## Abstract

Lattice discontinuities include lattice defects and surfaces both providing traps for hydrogen atoms. It will be discussed under which conditions discontinuities of a given distribution either release trapped hydrogen to become diffusible or capture diffusible H-atoms to become trapped. It will be shown that for any distribution, the self-diffusion coefficient of hydrogen is determined by the product of the H-diffusion in the perfect lattice times the fraction of hydrogen being diffusible. In this context, the quantities diffusible hydrogen, lattice hydrogen, thermodynamic activity of hydrogen and chemical potential of hydrogen are interchangeable in a general way. New discontinuities are generated during hydrogen embritllement (fracture surfaces, voids, dislocations) and dislocations move by kink pair formation. The production rate of these discontinuities depends on the chemical potential of hydrogen within the defactant concept or the generalized Gibbs adsorption isotherm. Thus, the chemical potential of hydrogen determines both the amount of trapping and the defect generation rate. For a crack propagating by dislocations generation, the chemical potential affects its velocity independent of the accompanying concentration enhancement in front of the crack tip or the related adsorption on the freshly generated crack surface.

This article is part of the themed issue ‘The challenges of hydrogen and metals’.

## 1. Introduction

Hydrogen embrittlement of metals or steels in particular is a complicated failure mechanism which is explained by a variety of different and often controversial models proposing different mechanisms of crack propagation. In this study, it is assumed that by the advancing crack, discontinuities are generated. These are two new surfaces right behind the crack tip, the surface of a void opening up in front of the crack tip, or dislocations generated and emitted from the crack tip. Hydride formation in front of the crack as induced by hydrostatic stress components will not be considered, because in steels hydrogen concentrations are too low to allow a corresponding H-enrichment with a concomitant hydride formation. If hydrogen segregates to the newly formed discontinuities, its transport to the crack tip either from external sources (gaseous hydrogen, corrosion reactions) or internal sources (traps) comes into play. Trap sites are lattice sites of lower potential energy existing near lattice defects or lattice discontinuities, respectively. The latter term is preferred as it includes surfaces. At discontinuities, hydrogen atoms in interstices will experience a different coordination when compared with normal lattice sites, as, for instance, tetrahedral sites in the body-centered cubic (BCC) lattice of α-iron. Thus, subsurface sites with a slightly different coordination, i.e. different distances to neighbouring metal atoms, may become traps. For dislocations with their long-ranging strain field, there are no normal lattice sites in principle. However, for practical purposes, sites may be defined as normal, if their potential energy differs by an amount of the order of the thermal energy *RT* when compared with undisturbed lattice sites. Hydrogen atoms occupying normal lattice sites will be called free in the following. It will be shown that a hydrogen flux induced by a gradient of the chemical potential of hydrogen is proportional to the concentration of free hydrogen. Thus, the free hydrogen may be also defined as the diffusible hydrogen. The chemical potential of hydrogen plays a central role in determining the distribution of hydrogen among various trap sites and their saturation which in turn determines the overall mobility of H-atoms. Its dependence on stress is given by a Maxwell equation. A different Maxwell equation reveals that the chemical potential of hydrogen also affects the generation of discontinuities. Thus, the creation of vacancies, dislocations and surfaces requires less and less energy with increasing chemical potential of hydrogen. In this study, the importance of the chemical potential in affecting hydrogen solubility, diffusion, trapping and defect generation is for the first time compiled and related to the fundamental laws of thermodynamics and statistical mechanics.

