ArticleNo Access

CHARACTERIZATION OF CHLORIDE IONS DIFFUSION IN CONCRETE USING FRACTIONAL BROWNIAN MOTION RUN WITH POWER LAW CLOCK

SHENGJIE YAN

College of Mechanics and Materials, Hohai University, Nanjing, P. R. China

Search for more papers by this author

YINGJIE LIANG

https://orcid.org/0000-0002-3201-6969

College of Mechanics and Materials, Hohai University, Nanjing, P. R. China

Institute of Physics & Astronomy, University of Potsdam, Potsdam-Golm, Germany

E-mail Address: liangyj@hhu.edu.cn

Corresponding author.

Search for more papers by this author

, and

WEI XU

College of Science, Qingdao University of Technology, Qingdao, Shandong, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0218348X22501778Cited by:9 (Source: Crossref)

Abstract

In this paper, we propose a revised fractional Brownian motion run with a nonlinear clock (fBm-nlc) model and utilize it to illustrate the microscopic mechanism analysis of the fractal derivative diffusion model with variable coefficient (VC-FDM). The power-law mean squared displacement (MSD) links the fBm-nlc model and the VC-FDM via the two-parameter power law clock and the Hurst exponent is 0.5. The MSD is verified by using the experimental points of the chloride ions diffusion in concrete. When compared to the linear Brownian motion, the results show that the power law MSD of the fBm-nlc is much better in fitting the experimental points of chloride ions in concrete. The fBm-nlc clearly interprets the VC-FDM and provides a microscopic strategy in characterizing different types of non-Fickian diffusion processes with more different nonlinear functions.

Keywords:

