# A mini-review on release oscillation in a hollow fiber

^{1}Ningbo Advanced Textile Technology & Fashion CAD Key Laboratory, Zhejiang Fashion Institute of Technology, Ningbo, China^{2}Zhejiang Sci-Tech University, College of Textile Science and Engineering (International Silk College), Hangzhou, China^{3}Laimei Technology Co., Ltd., Changxing, Huzhou, Zhejiang, China

This mini-review aims at strengthening the links among textile science, physics, and mathematics. The state-of-the-art technology for silver ions’ release from hollow fibers is reviewed, its bottleneck problems are identified, and some open problems are elucidated. The release oscillation opens a new era for modern applications of hollow fibers containing silver ions.

## 1 Introduction

Hollow fibers have obvious advantages in that they are low density and have good flexibility. Natural hollow fibers have even more amazing properties, for example, polar bear hairs have remarkable thermal properties [1, 2]. Wang et al. elucidated the biomechanism of the hollow hair of the polar bear using the fractal calculus with great success [3], Cui et al. designed a biomimetic textile with good thermal insulation [4], and Liu et al. found a new phenomenon of thermal oscillation in the thermal insulation [5]. Hollow-fiber liquid-phase microextractionare is highly efficient for extraction of heavy metals and pharmaceuticals [6–8]. The corresponding solvent, which should be of low polarity and immiscible with water, is immobilized in the pores in the wall of hollow fibers and serves as a supported liquid membrane. A larger number of reports have been published on the development of hollow fibers as a green sample preparation technique requiring only a few microliters of organic solvent per sample. Due to the protection of the acceptor phase by the supported liquid membrane, hollow fibers are amenable to highly complex samples such as plasma, whole blood, urine, saliva, breast milk, tap water, surface water, pond water, seawater, and soil slurries [9].

The physical process of hollow fiber spinning always involves four steps: solution formulation, extrusion, coagulation, and coagulated fiber treatment [10]. Thus far, the electrospinning technique has been considered as a versatile and efficient method for the fabrication of membranes with highly interconnected pore structures [11]. The flexibility of device construction for electrospinning and the diversity of the post-treatment process to electrospun membrane leaves vast scope for researchers to tailor the membrane structures and properties; thus, polymeric nano-scale hollow fibers via electrospinning technology have become popular, for example, bubble electrospinning might be a good candidate for hollow fiber fabrication [12, 13].

This paper focuses on artificial hollow fibers containing silver particles [14, 15], with an emphasis on the release oscillation [16–18].

## 2 Antibacterial mechanism

Hollow fibers containing silver ions are widely used for antibacterial and antifouling applications [19, 20]; when the fibers are submerged in water, silver ions are gradually released from the inner wall into the water.

Viruses and bacteria are generally quite small [21–23]. In particular, some deadly viruses (e.g., the COVID-19 virus) have a complex unsmooth surface, and a small surface has high surface energy (geometric potential) [24–26], which can easily absorb silver nanoparticles around the surface. The absorbed nanoparticles make viruses and bacteria inactive [24].

On the other hand, silver ions react with water when ions are adhered to the surface of bacteria or viruses:

Bacteria and viruses will be killed due to their exsiccosis and hydrogen ions can react with macromolecules, which is the mechanism of the antibacterial property of the hollow fibers containing silver ions. Of course, however, a high concentration of silver ions will be also harmful to human cells.

## 3 Capillary effect and diffusion process

The inner diameter of hollow fibers greatly affects the ions release. A smaller diameter implies a higher capillary rise [27, 28]; as a result, more ions can dissolve in water and the diffusion process makes the ions release into the outside of the hollow fiber. Han and He unlocked the secret of hollow fibers’ antifouling properties using the capillary effect [29]. Environmental temperature and saline water will affect the ions’ diffusion process [30–33].

Though hollow fibers with thinner diameters have better capillary effect, the corresponding fabrication needs more costs, meanwhile, the inner wall surface area is less, so there are less loaded ions. The effects of the temperature on the diffusion process and viruses and bacteria’s metabolism should also be considered, as well as additionally the nanofluid mechanics [34–37] being of paramount importance in studying the optimal design of the hollow fiber’s geometrical structure and its effect on its antibacterial properties.

