# The Fate of Molecular Species in Water Layers in the Light of Power-Law Time-Dependent Diffusion Coefficient

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

_{2}O

_{2}, HO

_{2}, NO

_{3}

^{-}, and NO

_{2}

^{-}via the calculation of the mean square displacement using the two methods.

## 1. Introduction

## 2. Numerical Modelling Using COMSOL Multiphysics

## 3. Anomalous Diffusion of Molecular Species: Fractional Approach

## 4. Results

_{2}O

_{2}, and HO

_{2}to study their behaviour in water layers and compare it to the results of MD simulation published before [11], where we used the following diffusion coefficients: 0.837, 0.134, and 0.068 Å${}^{2}$/ps for OH, H

_{2}O

_{2}, and HO

_{2}, respectively, that were used in the mentioned work. The numerical and analytical were solved using these values, and consequently, we calculated the MSD (see Figure 7). As it can be seen clearly from Figure 7, the diffusivity of the H

_{2}O

_{2}molecule is in between OH and HO

_{2}, and these results correspond well with the published data using the MD simulation [11]. Moreover, to test the validity of our models for other molecular species, we chose two more species (for their importance in plasma medicine): nitrite (NO

_{3}

^{-}) and nitrate (NO

_{2}

^{-}); they are common nitrogen oxyanions that constitute the basic forms of nitric (HNO

_{3}) and nitrous (HNO

_{2}) acids. To calculate MSD for NO

_{3}

^{-}and NO

_{2}

^{-}, we used the diffusion coefficients in a recently published MD simulation results as 0.3 and 0.27Å${}^{2}$/ps, respectively [38]. The MSD using both models can be seen clearly in Figure 8. However, though the values of diffusion coefficients for NO

