# Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{2}, as one of the most important plasma-generated reactive species, was reported to be produced by an argon-based radio frequency plasma jet and then diffused to the aqueous phase [11]. Moreover Wende et al. reported that plasma-generated atomic oxygen can initiate chemical reactions in a standard physiological buffer or liquid via reactions with chloride, yielding Cl

_{2}

^{−}or ClO

^{−}. Gorbanev et al. [12] reported that, by using isotopes, H

_{2}O

_{2}, O

_{2}

^{−}, OH, OOH, and H species in the plasma-treated liquid originate from the gas phase. Lukes et al. also evidenced the generation of some ROS like ONOOH, NO

_{2}, NO, and OH radicals and NO

^{+}by the air plasma at the gas–liquid interface because of post-discharge processes in plasma-activated liquid [13]. The plasma-generated reactive species can affect many biological targets directly (direct exposure to CAPs) or indirectly through plasma-activated liquid (PAL), where they have a prolonged effect on biological targets ranging from minutes and can reach several days after treatment with plasma [4,14,15,16,17].

^{1}8O

_{2}isotope, the ${O}_{(aqueous)}$ atoms react directly with phenol molecules, without dissociating water molecules in intermediate reactions [9]. Additionally, a 2D axially symmetric fluid model of the gas flow and diffusion-convective transport of plasma-generated reactive species combined with the basic kinetic model of the reaction of O atoms with O

_{2}molecules has been solved to study the transport and surface reactivity of O atoms.

_{2}and H

_{2}O

_{2}). They also illustrated different spatial distributions for three kinds of molecular species in the water layer. They found that not all species have the same distribution behavior (classical Gaussian behavior), and the values of the diffusion coefficients are not the same for all species. That means diffusivity is not the same for every molecular species, especially for O radical, which cannot travel deep into the liquid layer (slow-diffusion). Therefore, we will consider a new treatment by suggesting the case of subdiffusion, which is more suitable to verify the non-Gaussian distribution and cover all classes of diffusivity (classical and anomalous). Anomalous diffusion is described by non-Gaussian statistics, which encompasses probability densities with power-law tails [27,28,29]. The mean square displacement of a diffusive species is characterized by $\langle {x}^{2}(t)\rangle \sim {t}^{\gamma}$ (non-linear with time). The process is called sub-diffusive when $0<\gamma <1$, while it is called super-diffusive for $1<\gamma <2$.

## 2. Fractional Reaction–Telegraph Model

## 3. Discussion

_{2}O

_{2}and HO

_{2}. That is due to the contribution of the time-fractional order, which shows fatter tails than the reference case of the Gaussian distribution in the case of OH. This agrees with Yusupov et al. [26], who concluded that the OH radicals are more rapidly transported than the other two species, and this was validated through calculating the MSD and diffusion coefficients for these species as well.

_{2}O

_{2}molecules travel without reaction faster than HO

_{2}and slower than OH; however, both form a reaction through traveling in the water layer. Therefore, we can study the typical behavior of mean square displacement in the case of neglecting the reaction terms until the time-fractional order and relaxation effects can be demonstrated. In this case, the mean square displacement of the fractional telegraph equation can be expressed by the form:

_{2}O

_{2}because it moves slowly inside the water layer and almost reaches the biological targets. Besides that, the O species already failed to reach the target, and we suggest that this is because of its slow spread (sub-diffusive) with long waiting times inside the liquid layer. Therefore, it is more convenient to describe the motion of different molecular species to introduce the time-fractional order to switch between the normal and slow diffusion. In addition, the relaxation time plays an essential role in regulating the speed of propagation for each molecule and thus covers all forms of transport inside the liquid layer. From this point of view in our brief discussion, we believe now that it is essential to work with the fractional reaction–telegraph instead of the classical reaction–diffusion built up in many numerical simulations.