## 2. Trapped and free hydrogen

The potential energy of normal lattice sites may be *E*_{o}, whereas sites affected by discontinuities may have a distribution of potential energies *n*(*E*) being trap sites for *E* < *E*_{o} or anti-traps for *E* > *E*_{o}. For a total number of lattice sites *N*_{o} including a number *N*_{d} of sites disturbed by discontinuities, the following relation holds
2.1
Thus, the total energy distribution becomes
2.2
with *δ*(*E* − *E*_{o}) being the Dirac delta function. Introducing *N*_{H} hydrogen atoms and minimizing the Gibbs free energy under the condition that one site can be occupied by one H-atom only leads to [1]
2.3
or
2.4
where *μ* is the chemical potential of hydrogen. The procedure of minimizing the free energy is equivalent to the application of Fermi–Dirac statistics (FDS), where *μ* is the same as the Fermi-energy of hydrogen. The first term on the r.h.s. of equation (2.4) is the number of free H-atoms and the second term is the number of H-atoms trapped by discontinuities. Configurational entropy is included in equation (2.3) and other contributions to entropy like vibrational ones may be taken into account by replacing the energy *E* by the Gibbs free energy *G*.

If there are no traps (*N*_{d} = 0), the chemical potential is obtained from equation (2.4) in the known form as
2.5
For one type of traps of energy *E*_{t} or , equation (2.4) yields
2.6
where *N*_{Hf} is the number of free H-atoms and *N*_{Ht} the one of trapped H-atoms. Equation (2.6) also reveals that hydrogen is in an ideal dilute solution in both sites. Rearranging equation (2.6) leads to the Oriani relation [2]
2.7

For multiple traps with a fixed energy, a corresponding sum of delta functions has to be used for *N*(*E*) and inserted in equation (2.3). Further examples for hydrogen segregating to discontinuities and the related chemical potential are given in [3,4]. For extended discontinuities like dislocations, grain boundaries and surfaces and , the H-atoms are trapped in close distances and hydrogen/hydrogen-interaction has to be taken into account separately as it is not included in equation (2.3). Experimental evidence for H/H-interaction is provided for hydrogen at dislocations [5,6] and grain boundaries [7] in palladium, on surfaces of nickel [8] and by computational means for hydrogen at dislocations in nickel [9].

Gaussian distributions of site energies are suited to describe the chemical potential of solute entities in amorphous matrices. This is shown in figure 1, where the chemical potential of (i) atomic hydrogen in a metallic glass, (ii) sodium ions and hydrogen molecules in amorphous silica and various small molecules in glassy polymers is presented as a function of erf^{−1}(1 − 2*c*). The linear dependence between the two quantities is a consequence of equation (2.3) with and the step-like behaviour of the thermal occupancy [10]. In a kinetic Monte Carlo simulation [10,11], where sites are occupied by thermally activated hopping over energy barriers between the sites, the resulting occupancy of sites follows FDS (cf. figure 2), if a site can be occupied by one particle only.

## 3. Diffusion of hydrogen in the presence of traps

In the framework of irreversible thermodynamics, a gradient of the chemical potential of hydrogen is causing a flux of H-atoms given as
3.1
where *M* is the mobility which is related to the self-diffusion or tracer diffusion coefficient *D** via the Einstein–Smoluchowski relation
3.2

For an uncorrelated random walk of H-atoms through all lattice sites, the self-diffusion coefficient is defined as
3.3
where *R* is the distance covered by the migrating H-atom during time *t*. For an ensemble of H-atoms in an energy landscape, the average occupancy of a site is proportional to the average time of residence in this site as a consequence of the ergodic hypothesis. Following one H-atom migrating through the lattice visiting only empty sites with the other H-atoms occupying sites according to FDS leads to the following result [3] as derived in the appendix A:
3.4
where *τ*_{r} is the residence time in a reference site of energy *E*_{o} and *a* the jump distance. The choice of a reference site is arbitrary, but within the present context, having a majority of normal sites, these sites qualify being the reference sites. Note that the result is independent of the type of energy distribution. Combined with equation (2.6), it leads to
3.5
which simplifies for small H- and small trap concentrations to
3.6
where is the self-diffusion coefficient in a lattice free of discontinuities containing normal lattice sites only. For trapping by any distribution of site energies, equation (3.6) teaches that the self-diffusion coefficient is reduced by the fraction of H-atoms to be free. This fraction may be defined as the diffusible hydrogen.