## 4 Release oscillation and frequency property

Due to environmental perturbation, the water in hollow fibers is vibrated periodically, the mechanism of which was first found in [17]. The vibrating water accelerates the release process; however, the non-linear vibrations make it difficult to predict its frequency properties. The governing equation can be expressed as [17].

with initial conditions

where *u* is the capillary rise, a, b, and *A* is the initial velocity.

Solving Eq. 2 effectively is still an open problem. The possible methods to solve Eq. 2 with the initial conditions of Eq. 3 include mainly the homotopy perturbation method [38, 39], the Li-He method [40–42], frequency-amplitude formulation [43], and the variational principle [44].

For u, Eq. 2 can be approximately expressed as

or

This equation was studied in [45]; the quadratic non-linearity will gradually consume the vibrating energy, and finally the vibrating motion will stop (see the discussion in [46]).

## 5 Fractal-fractional model for ions release

The unsmooth surface of the inner wall of the hollow fiber is another important factor affecting the release process. Because any physical laws are scale-dependent, the same phenomenon may lead to debating theories if observed using different scales [47]. Capillary effect plays an important role in the heat transmission of porous media and capillary vibration significantly affects the capillary rise or capillary pressure; therefore, the mass transfer or heat transfer will be greatly affected [48]. Most capillary vibrations in the literatures have assumed that the capillary tube is small and uniform; however, capillary tubes are non-uniform in most porous media [24, 48]. The capillary fluid moves extremely slowly, and its vibrations near its equilibrium have an extremely low frequency [48]. Owing to two types of capillary pressures (positive capillary pressure and negative capillary pressure), the capillary pressure from porous media should be taken into consideration [11]. Furthermore, capillary pressure is affected by pore size, capillary pressure with different pore sizes has been analyzed for the hydrophobic-hydrophilic interface in detail, such as electrospun hollow nanofibers used in oil/water separation [11].

The capillary effect has wide applications for microelectromechanical systems and microfluidics devices, in which the capillary vibration significantly affects its mass transmission [48]. Nanofluid mechanics can be directly used for describing the releasing process for the smooth boundary, so the unsmooth boundary makes the release more difficult, but it is amazing Wolfgang Pauli (1900–1958) once said that “God made the bulk, the surface was invented by the devil”. The unsmooth surface determines the release process and it can be modeled by the two-scale fractal dimension [49] with ease. In the fractal space, Liu et al. established a fractional model for the silver ions’ release oscillation [50]. The fractal-fractional model offers a new window for studying the effect of the unsmooth boundary on the release process. Fan et al. concluded that the fractal calculus plays an important role in unlocking the mechanisms of natural fibers [51]. Lu et al. provided two numerical approaches for finding the approximated solutions of the time fractional Boussinesq-Burgers equations without any linearization or complicated computation, including the homotopy perturbation transform method and the method based on the fractional complex transform and homotopy perturbation method [52]. Afterwards, a numerical approach was proposed for finding the approximated solutions of a fractal modification of the Yao-Cheng oscillator based on the two-scale fractal transformation and the global residue harmonic balance method with He’s fractal derivative as well [53]. They also proposed a combined technique for solving the fractional modification of the non-linear oscillator with coordinate-dependent mass [54]. Meanwhile, the numerical sensitive analysis of the approximations were further considered with respect to different amplitudes and parameters, confirming their high efficiency and stability [53, 54]. Considering that two-scale thermodynamics observes the same phenomenon using two different scales, fractal calculus is adopted to establish governing equations, and fractal variational principles are discussed for 1-D fluid mechanics [47], modeling the ions’ release process from an unsmooth boundary of the inner wall of the hollow fibers might be possible.

## 6 Conclusion

Hollow fibers are now a research Frontier in textile engineering, nanofluid mechanics, material science, non-linear science, physics, and mathematics. This mini-review article provides a panoramic view of the recent studies in this meaningful direction. It is still an open problem to model the ions’ release process from an unsmooth boundary of the inner wall of the hollow fibers; a mathematical model for the fractal release oscillation might be more attractive and promising. There is much opportunity and challenge, so this article should be the beginning of future research, not only a review.

## Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

The authors thank the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ21E030016 and the China Postdoctoral Science Foundation under Grant No. 2021M692866.

