Stress-induced diffusion of hydrogen in metallic membranes: cylindrical vs. planar formulation. I

Abstract

The hydrogen chemical potential in metallic membranes is affected by the self-stresses generated by the interstitial transport within the lattice (stress-induced diffusion). This article provides analytical and numerical evidence that, in the presence of stress-induced diffusion, hydrogen transport in thin metallic cylindrical membranes can exhibit macroscopic features qualitatively different from those observed in planar structures. This is a consequence of the different ways diffusion-induced stresses propagate in planar and in the cylindrical structures. We focus on the investigation of the uphill diffusion effect (observed experimentally in Pd and I'd alloy membranes) originally explained as a consequence of the failure of the Fick law for solid-state diffusion. We show that the classical stress-induced diffusion (SID) model applied to a cylindrical structure leads to a Fickian-type transport equation, which indeed displays the uphill effect solely as a consequence of the non-linear, non-local time-dependent boundary conditions corresponding to permeation experiments. (C) 2003 Elsevier B.V. All rights reserved

Effects of self-stress on hydrogen diffusion in Pd membranes in the coexistence of α and β phases

“…(5) and (6) that both in the ␣ and ␤ phases the hydrogen flux is proportional to the concentration gradient. Therefore, in cylindrical geometry, even in the presence of stress-induced diffusion, the flux expression is strictly Fickian, whereas in the planar case, a non-local integral contribution to hydrogen flux is also present [10][11][12][13][14]. Consequently, the role of self-stresses in tubular membranes is solely that of increasing the effective (concentration dependent) diffusion coefficient both in the ␣ and in the ␤ phases.…”

“…In previous papers, Adrover et al [12][13][14] and Zhang et al [10,11] have made use of the SJD model to analyze the occurrence of the uphill-diffusion in planar and tubular Pd membranes, for hydrogen contents falling within the ␣ phase composition range. These authors have shown that: (i) the SID model does not display any non-local contribution to the flux in a cylindrical membrane; but (ii) the uphill effect occurs as a direct consequence of the non-linear, non-local, time-dependent boundary conditions, corresponding to a permeation experiment.…”

“…within the time interval (0, t * ) when the hydrogen concentration in the whole membrane lies in the ␣ phase composition range, the classical approximation of semi-infinite medium can be adopted [6,12], so that (n H − n H0 ) m (n av H − n H0 )/2, and Eq. (37) for n L can be approximated as follows:…”

“…More recently, Zhang et al [10,11] and Adrover et al [12][13][14] have provided analytical and numerical evidence that hydrogen transport exhibits different macroscopic features in thin cylindrical and planar membranes, as a consequence of the different propagation mechanisms of the stress field in each of these geometries.…”

Remarks on “Determination of hydrogen absorption isotherm and diffusion coefficient in Pd81Pt19 alloy” [F. Vigier et al., J. Electroanal. Chem. 588 (2006) 32]

“…Hence, the usefulness of these equations for modelling of experimental data and their quantitative interpretation is really limited [8,30,45,46]. Later on, for the discussed transport even more complex models have been also proposed [49][50][51].…”