Nature
**1964**, 204, 1153–1155. [Google Scholar] [CrossRef] - Egerton, G.S.; Shah, K.M. The Effect of Temperature on the Photochemical Degradation of Textile Materials Part I: Degradation Sensitized by Titanium Dioxide. Text. Res. J.
**1968**, 38, 130–135. [Google Scholar] [CrossRef] - Amor, N.; Noman, M.T.; Petru, M. Classification of Textile Polymer Composites: Recent Trends and Challenges. Polymers
**2021**, 13, 2592. [Google Scholar] [CrossRef] - Dave, S.; Jagtap, P.; Verma, S.; Nehra, R.; Dave, S.; Mohanty, P.; Das, J. Mathematical modeling and surface response curves for green synthesized nanomaterials and their application in dye degradation. Photocatal. Degrad. Dyes
**2021**, 2021, 571–591. [Google Scholar] [CrossRef] - Forsythe, P.J.E. Exudation of material from slip bands at the surface of fatigued crystals of an aluminium-copper alloy. Nature
**1953**, 171, 172–173. [Google Scholar] [CrossRef] - Suresh, S. Fatigue of Materials; Cambridge University Press: Cambridge, UK, 1992; p. 617. [Google Scholar]
- Kim, W.H.; Laird, C. Crack nucleation and stage I propagation in high strain fatigue—II. mechanism. Acta Metall.
**1978**, 26, 789–799. [Google Scholar] [CrossRef] - Rozumek, D. The Development of Fatigue Cracks in Metals. Mater. Res. Proc.
**2019**, 12, 124–130. [Google Scholar] - Schijve, J. Fatigue of Structures and Materials; Springer: Berlin/Heidelberg, Germany, 2009; p. 622. [Google Scholar]
- Draper, J. Modern Metal Fatigue Analysis; EMAS: Brussel, Belgium, 2008; p. 290. [Google Scholar]
- Kim, H.S. Mechanics of Solids and Fracture, 3rd ed.; Bookboon: London, UK, 2018; p. 223. [Google Scholar]
- Stepnov, M.N.; Naumkin, A.C. A computational and experimental method for plotting multicycle fatigue curves for structural elements with concentrated stress. J. Mach. Manuf. Reliab.
**2012**, 41, 34–38. [Google Scholar] [CrossRef] - McKeen, L. Fatigue and Tribological Properties of Plastics and Elastomers, 2nd ed.; William Andrew: Norwich, NY, USA, 2009; p. 463. [Google Scholar]
- Jäntschi, L. Structure—Property relationships for solubility of monosaccharides. Appl. Water Sci.
**2019**, 9, 38. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Ultimate tensile strength (ordinate in Newtons) of polypropylene fibers with TiO${}_{2}$ photocatalyst nanoparticles as a function of irradiation time (abscissa in minutes). The asterisks on the graph are the model data, and the curved line is the results of the approximation of the model data by Equation (10), whose coefficients are obtained by the Monte Carlo method.

**Figure 2.**Ultimate tensile strength (ordinate in Newtons) of polyester fibers with TiO${}_{2}$ as a function of irradiation time (abscissa in minutes). The asterisks on the graph are the model data, and the curved line is the results of the approximation of the model data by Equation (12), the coefficients of which are obtained by the Monte Carlo method.

**Figure 3.**Ultimate tensile strength (ordinate in Newtons) of polyamide fibers with TiO${}_{2}$ as a function of irradiation time (abscissa in minutes). The asterisks on the graph are the model data, and the curved line is the results of the approximation of the model data by the Equation (14), the coefficients of which are obtained by the Monte Carlo method.

**Table 1.**Ultimate tensile strength (UTS) value (in Newtons) of polypropylene fibers as a function of irradiation time (in minutes). Values of UTS are digitized graphs from the paper by Wiener et al. [22].

Minutes | UTS without TiO${}_{2}$ | UTS with TiO${}_{2}$ |
---|---|---|

0 | $24.55$ | $25.00$ |

30 | $24.55$ | $24.77$ |

60 | $24.50$ | $24.53$ |

90 | $24.45$ | $24.30$ |

120 | $24.30$ | $23.82$ |

150 | $24.05$ | $23.34$ |

180 | $23.80$ | $22.87$ |

210 | $23.45$ | $22.39$ |

240 | $22.90$ | $21.91$ |

270 | $22.35$ | $21.41$ |

300 | $21.80$ | $20.96$ |

330 | $21.33$ | $20.47$ |

360 | $20.86$ | $20.00$ |

390 | $20.39$ | $19.70$ |

420 | $19.92$ | $19.40$ |

450 | $19.45$ | $19.10$ |

480 | $18.92$ | $18.80$ |

510 | $18.40$ | $18.50$ |

540 | $17.85$ | $18.20$ |

570 | $17.20$ | $17.90$ |

600 | $16.60$ | $17.60$ |

**Table 2.**Ultimate tensile strength (UTS) value (in Newtons) of polyester fibers as a function of irradiation time (in minutes). Value of UTS are digitized graphs from the paper by Wiener et al. [22].