Most experimental methods determine hydrogen diffusion in a concentration gradient, yielding the chemical diffusion coefficient *D* defined by Fick's first law
3.7
with *V* being the volume. Comparison with equation (3.1) yields
3.8
where the term after *D** is called thermodynamic factor.

With the assumption that interstitial sites are left by thermally activated hopping over an energy barrier *Q* with and/or without a gradient of the chemical potential [1,12], equation (3.8) becomes
3.9
where is the self-diffusion coefficient in a lattice free of discontinuities expressed in an Arrhenius form. For a lattice containing a variety of different traps, equations (3.8) and (3.9) tell us that the effective activation energy for diffusion is . Thus with increasing chemical potential which for a single-phase material is equivalent with an increasing hydrogen concentration, the diffusion coefficient increases due to a decreased effective activation energy. This is exemplified for an energy landscape in figure 3.

For a known distribution of site energies *N*(*E*), equation (2.3) may be used to calculate the chemical potential as a function of H-content and insert that into equation (3.8). This implies uncertainties regarding entropy or H/H-interaction. Thus, in the examples presented in figures 4–6, measured values of the chemical potentials were used instead [3,4,10].

The agreement between measured and calculated diffusion coefficients as shown in figures 4–6 and 7 supports the random walk and related models for hydrogen—or more general interstitial—diffusion. The agreement was achieved within the framework of irreversible and statistical thermodynamics which implicitly assumes local equilibrium, i.e. the occupancy of a site is determined by the chemical potential. Thus, changes of the occupancy are due to changes of the chemical potential and there is no need to apply a formalism with a trapping and detrapping rate as introduced by McNabb & Forster [13] and others [14]. On the contrary, freely changing rates will on the one hand require unknown parameters to be introduced and on the other hand may violate the principle of local equilibrium, the basis of irreversible thermodynamics and, therefore, the basis of Fick's first law.

Changes of the chemical potential are caused by changing boundary conditions as in an electrochemical permeation experiment, where the application of an anodic potential on one side of a disc reduces the chemical potential at this side, whereas its value at the other side is kept constant with a constant cathodic current [15]. After a transient time, a steady state is reached with a flux independent of time corresponding to a constant gradient of the chemical potential according to equation (3.1). The gradient of H-concentration is less simple and will be determined from the chemical potential via equation (2.3). The transient behaviour during a permeation experiment is described by Fick's second law 3.10 where the diffusion coefficient depends on concentration as described by equations (3.8) and (2.3). Numerical solutions for a simple one-trap system are provided in [16]. It can be shown that values of trapping energies as determined by electrochemical permeation as given in the literature are often less reliable, because they are evaluated on the basis of non-saturated traps as given in [2].

During a TDS experiment (thermal desorption spectroscopy), a sample is placed in a vacuum chamber and the temperature is raised in a linear fashion. Here, the boundary condition is zero concentration on the surface and, as before, equations (3.9) and (3.10) have to be solved numerically. Again, a McNabb–Forster formalism will complicate the procedure and/or falsify results. Approximate analytical solutions of equations (3.9) and (3.10) in combination with equations (2.3) and (3.8) are given in [17].

## 4. Chemical potential of hydrogen in the presence of stress

Starting with the differential of the Gibbs free energy *G* and including the work due to mechanical stress *σ _{ik}* and related strain

*as*

_{ik}*d*

_{ik}*σ*leads to 4.1 where

_{ik}*S*is the entropy,

*T*the temperature,

*V*

_{m}the molar volume,

*μ*

_{M}the chemical potential of the metal (i.e. iron),

*μ*

_{H}the chemical potential of hydrogen and

*n*

_{M}and

*n*

_{H}are the number of moles of solvent and solute. As

*G*is a thermodynamic state function, its second derivatives should be independent of the sequence of differentiation leading to the so-called Maxwell equations. Based on equation (4.1), the following Maxwell equation is derived 4.2