_{3}

^{-}and NO

_{2}

^{-}are very close to each other, our models were able to distinguish between their MSD, as can be seen from Figure 8; a high similarity can also be observed between the numerical and analytical models.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Laroussi, M. Plasma medicine: A brief introduction. Plasma
**2018**, 1, 5. [Google Scholar] [CrossRef][Green Version] - Fridman, G.; Friedman, G.; Gutsol, A.; Shekhter, A.B.; Vasilets, V.N.; Fridman, A. Applied plasma medicine. Plasma Process. Polym.
**2008**, 5, 503–533. [Google Scholar] [CrossRef] - Adamovich, I.; Baalrud, S.; Bogaerts, A.; Bruggeman, P.; Cappelli, M.; Colombo, V.; Czarnetzki, U.; Ebert, U.; Eden, J.; Favia, P.; et al. The 2017 Plasma Roadmap: Low temperature plasma science and technology. J. Phys. D Appl. Phys.
**2017**, 50, 323001. [Google Scholar] [CrossRef] - Samukawa, S.; Hori, M.; Rauf, S.; Tachibana, K.; Bruggeman, P.; Kroesen, G.; Whitehead, J.C.; Murphy, A.B.; Gutsol, A.F.; Starikovskaia, S.; et al. The 2012 plasma roadmap. J. Phys. D Appl. Phys.
**2012**, 45, 253001. [Google Scholar] [CrossRef] - Yokoyama, T.; Kogoma, M.; Kanazawa, S.; Moriwaki, T.; Okazaki, S. The improvement of the atmospheric-pressure glow plasma method and the deposition of organic films. J. Phys. D Appl. Phys.
**1990**, 23, 374. [Google Scholar] [CrossRef] - Fridman, A. Plasma Chemistry; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- El-Kalliny, A.S.; Abd-Elmaksoud, S.; El-Liethy, M.A.; Abu Hashish, H.M.; Abdel-Wahed, M.S.; Hefny, M.M.; Hamza, I.A. Efficacy of Cold Atmospheric Plasma Treatment on Chemical and Microbial Pollutants in Water. ChemistrySelect
**2021**, 6, 3409–3416. [Google Scholar] [CrossRef] - Hefny, M.M.; Nečas, D.; Zajíčková, L.; Benedikt, J. The transport and surface reactivity of O atoms during the atmospheric plasma etching of hydrogenated amorphous carbon films. Plasma Sources Sci. Technol.
**2019**, 28, 035010. [Google Scholar] [CrossRef] - Kaushik, N.K.; Kaushik, N.; Linh, N.N.; Ghimire, B.; Pengkit, A.; Sornsakdanuphap, J.; Lee, S.J.; Choi, E.H. Plasma and nanomaterials: Fabrication and biomedical applications. Nanomaterials
**2019**, 9, 98. [Google Scholar] [CrossRef][Green Version] - Benedikt, J.; Hefny, M.M.; Shaw, A.; Buckley, B.; Iza, F.; Schäkermann, S.; Bandow, J. The fate of plasma-generated oxygen atoms in aqueous solutions: Non-equilibrium atmospheric pressure plasmas as an efficient source of atomic O (aq). Phys. Chem. Chem. Phys.
**2018**, 20, 12037–12042. [Google Scholar] [CrossRef][Green Version] - Yusupov, M.; Neyts, E.; Simon, P.; Berdiyorov, G.; Snoeckx, R.; Van Duin, A.; Bogaerts, A. Reactive molecular dynamics simulations of oxygen species in a liquid water layer of interest for plasma medicine. J. Phys. D Appl. Phys.
**2013**, 47, 025205. [Google Scholar] [CrossRef][Green Version] - Bogaerts, A.; Yusupov, M.; Razzokov, J.; Van der Paal, J. Plasma for cancer treatment: How can RONS penetrate through the cell membrane? Answers from computer modeling. Front. Chem. Sci. Eng.
**2019**, 13, 253–263. [Google Scholar] [CrossRef] - Amhamed, A.; Atilhan, M.; Berdiyorov, G. Permeabilities of CO2, H2S and CH4 through choline-based ionic liquids: Atomistic-scale simulations. Molecules
**2019**, 24, 2014. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rais, D.; Menšik, M.; Paruzel, B.; Toman, P.; Pfleger, J. Concept of the time-dependent diffusion coefficient of polarons in organic semiconductors and its determination from time-resolved spectroscopy. J. Phys. Chem. C
**2018**, 122, 22876–22883. [Google Scholar] [CrossRef] - Cherstvy, A.G.; Safdari, H.; Metzler, R. Anomalous diffusion, nonergodicity, and ageing for exponentially and logarithmically time-dependent diffusivity: Striking differences for massive versus massless particles. J. Phys. D Appl. Phys.
**2021**, 54, 195401. [Google Scholar] [CrossRef] - Crank, J. The Mathematics of Diffusion; Oxford University Press: Oxford, UK, 1979. [Google Scholar]
- Garra, R.; Giusti, A.; Mainardi, F. The fractional Dodson diffusion equation: A new approach. Ric. Mat.
**2018**, 67, 899–909. [Google Scholar] [CrossRef][Green Version] - Tawfik, A.M.; Hefny, M.M. Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach. Fractal Fract.
**2021**, 5, 51. [Google Scholar] [CrossRef] - Hefny, M.M.; Pattyn, C.; Lukes, P.; Benedikt, J. Atmospheric plasma generates oxygen atoms as oxidizing species in aqueous solutions. J. Phys. D Appl. Phys.
**2016**, 49, 404002. [Google Scholar] [CrossRef] - Batchelor, G. Diffusion in a field of homogeneous turbulence: II. The relative motion of particles. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1952; Volume 48, pp. 345–362. [Google Scholar]
- Nagy, Á.; Omle, I.; Kareem, H.; Kovács, E.; Barna, I.F.; Bognar, G. Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation. Computation
**2021**, 9, 92. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. From a generalized chapman- kolmogorov equation to the fractional klein- kramers equation. J. Phys. Chem. B
**2000**, 104, 3851–3857. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] [CrossRef] - Evangelista, L.R.; Lenzi, E.K. Fractional Diffusion Equations and Anomalous Diffusion; Cambridge University Press: Cambridge, UK, 2018. [Google Scholar]
- Bologna, M.; Svenkeson, A.; West, B.J.; Grigolini, P. Diffusion in heterogeneous media: An iterative scheme for finding approximate solutions to fractional differential equations with time-dependent coefficients. J. Comput. Phys.
**2015**, 293, 297–311. [Google Scholar] [CrossRef] - Fa, K.S.; Lenzi, E. Time-fractional diffusion equation with time dependent diffusion coefficient. Phys. Rev. E
**2005**, 72, 011107. [Google Scholar] [CrossRef] [PubMed] - Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 44, 460–481. [Google Scholar] [CrossRef][Green Version] - Tawfik, A.M.; Abdelhamid, H.M. Generalized fractional diffusion equation with arbitrary time varying diffusivity. Appl. Math. Comput.
**2021**, 410, 126449. [Google Scholar] [CrossRef] - Rabanimehr, F.; Farhadian, M.; Solaimany Nazar, A.R.; Behineh, E.S. Simulation of photocatalytic degradation of methylene blue in planar microreactor with integrated ZnO nanowires. J. Appl. Res. Water Wastewater
**2021**, 8, 36–40. [Google Scholar] - Janczarek, M.; Kowalska, E. Computer simulations of photocatalytic reactors. Catalysts
**2021**, 11, 198. [Google Scholar] [CrossRef] - Hasanpour, M.; Motahari, S.; Jing, D.; Hatami, M. Numerical modeling for the photocatalytic degradation of methyl orange from aqueous solution using cellulose/zinc oxide hybrid aerogel: Comparison with experimental data. Top. Catal.
**2021**, 1–14. [Google Scholar] [CrossRef] - Klafter, J.; Silbey, R. Derivation of the continuous-time random-walk equation. Phys. Rev. Lett.
**1980**, 44, 55. [Google Scholar] [CrossRef][Green Version] - Lutz, E. Fractional langevin equation. In Fractional Dynamics: Recent Advances; World Scientific: Singapore, 2012; pp. 285–305. [Google Scholar]
- Gorenflo, R.; Kilbas, A.A.; Mainardi, F.; Rogosin, S.V. Mittag-Leffler Functions, Related Topics and Applications; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Mainardi, F.; Pagnini, G. The Wright functions as solutions of the time-fractional diffusion equation. Appl. Math. Comput.
**2003**, 141, 51–62. [Google Scholar] [CrossRef][Green Version] - Wang, K.; Lung, C. Long-time correlation effects and fractal Brownian motion. Phys. Lett. A
**1990**, 151, 119–121. [Google Scholar] [CrossRef] - Wang, W.; Cherstvy, A.G.; Liu, X.; Metzler, R. Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise. Phys. Rev. E
**2020**, 102, 012146. [Google Scholar] [CrossRef] [PubMed] - Cordeiro, R.M.; Yusupov, M.; Razzokov, J.; Bogaerts, A. Parametrization and molecular dynamics simulations of nitrogen oxyanions and oxyacids for applications in atmospheric and biomolecular sciences. J. Phys. Chem. B
**2020**, 124, 1082–1089. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Model geometry presents the extremely fine mesh for water reactor, and (