Minutes | UTS without TiO${}_{2}$ | UTS with TiO${}_{2}$ |
---|---|---|

0 | $9.250$ | $7.450$ |

30 | $9.000$ | $7.400$ |

60 | $8.785$ | $7.350$ |

90 | $8.550$ | $7.300$ |

120 | $8.325$ | $7.225$ |

150 | $8.100$ | $7.150$ |

180 | $7.990$ | $7.075$ |

210 | $7.880$ | $7.010$ |

240 | $7.770$ | $6.910$ |

270 | $7.660$ | $6.810$ |

300 | $7.550$ | $6.710$ |

330 | $7.470$ | $6.610$ |

360 | $7.390$ | $6.510$ |

390 | $7.310$ | $6.410$ |

420 | $7.240$ | $6.310$ |

450 | $7.240$ | $6.200$ |

480 | $7.240$ | $6.090$ |

510 | $7.239$ | $5.980$ |

540 | $7.239$ | $5.870$ |

570 | $7.239$ | $5.760$ |

600 | $7.239$ | $5.650$ |

**Table 3.**Ultimate tensile strength (UTS) value (in Newtons) of polyamide fibers as a function of irradiation time (in minutes). Values of UTS are digitized graphs from the paper by Wiener et al. [22].

Minutes | UTS without TiO${}_{2}$ | UTS with TiO${}_{2}$ |
---|---|---|

0 | $3.250$ | $3.450$ |

30 | $3.500$ | $3.650$ |

60 | $3.750$ | $3.825$ |

90 | $3.900$ | $3.950$ |

120 | $4.050$ | $4.100$ |

150 | $4.200$ | $4.200$ |

180 | $4.300$ | $4.300$ |

210 | $4.400$ | $4.375$ |

240 | $4.505$ | $4.400$ |

270 | $4.600$ | $4.375$ |

300 | $4.650$ | $4.300$ |

330 | $4.700$ | $4.250$ |

360 | $4.700$ | $4.150$ |

390 | $4.650$ | $4.100$ |

420 | $4.600$ | $3.975$ |

450 | $4.550$ | $3.850$ |

480 | $4.500$ | $3.725$ |

510 | $4.445$ | $3.550$ |

540 | $4.375$ | $3.400$ |

570 | $4.300$ | $3.150$ |

600 | $4.200$ | $2.900$ |

**Table 4.**Table of experimental parameters with confidence intervals for phenomenological equations of polypropylene, polyester, and polyamide.

Equation | $\mathit{\alpha}$ | a | b | c |
---|---|---|---|---|

(10) | $1.2696\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.0995$ | $0.1212\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.0574$ | $5.8923\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.6032$ | $25.3581\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}5.4543$ |

(12) | $2.4885\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.4311$ | $0.3676\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.0151$ | $3.8004\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.1181$ | $6.3235\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.8956$ |

(14) | $17.3862\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.5428$ | $1.2817\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.0138$ | $1.0190\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}0.0174$ | $187.5523\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{0.277778em}{0ex}}8.7477$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Orekhov, A.V.; Artemev, Y.M.; Pavilaynen, G.V.
The Mathematical Simulation for the Photocatalytic Fatigue of Polymer Nanocomposites Using the Monte Carlo Methods. *Mathematics* **2022**, *10*, 1613.
https://doi.org/10.3390/math10091613

**AMA Style**

Orekhov AV, Artemev YM, Pavilaynen GV.
The Mathematical Simulation for the Photocatalytic Fatigue of Polymer Nanocomposites Using the Monte Carlo Methods. *Mathematics*. 2022; 10(9):1613.
https://doi.org/10.3390/math10091613

**Chicago/Turabian Style**

Orekhov, Andrey V., Yurii M. Artemev, and Galina V. Pavilaynen.
2022. "The Mathematical Simulation for the Photocatalytic Fatigue of Polymer Nanocomposites Using the Monte Carlo Methods" *Mathematics* 10, no. 9: 1613.
https://doi.org/10.3390/math10091613