Anomalous diffusion is the rule in concentration-dependent diffusion processes

Abstract

In this note, we re-examine one-dimensional diffusion experiments of high concentration aqueous CuSO4 into deionized water carried out by Carey A E et al (1995 Water Resources Res. 31 2213–18). We show that spreading of concentrated CuSO4 fronts does not conform to the t1/2 relation expected from Fick's hypothesis but is subdiffusive; i.e. the cumulated concentration of CuSO4 increases much slower than t1/2. Occurrence of subdiffusion of CuSO4 fronts can be related to the fact that the diffusion coefficient, D(C), of aqueous CuSO4 decreases with increasing C, as previously predicted in Küntz M and Lavallée P (2003 J. Phys. D: Appl. Phys. 36 1135–42). This result supports our initial assumption that non-Fickian diffusion is the rule in all concentration-dependent diffusivity processes, i.e. in diffusion processes involving variation of the diffusion coefficient, D(C), with the concentration of the transported quantity, C. A simple ‘offer and demand’ model is proposed that accounts qualitatively well for all the available experimental data. Spreading fronts are subdiffusive for D(C) decreasing with C, superdiffusive for increasing D(C) and scale as t1/2 only for constant D.

Sorption and Diffusional Studies of Extruded Waxy Maize Starch‐Glycerol Systems

“…It is well established from the literature [16,17] that the concentration gradient within the polymer is not the only variable that may describe the diffusivity when the matrix approaches T g . The deviation from Fickian behaviour can be associated with the relaxation time of the polymer structure with a similar order of magnitude as that of the sorption kinetics of the sorbant molecules [17,[36][37][38]. It is possible that for high RH, a different diffusional mechanism is present suggesting that the Fickian model used to obtain D is not entirely appropriate.…”

Solution of a Cattaneo-Maxwell diffusion model using a Spectral element least-squares method

“…Evidence of superdiffusion has been found observing water infiltration profiles in porous building materials (Küntz and Lavallée, 2001) and also subdiffusion has been observed in one-dimensional diffusion in high concentration aqueous CuSO 4 solutions (Küntz and Lavallée, 2004). Extended versions of Cattaneo's equation including fractional derivatives are able to predict this behavior in a consistent way.…”

“…The cases with g < 1 and g > 1, which are not properly accounted by Fick's law, are termed subdiffusion (Küntz and Lavallée, 2004) and superdiffusion (Küntz and Lavallée, 2001) respectively. An alternative model to deal with these inherent problems of Fick's law is Maxwell-Cattaneo's model.…”

Least-Squares Spectral Method for the solution of a fractional advection–dispersion equation

“…(2)) which is appropriate only in a very limited number of cases [1,2]. Diffusive flux is then modeled as Jðx; tÞ ¼ ÀlðxÞ @Cðx; tÞ @x ; ð2Þ…”

“…However, as has been stated many times and supported by new experimental evidence, this model can be insufficient in many cases [1,2]. Numerous approaches for deriving a more general formulation have been attempted such as hyperbolic diffusion theories [3][4][5][6] and Continuous Time Random Walks (CTRW) [7].…”