References

1. J. P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195(4–5) (1990) 127–293. Crossref, Web of Science, Google Scholar
2. S. Abe and S. Thurner, Anomalous diffusion in view of Einstein’s 1905 theory of Brownian motion, Physica A 356(2–4) (2005) 403–407. Crossref, Web of Science, Google Scholar
3. A. Masuda, K. Ushida and T. Okamoto, Direct observation of spatiotemporal dependence of anomalous diffusion in inhomogeneous fluid by sampling-volume-controlled fluorescence correlation spectroscopy, Phys. Rev. E 72(6) (2005) 060101. Crossref, Web of Science, Google Scholar
4. A. M. Reynolds, Lagrangian stochastic modeling of anomalous diffusion in two-dimensional turbulence, Phys. Fluids 14(4) (2002) 1442–1449. Crossref, Web of Science, Google Scholar
5. A. Caspi, R. Granek and M. Elbaum, Enhanced diffusion in active intracellular transport, Phys. Rev. Lett. 85(26) (2000) 5655–5663. Crossref, Web of Science, Google Scholar
6. X. Bao and X. Hua, Anomalous diffusion of implanted boron in preamorphized silicon, Phys. Status Solidi A 123(2) (1991) 1–10. Crossref, Google Scholar
7. R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339(1) (2000) 1–77. Crossref, Web of Science, Google Scholar
8. B. Yu and X. Jiang, A fractional anomalous diffusion model and numerical simulation for sodium Ion transport in the intestinal wall, Adv. Math. Phys. 2013 (2013) 479634. Crossref, Web of Science, Google Scholar
9. G. Rangarajan et al., First passage time problem for biased continuous-time random walks, Fractals 8(2) (2000) 139–145. Link, Web of Science, Google Scholar
10. K. Wang, A novel perspective to the local fractional Zakharov–Kuznetsov-modified equal width dynamical model on Cantor sets, Math. Methods Appl. Sci. 2002 (2022) 1–9, https://doi.org/10.1002/mma.8533. Google Scholar
11. W. Chen et al., Fractional derivative anomalous diffusion equation modeling prime number distribution, Frac. Calc. Appl. Anal. 18(3) (2015) 789–798. Crossref, Web of Science, Google Scholar
12. Y. Liang and W. Chen, A non-local structural derivative model for characterization of ultraslow diffusion in dense colloids, Commun. Nonlinear Sci. Numer. Simul. 56 (2018) 131–137. Crossref, Web of Science, Google Scholar
13. X. Yang et al., Fundamental solutions of the general fractional order diffusion equations, Math. Methods Appl. Sci. 41 (2018) 9312–9320. Crossref, Web of Science, Google Scholar
14. W. Chen et al., Time-space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 (2006) 923–929. Crossref, Web of Science, Google Scholar
15. J. Li and M. Ostoja-Starzewski, Fractal solids, product measures and fractional wave equations, Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 465 (2009) 2521–2536. Crossref, Web of Science, Google Scholar
16. A. Balankin, J. Bory-Reyes and M. Shapiro, Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric, Physica A 444 (2016) 345–359. Crossref, Web of Science, Google Scholar
17. H. Sun et al., A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media, Adv. Water Resour. 52 (2013) 292–295. Crossref, Web of Science, Google Scholar
18. W. Cai et al., The fractal derivative wave equation: Application to clinical amplitude velocity reconstruction imaging, J. Acoust. Soc. Amer. 143 (2018) 1559–1566. Crossref, Web of Science, Google Scholar
19. F. Wang et al., Kansa method based on the Hausdorff fractal distance for Hausdorff derivative Poisson equations, Fractals 26 (2018) 1850084. Link, Web of Science, Google Scholar
20. F. Wang et al., A simple empirical formula of origin intensity factor in singular boundary method for two-dimensional Hausdorff derivative Laplace equations with Dirichlet boundary, Comput. Math. Appl. 76 (2018) 1075–1084. Crossref, Web of Science, Google Scholar
21. G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 374 (2002) 461–580. Crossref, Web of Science, Google Scholar
22. M. Nokken et al., Time dependent diffusion in concrete-three laboratory studies, Cem. Concr. Res. 36 (2006) 200–207. Crossref, Web of Science, Google Scholar
23. S. L. Amey et al., Predicting the service life of concrete marine structures: An environmental methodology, Amer. Concr. Inst. Struc. J. 95 (1998) 205–214. Google Scholar
24. S. W. Pack et al., Prediction of time dependent chloride transport in concrete structures exposed to a marine environment, Cem. Concr. Res. 40(2) (2010) 302–312. Crossref, Web of Science, Google Scholar
25. M. K. Kassir and M. Ghosn, Chloride-induced corrosion of reinforced concrete bridge decks, Cem. Concr. Res. 32(1) (2002) 139–143. Crossref, Web of Science, Google Scholar
26. S. Y. Reutskiy, A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients, Appl. Math. Model. 45 (2017) 238–254. Crossref, Web of Science, Google Scholar
27. M. Garshasbi and F. Sanaei, A variable time-step method for a space fractional diffusion moving boundary problem: An application to planar drug release devices, Int. J. Numer. Model. 34(3) (2021) e2852. Crossref, Web of Science, Google Scholar
28. Y. Liang, N. Su and W. Chen, A time-space Hausdorff derivative model for anomalous transport in porous media, Fract. Calc. Appl. Anal. 22(6) (2019) 1517–1536. Crossref, Web of Science, Google Scholar
29. F. Biagini, Stochastic Calculus for Fractional Brownian Motion and Applications (Springer, London, 2008). Crossref, Google Scholar
30. W. Chen et al., Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion, Fract. Calc. Appl. Anal. 19(5) (2016) 1250–1261. Crossref, Web of Science, Google Scholar
31. W. Xu et al., Ultrafast dynamics modeling via fractional Brownian motion run with Mittag-Leffler clock in porous media, Int. J. Heat Mass Transf. 151 (2020) 119402. Crossref, Web of Science, Google Scholar
32. M. Park, J. H. Cushman and D. O’Malley, Fractional Brownian motion run with a multi-scaling clock mimics diffusion of spherical colloids in micro structural fluids, Langmuir 30(38) (2014) 11263–11266. Crossref, Web of Science, Google Scholar
33. M. Park and J. H. Cushman, The complexity of Brownian processes run with nonlinear clocks, Mod. Phys. Lett. B 25(1) (2011) 1–10. Link, Web of Science, Google Scholar
34. D. OMalley and J. H. Cushman, Fractional Brownian motion run with a nonlinear clock, Phys. Rev. E 82 (2010) 032102. Crossref, Web of Science, Google Scholar
35. J. H. Cushman et al., Anomalous diffusion as modeled by a nonstationary extension of Brownian motion, Phys. Rev. E 73 (2009) 032101. Crossref, Web of Science, Google Scholar
36. D. OMalley, Diffusion processes run with non-linear clocks, Purdue University, West Lafayette (2011). Google Scholar
37. W. Zhang et al., Multi-scale study water and ions transport in the cement-based materials: From molecular dynamics to random walk, Microporous Mesoporous Mater. 325 (2021) 111330. Crossref, Web of Science, Google Scholar
38. Y. Tu et al., Molecular dynamics simulation of coupled water and ion adsorption in the nano-pores of a realistic calcium-silicate-hydrate gel, Constr. Build. Mater. 299 (2021) 123961. Crossref, Web of Science, Google Scholar
39. Y. Liang and W. Chen, Continuous time random walk model with asymptotical probability density of waiting times via inverse Mittag-Leffler function, Commun. Nonlinear Sci. Numer. Simul. 57 (2018) 439–448. Crossref, Web of Science, Google Scholar
40. X. Liu et al., A variable-order fractal derivative model for anomalous diffusion, Therm. Sci. 21 (2017) 51–59. Crossref, Web of Science, Google Scholar