Abstract
The diffusion kinetics may experience either standard or anomalous behavior. In a porous medium, a transport process may distinguish between the standard and anomalous diffusion. Herein, we perform an analysis of the diffusion concentration profiles in a hierarchically porous material using the standard and the time-fractional diffusion equations. It is shown that the mass transfer regime changes from the anomalous to the standard one regarding the temporal scale of the concentration profile. Particularly, at relatively short times, the mass transfer regime corresponds to the anomalous diffusion with the superdiffusive anomalous diffusion exponent. In contrast, at relatively long times, the standard Fickian regime is obeyed. The investigated phenomenon is assigned to the diffusion in different components of a porous material. The relevant physical model of the corresponding transition is discussed based on the Cattaneo-type diffusion equation. The relaxation time for the transport of the gaseous molecules in the porous medium was found to be in the range of 10–8–10–6 s.
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Ahmadi, S., Bowles, R.K.: Diffusion in quasi-one-dimensional channels: a small system n, p, T, transition state theory for hopping times. J. Chem. Phys. 146, 154505 (2017). https://doi.org/10.1063/1.4981010
Ahmed, E., Hassanb, S.Z.: On diffusion in some biological and economic systems. Z. Natur. A. 55, 669–672 (2000). https://doi.org/10.1515/zna-2000-0801
Alvarez-Ramírez, J., Valdés-Parada, F.J., Alberto Ochoa-Tapia, J.: A lattice-Boltzmann scheme for Cattaneo’s diffusion equation. Phys. A Stat. Mech. Its Appl. 387, 1475–1484 (2008). https://doi.org/10.1016/J.PHYSA.2007.10.051
Awad, E.: On the time-fractional Cattaneo equation of distributed order. Phys. A Stat. Mech. Its Appl. 518, 210–233 (2019). https://doi.org/10.1016/J.PHYSA.2018.12.005
Balogh, Z., Erdélyi, Z., Beke, D.L., Langer, G.A., Csik, A., Boyen, H.-G., Wiedwald, U., Ziemann, P., Portavoce, A., Girardeaux, C.: Transition from anomalous kinetics toward Fickian diffusion for Si dissolution into amorphous Ge. Appl. Phys. Lett. 92, 143104 (2008). https://doi.org/10.1063/1.2908220
Bates, S.P., Van Well, W.J.M., Van Santen, R.A., Smit, B.: Energetics of n-alkanes in zeolites: a configurational-bias Monte Carlo investigation into pore size dependence. J. Am. Chem. Soc. 118, 6753–6759 (1996). https://doi.org/10.1021/ja953856q
Bovet, A., Gamarino, M., Furno, I., Ricci, P., Fasoli, A., Gustafson, K., Newman, D.E., Sánchez, R.: Transport equation describing fractional Lévy motion of suprathermal ions in TORPEX. Nucl. Fusion. 54, 104009 (2014). https://doi.org/10.1088/0029-5515/54/10/104009
Brandani, S., Ruthven, D.M.: Analysis of ZLC desorption curves for gaseous systems. Adsorption. 2, 133–143 (1996). https://doi.org/10.1007/BF00127043
Burnecki, K., Kepten, E., Garini, Y., Sikora, G., Weron, A.: Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors—an alternative approach. Sci. Rep. 5, 11306 (2015). https://doi.org/10.1038/srep11306
Cattaneo, C.: Sur une forme de l’equation de la chaleur elinant le paradoxe d’une propagation instantance. Comptes Rendus Acad. Sci. 247, 431 (1958)
Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev E. Stat. Phys. Plasmas Fluids Relat. Interdiscipl. Top. 66, 7 (2002). https://doi.org/10.1103/PhysRevE.66.046129
Chen, Q., Moore, J.D., Liu, Y.-C., Roussel, T.J., Wang, Q., Wu, T., Gubbins, K.E.: Transition from single-file to Fickian diffusion for binary mixtures in single-walled carbon nanotubes. J. Chem. Phys. 133, 094501 (2010). https://doi.org/10.1063/1.3469811
Cherstvy, A.G., Thapa, S., Wagner, C.E., Metzler, R.: Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels. Soft Matter 15, 2526–2551 (2019). https://doi.org/10.1039/C8SM02096E
Chubynsky, M.V., Slater, G.W.: Diffusing diffusivity: a model for anomalous, yet Brownian, diffusion. Phys. Rev. Lett. 113, 098302 (2014). https://doi.org/10.1103/PhysRevLett.113.098302
Compte, A., Metzler, R.: The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A. Math. Gen. 30, 7277–7289 (1997). https://doi.org/10.1088/0305-4470/30/21/006
de Azevedo, E.N., de Sousa, P.L., de Souza, R.E., Engelsberg, M., Miranda, M.D.N.D.N., Silva, M.A.: Concentration-dependent diffusivity and anomalous diffusion: A magnetic resonance imaging study of water ingress in porous zeolite. Phys. Rev. E. 73, 11204 (2006)
Dutta, A.R., Sekar, P., Dvoyashkin, M., Bowers, C., Ziegler, K.J., Vasenkov, S.: Possible role of molecular clustering in single-file diffusion of mixed and pure gases in dipeptide nanochannels. Microp. Mesop. Mater. 269, 83–87 (2018). https://doi.org/10.1016/j.micromeso.2017.05.025
Ellery, A.J., Baker, R.E., Simpson, M.J.: Communication: Distinguishing between short-time non-Fickian diffusion and long-time Fickian diffusion for a random walk on a crowded lattice. J. Chem. Phys. 144, 171104 (2016). https://doi.org/10.1063/1.4948782
Evangelista, L.R., Lenzi, E.K.: Fractional Diffusion Equations and Anomalous Diffusion. Cambridge University Press, Cambridge (2018)
Jain, R., Sebastian, K.L.: Diffusing diffusivity: Rotational diffusion in two and three dimensions. J. Chem. Phys. 146, 214102 (2017). https://doi.org/10.1063/1.4984085
Kärger, J., Petzold, M., Pfeifer, H., Ernst, S., Weitkamp, J.: Single-file diffusion and reaction in zeolites. J. Catal. 136, 283–299 (1992). https://doi.org/10.1016/0021-9517(92)90062-M
Kärger, J., Ruthven, D.M.: Diffusion in nanoporous materials: fundamental principles, insights and challenges. New J. Chem. 40, 4027–4048 (2016). https://doi.org/10.1039/C5NJ02836A
Liang, Y., Ye, A.Q., Chen, W., Gatto, R.G., Colon-Perez, L., Mareci, T.H., Magin, R.L.: A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging. Commun. Nonlinear Sci. Numer. Simul. 39, 529–537 (2016). https://doi.org/10.1016/j.cnsns.2016.04.006
Liang, Y., Chen, W., Akpa, B.S., Neuberger, T., Webb, A.G., Magin, R.L.: Using spectral and cumulative spectral entropy to classify anomalous diffusion in SephadexTM gels. Comput. Math. Appl. 73, 765–774 (2017). https://doi.org/10.1016/j.camwa.2016.12.028
Liu, J.Y., Simpson, W.T.: Solutions of diffusion equation with constant diffusion and surface emission coefficients. Dry. Technol. 15, 2459–2477 (1997). https://doi.org/10.1080/07373939708917370
Lizana, L., Lomholt, M.A., Ambjörnsson, T.: Single-file diffusion with non-thermal initial conditions. Phys. A Stat. Mech. Its Appl. 395, 148–153 (2014). https://doi.org/10.1016/J.PHYSA.2013.10.025