## 4. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Fridman, A. Plasma Chemistry; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Kong, M.G.; Kroesen, G.; Morfill, G.; Nosenko, T.; Shimizu, T.; Van Dijk, J.; Zimmermann, J. Plasma medicine: An introductory review. New J. Phys.
**2009**, 11, 115012. [Google Scholar] [CrossRef] - Laroussi, M.; Kong, M.; Morfill, G. Plasma Medicine: Applications of Low-Temperature Gas Plasmas in Medicine and Biology; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Hefny, M.M.; Kairo, A. Experimental Study of Cold Atmospheric Plasma for Plasma Medicine Research and Applications. Ph.D. Thesis, Ruhr-Universität Bochum, Bochum, Germany, 2019. [Google Scholar]
- 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] - Bekeschus, S.; Wende, K.; Hefny, M.M.; Rödder, K.; Jablonowski, H.; Schmidt, A.; von Woedtke, T.; Weltmann, K.D.; Benedikt, J. Oxygen atoms are critical in rendering THP-1 leukaemia cells susceptible to cold physical plasma-induced apoptosis. Sci. Rep.
**2017**, 7, 1–12. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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.; 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] - Benedikt, J.; Hefny, M.M.; Shaw, A.; Buckley, B.R.; 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] - Blackert, S.; Haertel, B.; Wende, K.; von Woedtke, T.; Lindequist, U. Influence of non-thermal atmospheric pressure plasma on cellular structures and processes in human keratinocytes (HaCaT). J. Dermatol. Sci.
**2013**, 70, 173–181. [Google Scholar] [CrossRef] - Wende, K.; Straßenburg, S.; Haertel, B.; Harms, M.; Holtz, S.; Barton, A.; Masur, K.; von Woedtke, T.; Lindequist, U. Atmospheric pressure plasma jet treatment evokes transient oxidative stress in HaCaT keratinocytes and influences cell physiology. Cell Biol. Int.
**2014**, 38, 412–425. [Google Scholar] [CrossRef] - Gorbanev, Y.; O’Connell, D.; Chechik, V. Non-thermal plasma in contact with water: The origin of species. Chem. A Eur. J.
**2016**, 22, 3496–3505. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lukes, P.; Dolezalova, E.; Sisrova, I.; Clupek, M. Aqueous-phase chemistry and bactericidal effects from an air discharge plasma in contact with water: Evidence for the formation of peroxynitrite through a pseudo-second-order post-discharge reaction of H
_{2}O_{2}and HNO_{2}. Plasma Sources Sci. Technol.**2014**, 23, 015019. [Google Scholar] [CrossRef] - Duan, J.; Ma, M.; Yusupov, M.; Cordeiro, R.M.; Lu, X.; Bogaerts, A. The penetration of reactive oxygen and nitrogen species across the stratum corneum. Plasma Process. Polym.
**2020**, 17, 2000005. [Google Scholar] [CrossRef] - Yusupov, M.; Razzokov, J.; Cordeiro, R.M.; Bogaerts, A. Transport of reactive oxygen and nitrogen species across aquaporin: A molecular level picture. Oxidative Med. Cell. Longev.
**2019**, 2019. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Van der Paal, J.; Hong, S.H.; Yusupov, M.; Gaur, N.; Oh, J.S.; Short, R.D.; Szili, E.J.; Bogaerts, A. How membrane lipids influence plasma delivery of reactive oxygen species into cells and subsequent DNA damage: An experimental and computational study. Phys. Chem. Chem. Phys.
**2019**, 21, 19327–19341. [Google Scholar] [CrossRef] - Schneider, S.; Lackmann, J.W.; Ellerweg, D.; Denis, B.; Narberhaus, F.; Bandow, J.E.; Benedikt, J. The role of VUV radiation in the inactivation of bacteria with an atmospheric pressure plasma jet. Plasma Process. Polym.
**2012**, 9, 561–568. [Google Scholar] [CrossRef] [Green Version] - 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] - Sene, N. Fractional model for a class of diffusion-reaction equation represented by the fractional-order derivative. Fractal Fract.
**2020**, 4, 15. [Google Scholar] [CrossRef] - Hadeler, K.P. Reaction transport systems in biological modelling. In Mathematics Inspired by Biology; Springer: New York, NY, USA, 1999; pp. 95–150. [Google Scholar]
- Boon, J.P.; Lutsko, J.F.; Lutsko, C. Microscopic approach to nonlinear reaction-diffusion: The case of morphogen gradient formation. Phys. Rev. E
**2012**, 85, 021126. [Google Scholar] [CrossRef] [Green Version] - Evans, M.R.; Majumdar, S.N. Diffusion with stochastic resetting. Phys. Rev. Lett.
**2011**, 106, 160601. [Google Scholar] [CrossRef] [Green Version] - dos Santos, M.A. Fractional Prabhakar derivative in diffusion equation with non-static stochastic resetting. Physics
**2019**, 1, 40–58. [Google Scholar] [CrossRef] [Green Version] - Mendez, V.; Fedotov, S.; Horsthemke, W. Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities; Springer Science & Business Media: Berlin, Germany, 2010. [Google Scholar]
- Alharbi, W.; Petrovskii, S. Critical domain problem for the reaction–telegraph equation model of population dynamics. Mathematics