**b**) the initial density profile using Dirac delta function.

**Figure 2.**A 2D colour-coded map represents the propagation of molecular species in water layers at (

**a**) time = 0 s and (

**b**) time = 1 s.

**Figure 3.**Density profiles of molecular species through y-axis for different times at $\gamma =0.25$ using the numerical method.

**Figure 5.**Density profiles of molecular species for different times at $\gamma =0.25$ and $\alpha =1$ using the analytical method.

**Figure 6.**MSD using the analytical method for different values of $\gamma $ in the case of $\alpha =1$.

**Figure 7.**MSD for the selected molecular species: OH, H

_{2}O

_{2}, and HO

_{2}using the numerical method (left panel) and analytical method (right panel).

**Figure 8.**MSD for NO

_{3}

^{-}and NO

_{2}

^{-}using the numerical method (left panel) and analytical method (right panel).

**Figure 9.**MSD for different values of $\gamma $ in the case of $\alpha =1/2$ (left panel) with the double logarithmic plot (right panel).

Element Type | Number |
---|---|

Triangles | 5034 |

Edge elements | 240 |

Vertex element | 4 |

Number of elements | 5034 |

Element area ratio | 0.3717 |

Mesh area | 5 m${}^{2}$ |

Average element quality | 0.9457 |

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**MDPI and ACS Style**

Hefny, M.M.; Tawfik, A.M.
The Fate of Molecular Species in Water Layers in the Light of Power-Law Time-Dependent Diffusion Coefficient. *Symmetry* **2022**, *14*, 1146.
https://doi.org/10.3390/sym14061146

**AMA Style**

Hefny MM, Tawfik AM.
The Fate of Molecular Species in Water Layers in the Light of Power-Law Time-Dependent Diffusion Coefficient. *Symmetry*. 2022; 14(6):1146.
https://doi.org/10.3390/sym14061146

**Chicago/Turabian Style**

Hefny, Mohamed Mokhtar, and Ashraf M. Tawfik.
2022. "The Fate of Molecular Species in Water Layers in the Light of Power-Law Time-Dependent Diffusion Coefficient" *Symmetry* 14, no. 6: 1146.
https://doi.org/10.3390/sym14061146