Lucena, D., Tkachenko, D.V., Nelissen, K., Misko, V.R., Ferreira, W.P., Farias, G.A., Peeters, F.M.: Transition from single-file to two-dimensional diffusion of interacting particles in a quasi-one-dimensional channel. Phys. Rev. E. 85, 031147 (2012). https://doi.org/10.1103/PhysRevE.85.031147
Mainardi, F. (2014) On some properties of the Mittag–Leffler function Eα (−tα), completely monotone for t > 0 with 0 < α < 1. Discret. Contin. Dyn. Syst. Ser. B. 19: 2267–2278. 10.3934/dcdsb.2014.19.2267
Mainardi, F., Pagnini, G., Saxena, R.K.: Fox H functions in fractional diffusion. J. Comput. Appl. Math. 178, 321–331 (2005). https://doi.org/10.1016/j.cam.2004.08.006
Metzler, R., Jeon, J.-H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164 (2014). https://doi.org/10.1039/c4cp03465a
Molina-Garcia, D., Sandev, T., Safdari, H., Pagnini, G., Chechkin, A., Metzler, R.: Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers. New J. Phys. 20, 103027 (2018). https://doi.org/10.1088/1367-2630/aae4b2
Nelson, P.H., Auerbach, S.M.: Self-diffusion in single-file zeolite membranes is Fickian at long times. J. Chem. Phys. 110, 9235–9243 (1999). https://doi.org/10.1063/1.478847
Pachepsky, Y., Benson, D., Rawls, W.: Simulating scale-dependent solute transport in soils with the fractional advective–dispersive equation. Soil Sci. Soc. Am. J. 64, 1234–1243 (2000). https://doi.org/10.2136/sssaj2000.6441234x
Palombo, M., Gabrielli, A., Servedio, V.D.P., Ruocco, G., Capuani, S.: Structural disorder and anomalous diffusion in random packing of spheres. Sci. Rep. 3, 2631 (2013). https://doi.org/10.1038/srep02631
Qi, H.T., Xu, H.Y., Guo, X.W.: The Cattaneo-type time fractional heat conduction equation for laser heating. Comput. Math. with Appl. 66, 824–831 (2013). https://doi.org/10.1016/j.camwa.2012.11.021
Regner, B.M., Vučinić, D., Domnisoru, C., Bartol, T.M., Hetzer, M.W., Tartakovsky, D.M., Sejnowski, T.J.: Anomalous diffusion of single particles in cytoplasm. Biophys. J. 104, 1652–1660 (2013). https://doi.org/10.1016/J.BPJ.2013.01.049
Ruthven, D.M., Vidoni, A.: ZLC diffusion measurements: combined effect of surface resistance and internal diffusion. Chem. Eng. Sci. 71, 1–4 (2012). https://doi.org/10.1016/J.CES.2011.11.040
Sales Teodoro, G., Tenreiro Machado, J.A., Capelas de Oliveira, E.: A review of definitions of fractional derivatives and other operators. J. Comput. Phys. 388, 195–208 (2019). https://doi.org/10.1016/J.JCP.2019.03.008
Sané, J., Padding, J.T., Louis, A.A.: The crossover from single file to Fickian diffusion. Faraday Discuss. 144, 285–299 (2010). https://doi.org/10.1039/b905378f
So, K., Sakai, K., Kano, K.: Gas diffusion bioelectrodes. Curr. Opin. Electrochem. 5, 173–182 (2017). https://doi.org/10.1016/J.COELEC.2017.09.001
Stern, R., Effenberger, F., Fichtner, H., Schäfer, T.: The space-fractional diffusion-advection equation: analytical solutions and critical assessment of numerical solutions. Fract. Calc. Appl. Anal. 17, 171–190 (2014). https://doi.org/10.2478/s13540-014-0161-9
Titze, T., Lauerer, A., Heinke, L., Chmelik, C., Zimmermann, N.E.R., Keil, F.J., Ruthven, D.M., Kärger, J.: Transport in nanoporous materials including MOFs: the applicability of Fick’s laws. Angew. Chemie Int. Ed. 54, 14580–14583 (2015). https://doi.org/10.1002/anie.201506954