The strain induced by dissolving hydrogen in BCC-iron is unknown. For hydrogen atoms dissolved in tetrahedral sites of BCC-niobium, it is shown to be isotropic [18], i.e. , where *V*_{H} is the partial molar volume of hydrogen. Then equation (4.2) can be simplified to
4.3
Thus, the chemical potential of hydrogen is changed by hydrostatic stress components only. Hydrogen in palladium is an ideal model system to reveal the validity of equation (4.3), because changes of *μ*_{H} for applying tensile stress or torsion can be measured *in situ* via changes of the electromotive force. In addition, the partial molar volume of hydrogen is known [18] and H-atoms are dissolved in octahedral sites of face-centred cubic (FCC)-Pd which leads to isotropic strain due to the symmetry of these interstices. Corresponding experiments are published in [19,20].

Equation (4.3) is used to calculate trapping energies in the stress field of dislocations. As the hydrostatic stress around screw dislocations is zero, these dislocations will provide trap sites for H-atoms in their core only. Note that the scenario is different for carbon atoms in BCC-iron inducing a tetragonal distortion and corresponding changes of its chemical potential, i.e. trapping of C-atoms in the strain field of screw dislocations occurs (cf. equation (4.2)). Positive hydrostatic stresses exist below the glide plane of an edge dislocation, lowering *μ*_{H} with a concomitant flux of H-atoms towards the dislocation core and the related increase in the local hydrogen concentration until the differences of *μ*_{H} are annihilated. The H-enrichment and the concomitant decrease in H–H distances may become significant to cause deviations from the ideal dilute behaviour with increasing H/H-interaction and a final formation of a hydride. Within this scenario, a cylinder of hydride is formed along the dislocation line. For the Pd–H system, this has been experimentally proven by small angle neutron scattering [6] with diameters of the cylinder varying with H-concentration from zero up to 2 nm. Corresponding computational studies have also shown the existence of these so-called nanohydrides in the Ni–H system [9].

Equation (4.3) is also often used in the context of hydrogen embrittlement for calculating the enrichment of hydrogen in front of a crack tip under tensile load by assuming an ideal dilute behaviour of H-atoms . The change of *μ*_{H} caused by stress will induce a flux of H-atoms due to equation (3.1) towards regions of tensile hydrostatic stress until equilibrium is reached with a chemical potential being constant and an H-concentration increasing towards the crack tip. The H-enrichment and the concomitant decrease in H–H distances may like in the case of edge dislocations become significant to cause H/H-interaction and a final formation of a hydride. This has been shown for metals with a high hydrogen solubility like niobium [21] where a brittle hydride forms in front of the crack tip allowing the crack to propagate by brittle fracture. In the case of BCC-iron and ferritic steels, the H-solubility is so low that the changes of the chemical potential become not large enough to lead to iron hydride formation. In addition, part of the stresses in front of the crack tip relax within the plastic zone leading to a decreased H-enrichment. In some of the models on hydrogen embrittlement like hydrogen enhanced local decohesion [22] or hydrogen enhanced local plasticity (HELP) [23], it is assumed that the increased H-concentration in front of the crack tip is responsible for accelerated crack propagation when compared with the hydrogen free case. However, it will be shown in the following that the chemical potential of hydrogen decreases the formation energy of discontinuities and, therefore, independent of the concentration increase hydrogen increases crack velocity. This applies for all discontinuities as, for instance, by (i) the formation of the fracture surface, (ii) the formation of voids and void coalescence, or (iii) the emission of dislocation. Then the onset of fracture is not determined by reaching a critical H-concentration but by reaching a critical chemical potential with an induced critical density of discontinuities, i.e. critical crack length, void size or dislocation density. To reach these critical densities of discontinuities may take a long time leading to delayed fracture as often observed in steels.