## Conflict of interest

Author YL works at Zhejiang Sci-Tech University, and also is a joint postdoctoral fellow of Laimei Technology Co., Ltd., Changxing

The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## References

1. He JH, Wang QL, Sun J. Can polar bear hairs absorb environmental energy? *Therm Sci* (2011) 15:911–3. doi:10.2298/TSCI1103911H

2. Wang QL, He JH, Liu Z. Intelligent nanomaterials for solar energy harvesting: From polar bear hairs to unsmooth nanofiber fabrication. *Front Bioeng Biotech* (2022) 10:926253. doi:10.3389/fbioe.2022.926253

3. Wang QL, Shi XY, He JH, Li ZB. Fractal calculus and its application to explanation of biomechanism of polar bear hairs. *Fractals* (2018) 26(6):1850086. doi:10.1142/S0218348X1850086X

4. Cui Y, Gong HX, Wang YJ, Li DW, Bai H. A thermally insulating textile inspired by polar bear hair. *Adv Mater* (2018) 30(14):1706807. doi:10.1002/adma.201706807

5. Liu FJ, Zhang T, He CH, Tian D. Thermal oscillation arising in a heat shock of a porous hierarchy and its application. *Facta Univ.-Ser Mech* (2022) 20(3):633–45. doi:10.22190/FUME210317054L

6. Abulhassani J, Manzoori JL, Amjadi M. Hollow fiber based-liquid phase microextraction using ionic liquid solvent for preconcentration of lead and nickel from environmental and biological samples prior to determination by electrothermal atomic absorption spectrometry. *J Hazard Mater* (2010) 176(1-3):481–6. doi:10.1016/j.jhazmat.2009.11.054

7. Esrafili A, Baharfar M, Tajik M, Yamini Y, Ghambarian M. Two-phase hollow fiber liquid-phase microextraction. *Trac-trend Anal Chem* (2018) 108:314–22. doi:10.1016/j.trac.2018.09.015

8. Shariati S, Yamini Y, Esrafili A. Carrier mediated hollow fiber liquid phase microextraction combined with HPLC-UV for preconcentration and determination of some tetracycline antibiotics. *J Chromatogr B* (2009) 877(4):393–400. doi:10.1016/j.jchromb.2008.12.042

9. Khan WA, Arain MB, Yamini Y, Shah N, Kazi TG, Pedersen-Bjergaard S, et al. Hollow fiber-based liquid phase microextraction followed by analytical instrumental techniques for quantitative analysis of heavy metal ions and pharmaceuticals. *J Pharm Anal* (2020) 10(2):109–22. doi:10.1016/j.jpha.2019.12.003

10. Feng CY, Khulbe KC, Matsuura T, Ismail AF. Recent progresses in polymeric hollow fiber membrane preparation, characterization and applications. *Sep Purif Technol* (2013) 111:43–71. doi:10.1016/j.seppur.2013.03.017

11. Wang HX, Zhou H, Niu HT, Zhang J, Du Y, Lin T. Dual-layer superamphiphobic/superhydrophobic-oleophilic nanofibrous membranes with unidirectional oil-transport ability and strengthened oil-water separation performance. *Adv Mater Inter* (2015) 2(4):1400506. doi:10.1002/admi.201400506

12. Huang Y, Xiao C, Huang Q, Liu H, Zhao J. Progress on polymeric hollow fiber membrane preparation technique from the perspective of green and sustainable development. *Chem Eng J* (2021) 403:126295. doi:10.1016/j.cej.2020.126295

13. Li Y, Chen RX, Liu FJ. Comparison between electrospun and Bubbfil-spun Polyether sulfone fibers. *Matéria-Brazil* (2014) 19:363–9. doi:10.1590/S1517-70762014000400006

14. Lin L, Gong WZ, Wang X, Li XG, Wang SY. Preparation and characterizations of antibacterial PET-based hollow fibers containing silver particles. *Mater Lett* (2011) 65(9):1375–7. doi:10.1016/j.matlet.2011.02.006

15. Lin L, Gong WZ, Wang SY. Hollow PET fibers containing silver particles as antibacterial materials. *J Text Inst* (2011) 102(5):419–23. doi:10.1080/00405000.2010.486185

16. Lin L, Yao SW, Li HG. Silver ion release from Ag/PET hollow fibers: Mathematical model and its application to food packing. *J Eng Fiber Fabr* (2020) 15:1558925020935448. doi:10.1177/1558925020935448