Tournier, J.-D.: Diffusion MRI in the brain—theory and concepts. Prog. Nucl. Magn. Reson. Spectrosc. 112–113, 1–16 (2019). https://doi.org/10.1016/J.PNMRS.2019.03.001
Tyukodi, B., Vandembroucq, D., Maloney, C.E.: Diffusion in mesoscopic lattice models of amorphous plasticity. Phys. Rev. Lett. 121, 145501 (2018). https://doi.org/10.1103/PhysRevLett.121.145501
Vidoni, A., Ruthven, D.: Diffusion of methane in DD3R zeolite. Microp. Mesop. Mater. 159, 57–65 (2012). https://doi.org/10.1016/J.MICROMESO.2012.04.008
Villa, I.M.: Diffusion in mineral geochronometers: present and absent. Chem. Geol. 420, 1–10 (2016). https://doi.org/10.1016/J.CHEMGEO.2015.11.001
Wang, B., Anthony, S.M., Bae, S.C., Granick, S.: Anomalous yet Brownian. Proc. Natl. Acad. Sci. 106, 15160–15164 (2009). https://doi.org/10.1073/pnas.0903554106
Wang, B., Kuo, J., Bae, S.C., Granick, S.: When Brownian diffusion is not Gaussian. Nat. Mater. 11, 481–485 (2012). https://doi.org/10.1038/nmat3308
Wang, J., Yuan, Q., Dong, M., Cai, J., Yu, L.: Experimental investigation of gas mass transport and diffusion coefficients in porous media with nanopores. Int. J. Heat Mass Transf. 115, 566–579 (2017). https://doi.org/10.1016/j.ijheatmasstransfer.2017.08.057ss
Xu, G., Wang, J.: Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux. Appl. Math. Mech. 39, 1465–1476 (2018). https://doi.org/10.1007/s10483-018-2375-8
Yu, X., Zhang, Y., Sun, H.G., Zheng, C.: Time fractional derivative model with Mittag–Leffler function kernel for describing anomalous diffusion: analytical solution in bounded-domain and model comparison. Chaos, Solitons Fractals 115, 306–312 (2018). https://doi.org/10.1016/j.chaos.2018.08.026
Zhang, Y., Jiang, J., Bai, Y., Liu, J., Shao, H., Wu, C., Guo, Z.: A fractional mass transfer model for simulating VOC emissions from porous, dry building material. Build. Environ. 152, 182–191 (2019). https://doi.org/10.1016/J.BUILDENV.2019.01.053
Zhao, W., Cheng, Y., Pan, Z., Wang, K., Liu, S.: Gas diffusion in coal particles: a review of mathematical models and their applications. Fuel 252, 77–100 (2019). https://doi.org/10.1016/J.FUEL.2019.04.065
Zhokh, A., Strizhak, P.: Non-Fickian diffusion of methanol in mesoporous media: geometrical restrictions or adsorption-induced? J. Chem. Phys. 146, 124704 (2017). https://doi.org/10.1063/1.4978944
Zhu, X.G., Yuan, Z.B., Liu, F., Nie, Y.F.: Differential quadrature method for space-fractional diffusion equations on 2D irregular domains. Numer. Algorithms. 79, 853–877 (2018). https://doi.org/10.1007/s11075-017-0464-0
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This research was partially supported by the Target Integrated Programs of Fundamental Researches of the National Academy of Sciences of Ukraine “Fundamental Problems of New Nanomaterials and Nanotechnologies”.
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Zhokh, A.A., Strizhak, P.E. Investigation of the Time-Dependent Transitions Between the Time-Fractional and Standard Diffusion in a Hierarchical Porous Material. Transp Porous Med 133, 497–508 (2020). https://doi.org/10.1007/s11242-020-01435-8
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DOI: https://doi.org/10.1007/s11242-020-01435-8