**2018**, 6, 59. [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] - Dorea, C.C.; Medino, A.V. Anomalous diffusion index for Lévy motions. J. Stat. Phys.
**2006**, 123, 685–698. [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] - Antonio Faustino dos Santos, M. Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion. Fractal Fract.
**2020**, 4, 28. [Google Scholar] [CrossRef] - Klages, R.; Radons, G.; Sokolov, I.M. Anomalous Transport: Foundations and Applications; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Datsko, B.; Kutniv, M.; Włoch, A. Mathematical modelling of pattern formation in activator–inhibitor reaction—Diffusion systems with anomalous diffusion. J. Math. Chem.
**2020**, 58, 612–631. [Google Scholar] [CrossRef] - Compte, A.; Metzler, R. The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A Math. Gen.
**1997**, 30, 7277. [Google Scholar] [CrossRef] - Dos Santos, M.A. Non-Gaussian distributions to random walk in the context of memory kernels. Fractal Fract.
**2018**, 2, 20. [Google Scholar] [CrossRef] [Green Version] - Sandev, T.; Tomovski, Ž. Fractional Equations and Models; Springer: New York, NY, USA, 2019. [Google Scholar]
- Lenzi, E.K.; Evangelista, L.R.; Zola, R.S.; Petreska, I.; Sandev, T. Fractional Schrödinger equation and anomalous relaxation: Nonlocal terms and delta potentials. Mod. Phys. Lett. A
**2021**, 36, 2140004. [Google Scholar] [CrossRef] - Baleanu, D.; Jassim, H.K. Exact Solution of Two-Dimensional Fractional Partial Differential Equations. Fractal Fract.
**2020**, 4, 21. [Google Scholar] [CrossRef] - Tawfik, A.M. On fractional approximations of the Fokker–Planck equation for energetic particle transport. Eur. Phys. J. Plus
**2020**, 135, 1–19. [Google Scholar] [CrossRef] - Tawfik, A.M.; Abdou, M.; Gepreel, K.A. An analytical solution of the time-fractional telegraph equation describing neutron transport in a nuclear reactor. Indian J. Phys.
**2021**, 1–6. [Google Scholar] [CrossRef] - Klafter, J.; Sokolov, I.M. First Steps in Random Walks: From Tools to Applications; Oxford University Press: Oxford, UK, 2011. [Google Scholar]
- Chechkin, A.; Gorenflo, R.; Sokolov, I. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E
**2002**, 66, 046129. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Caputo, M. Linear models of dissipation whose Q is almost frequency independent—II. Geophys. J. Int.
**1967**, 13, 529–539. [Google Scholar] [CrossRef] - Huang, F.; Liu, F. The space-time fractional diffusion equation with Caputo derivatives. J. Appl. Math. Comput.
**2005**, 19, 179. [Google Scholar] [CrossRef] [Green Version] - Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Academic Press: Cambridge, MA, USA, 1998; Volume 198. [Google Scholar]
- Zimbardo, G.; Perri, S.; Effenberger, F.; Fichtner, H. Fractional Parker equation for the transport of cosmic rays: Steady-state solutions. Astron. Astrophys.
**2017**, 607, A7. [Google Scholar] [CrossRef] - Samko, S.; Kilbas, A.; Marichev, O. Fractional Integrals and Derivatives—The Ory and Appli Cations; Gordon and Breach: Longhorne, PA, USA, 1993. [Google Scholar]
- Górska, K.; Horzela, A.; Lenzi, E.; Pagnini, G.; Sandev, T. Generalized Cattaneo (telegrapher’s) equations in modeling anomalous diffusion phenomena. Phys. Rev. E
**2020**, 102, 022128. [Google Scholar] [CrossRef] - Halliwell, B.; Gutteridge, J.M. Free Radicals in Biology and Medicine; Oxford University Press: Oxford, UK, 2015. [Google Scholar]
- Mainardi, F. Why the Mittag-Leffler function can be considered the Queen function of the Fractional Calculus? Entropy
**2020**, 22, 1359. [Google Scholar] [CrossRef] - Wright, E.M. On the coefficients of power series having exponential singularities. J. Lond. Math. Soc.
**1933**, 1, 71–79. [Google Scholar] [CrossRef] - Mainardi, F.; Mura, A.; Pagnini, G. The M-Wright function in time-fractional diffusion processes: A tutorial survey. Int. J. Differ. Equations
**2010**, 2010. [Google Scholar] [CrossRef] [Green Version] - Gorenflo, R.; Kilbas, A.A.; Mainardi, F.; Rogosin, S.V. Mittag-Leffler Functions, Related Topics and Applications; Springer: New York, NY, USA, 2014. [Google Scholar]