17. Lin L, Yao SW. Release oscillation in a hollow fiber - Part 1: Mathematical model and fast estimation of its frequency. *J Low Freq N A* (2019) 38(3-4):1703–7. doi:10.1177/1461348419836347

18. Lin L, Li HG, Liu YP. Release oscillation in a hollow fiber - Part 2: The effect of its frequency on ions release and experimental verification. *J Low Freq N A* (2021) 40(2):1067–71. doi:10.1177/1461348419874973

19. Yu DG, Teng MY, Chou WL, Yang MC. Characterization and inhibitory effect of antibacterial PAN-based hollow fiber loaded with silver nitrate. *J Membr Sci.* (2003) 225(1-2):115–23. doi:10.1016/j.memsci.2003.08.010

20. Behboudi A, Jafarzadeh Y, Yegani R. Enhancement of antifouling and antibacterial properties of PVC hollow fiber ultrafiltration membranes using pristine and modified silver nanoparticles. *J Environ Chem Eng* (2018) 6(2):1764–73. doi:10.1016/j.jece.2018.02.031

21. He JH. Fatalness of virus depends upon its cell fractal geometry. *Chaos Soliton Fract* (2008) 38(5):1390–3. doi:10.1016/j.chaos.2008.04.018

22. Liu YP, Yan RY, Zhang XH. A possible way for preventing the novel coronavirus. *Therm Sci* (2022) 26(3):2677–81. doi:10.2298/TSCI200308331L

23. Li LJ. Thermal therapy for eye diseases. *Therm Sci* (2020) 24(4):2319–24. doi:10.2298/TSCI2004319L

24. Li XX, He JH. Nanoscale adhesion and attachment oscillation under the geometric potential. Part 1: The formation mechanism of nanofiber membrane in the electrospinning. *Result Phys* (2019) 12:1405–10. doi:10.1016/j.rinp.2019.01.043

25. Fan J, Zhang YR, Liu Y. Explanation of the cell orientation in a nanofiber membrane by the geometric potential theory. *Result Phys* (2019) 15:102537. doi:10.1016/j.rinp.2019.102537

26. Tian D, Li XX, He JH. Geometrical potential and nanofiber membrane's highly selective adsorption property. *Adsorpt Sci Technol* (2019) 37(5-6):367–88. doi:10.1177/0263617418813826

27. Jin X, Liu MN, Pan F. Low frequency of a deforming capillary vibration, part 1: Mathematical model. *J Low Freq N A* (2019) 38:1676–80. doi:10.1177/1461348419856227

28. Bin C, Lu JF, Xia ZZ. Numerical investigation of the fractal capillary oscillator. *J Low Freq N A* (2023) 2023. doi:10.1177/14613484221131245

29. Han CY, He JH. Effect of fabric surface’s cleanliness on its moisture/air permeability. *Therm Sci* (2021) 25(2):1517–21. doi:10.2298/TSCI2102517H

30. Zhou CJ, Tian D, He JH. Highly selective penetration of red ink in a saline water. *Therm Sci* (2019) 23(4):2265–70. doi:10.2298/TSCI1904265Z

31. He JH, Qian MY. A fractal approach to the diffusion process of red ink in a saline water. *Therm Sci* (2022) 26(3B):2447–51. doi:10.2298/TSCI2203447H

32. Qian MY, He JH. Two-scale thermal science for modern life-Making the Impossible Possible. *Therm Sci* (2022) 26(3B):2409–12.

33. Liu YP, Wang CC, Li SJ. A fractal Langmuir kinetic equation and its solution structure. *Therm Sci* (2021) 25(2):1351–4. doi:10.2298/TSCI200320033L

34. Kumar K, Chauhan PR, Kumar R, Bharj RS. Irreversibility analysis in Al_{2}O_{3}-water nanofluid flow with variable property. *Facta Univ.-Ser Mech* (2022) 20(3):503–18. doi:10.22190/FUME210308050K

35. He J, Elgazery NS, Elagamy K. Efficacy of a modulated viscosity-dependent temperature/nanoparticles concentration parameter on a nonlinear radiative electromagneto-nanofluid flow along an elongated stretching sheet. *J Appl Computat Mech* (2023) 9(3):848–60. doi:10.22055/jacm.2023.42294.3905

36. He JH, Abd-Elazem NY. The carbon nanotube-embedded boundary layer theory for energy harvesting. *Facta Univ.-Ser Mech* (2022) 20(2):211–35. doi:10.22190/FUME220221011H

37. Zhang H, Nikolov A, Wasan D. Dewetting film dynamics inside a capillary using a micellar nanofluid. *Langmuir* (2014) 30(31):9430–5. doi:10.1021/la502387j

38. He CH, Amer TS, Tian D. Controlling the kinematics of a spring-pendulum system using an energy harvesting device. *J Low Freq N A* (2022) 41(3):1234–57. doi:10.1177/14613484221077474

39. He CH, El-Dib YO. A heuristic review on the homotopy perturbation method for non-conservative oscillators. *J Low Freq N A* (2022) 41(2):572–603. doi:10.1177/14613484211059264

40. Anjum N, He JH, Ain QT, Tian D. Li-He’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system. *Facta Univ.-Ser Mech* (2021) 19(4):601–12. doi:10.22190/FUME210112025A

41. He JH, El-Dib YO. The enhanced homotopy perturbation method for axial vibration of strings. *Facta Univ.-Ser Mech* (2021) 19(4):735–50. doi:10.22190/FUME210125033H

42. Ji QP, Wang J, Lu LX, Ge CF. Li-He's modified homotopy perturbation method coupled with the energy method for the dropping shock response of a tangent nonlinear packaging system. *J Low Freq N A* (2021) 40(2):675–82. doi:10.1177/1461348420914457

43. He CH, Liu C. A modified frequency-amplitude formulation for fractal vibration systems. *Fractals* (2022) 30(3):2250046. doi:10.1142/S0218348X22500463

44. He CH. A variational principle for a fractal nano/microelectromechanical (N/MEMS) system. *Int J Numer Method H* (2023) 33(1):351–9. doi:10.1108/HFF-03-2022-0191

45. He CH, Tian D, Moatimid GM. Hybrid Rayleigh-van der pol-duffing oscillator: Stability analysis and controller. *J Low Freq N A* (2022) 41(1):244–68. doi:10.1177/14613484211026407

46. He JH, Yang Q, He CH. *Pull-down instability of the quadratic nonlinear oscillators*. Serbia: Facta Univ.-Ser. Mech. (2023). doi:10.22190/FUME230114007H

47. He JH, Ain QT. New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle. *Therm Sci* (2020) 24(2A):659–81. doi:10.2298/TSCI200127065H

48. Xiao J, Liu MN, Pan F, Li YP, Fan J. Low frequency of a deforming capillary vibration, part 1: Mathematical model. *J Low Freq Noise V A* (2019) 38(3-4):1676–80. doi:10.1177/14613484198562

49. He CH, Liu C. Fractal dimensions of a porous concrete and its effect on the concrete’s strength. *Facta Univ.-Ser Mech* (2023) 2023. doi:10.22190/FUME221215005H

50. Liu HY, Li ZM, Yao YJ. A fractional nonlinear system for release oscillation of silver ions from hollow fibers. *J Low Freq N A* (2019) 38(1):88–92. doi:10.1177/1461348418814122

51. Fan J, Yang X, Liu Y. Fractal calculus for analysis of wool fiber: Mathematical insight of its biomechanism. *J Eng Fiber Fabr* (2019) 2019:14. doi:10.1177/1558925019872200

52. Lu J, Sun Y. Numerical approaches to time fractional boussinesq-burgers equations. *Fractals* (2021) 29(08):2150244. doi:10.1142/S0218348X21502443

53. Lu J, Chen L. Numerical analysis of a fractal modification of Yao-Cheng oscillator. *Result Phys* (2022) 38:105602. doi:10.1016/j.rinp.2022.105602

Keywords: hollow fiber, ions release, capillary rise, antibacterial property, nanofluid, fractal, fractional calculus (FC)

Citation: Lin L and Li Y (2023) A mini-review on release oscillation in a hollow fiber. *Front. Phys.* 11:1210400. doi: 10.3389/fphy.2023.1210400

Received: 22 April 2023; Accepted: 16 May 2023;

Published: 31 May 2023.

Edited by:

Chun-Hui He, Xi’an University of Architecture and Technology, ChinaCopyright © 2023 Lin and Li. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Ya Li, liya@zstu.edu.cn