**Figure 1.**Schematic figure of reactive oxygen species (ROS) interaction with a liquid layer: a representation of molecular species transport in one dimension (semi-finite domain).

**Figure 2.**Spatial distribution along the z-axis in the case of $\alpha =1$ (

**left panel**) and $\alpha =1/2$ (

**right panel**) for different values of relaxation time $\tau $ at $\kappa =1$, $p-r=0.1$ and $t=1$.

**Figure 3.**Spatial distribution along the z-axis in the case of $\alpha =1$ (

**left panel**) and $\alpha =1/2$ (

**right panel**) for different values of reaction rate $p-r$ at $\kappa =1$, $\tau =0.2$ and $t=1$.

**Figure 4.**MSD in the case of $\alpha =1$ (

**left panel**) and $\alpha =1/2$ (

**right panel**) with different values of relaxation time $\tau $ at $\kappa =1$.

**Figure 5.**MSD in the case of $\tau =0.5$ (

**left panel**) and $\tau =0$ (

**right panel**) for different values of the time fractional order at $\kappa =1$.

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**MDPI and ACS Style**

Tawfik, A.M.; Hefny, M.M.
Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach. *Fractal Fract.* **2021**, *5*, 51.
https://doi.org/10.3390/fractalfract5020051

**AMA Style**

Tawfik AM, Hefny MM.
Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach. *Fractal and Fractional*. 2021; 5(2):51.
https://doi.org/10.3390/fractalfract5020051

**Chicago/Turabian Style**

Tawfik, Ashraf M., and Mohamed Mokhtar Hefny.
2021. "Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach" *Fractal and Fractional* 5, no. 2: 51.
https://doi.org/10.3390/fractalfract5020051