## 5. Generation and annihilation of discontinuities in the presence of hydrogen

Since Gibbs [24], it is known that with increasing chemical potential *μ* of a solute entity, the energy of generation of a new surface *γ* is reduced in the presence of a positive solute excess *Γ* at the surface. The corresponding solute atoms or molecules are called surfactants. The related Gibbs adsorption equation is
5.1
This relation has been used to show that the ideal work of intercrystalline fracture is decreased by hydrogen [8,25,26] and phosphorous [27] in iron. It was shown further on that equation (5.1) is also valid for other solutes and discontinuities besides hydrogen, surfaces and grain boundaries [28–31]. Like in the previous chapter, the corresponding derivation is based on a Maxwell equation. Here, a new thermodynamic state function Φ is used as introduced by Wagner [32]
5.2
where *F* is the Helmholtz free energy, *n*_{H} and *μ*_{H} are the number of moles hydrogen and the corresponding chemical potential. For the sake of simplicity, hydrogen is used as the solute, despite the fact that equation (5.2) and the following are also valid for other solutes. The differential of the state function is
5.3
where d*W* is a work term like the stress/strain-term in equation (4.1) or in the present context, it is the differential work to create a discontinuity
5.4

The formation energy of the discontinuity is *γ* (for instance, the surface energy) and the volumetric density of discontinuities is expressed by *ρ*. Then *V*d*ρ* corresponds to a differential change of the surface area, dislocation length or number of vacancies, etc. Taking second derivatives of the state function *Φ* leads to the following Maxwell equation
5.5
The last derivative on the r.h.s. is the excess hydrogen *Γ*, because it describes how much hydrogen d*n*_{H} is taken from a reservoir of constant chemical potential *μ*_{H}, if the volume density of discontinuities is changed by d*ρ* and by keeping the volume, temperature and number of metal atoms constant. Thus, equation (5.5) can be presented in the simplified form of equation (5.1); but now also being valid for zero-dimensional and one-dimensional discontinuities. For surfaces with *V*d*ρ* = d*a*, equation (5.5) is also derived differently by Cahn [33]. There are many experiments showing that surfactant molecules (SURFace ACTing agANT) reduce the surface energy of water or the interfacial energy between oil and water. A new term defactants (DEFect ACTing agANT) [31] was coined for atoms segregating at discontinuities reducing their formation energy according to equations (5.1) and (5.5). Among the two-dimensional discontinuities, grain boundaries are known to reduce their formation energy by segregation of defactants. Then in many cases, large grain boundary areas or small grain sizes, respectively, are obtained by adding defactants to a metal [34–37].

For vacancies in metals, it is known that hydrogen atoms segregate at these defects and increase their concentration [38]. However, these experimental findings have not been explained or evaluated in the context of the generalized Gibbs equation (equation (5.1)). For other solute atoms acting as defactants for vacancies, equation (5.1) has been validated [30]. During plastic deformation of crystalline materials, vacancies may affect creep or the formation of voids in front of an advancing crack tip. For hydrogen, the latter effect is part of a model on hydrogen embrittlement proposed by Nagumo [39], assuming that the increased concentration of vacancies gives rise to a higher rate of void formation and concomitant void coalescence in front of an advancing crack tip.

For dislocations, it has been shown only recently that hydrogen decreases the formation of new dislocation segments. This can be shown by nanoindentation [28,40–42], where the onset of plastic deformation starts at lower stresses in the presence of hydrogen. In addition, cold rolling of palladium leads to higher dislocation densities when performed with dissolved hydrogen. The ease of dislocation generation by the defactant hydrogen can be used as a basis for both of the H-embrittlement models HELP [23] and AIDE (adsorption-induced dislocation emission) [43]. In the HELP model, a crack propagates by emitting dislocation with preferred planar slip. Although increased crack velocity is assumed to be due to accelerated dislocation motion caused by internal hydrogen (see the following paragraph), it may also be that just the dislocation generation at the crack tip is enhanced by the defactant hydrogen. Note that this is controlled by the chemical potential of hydrogen and not by the increased H-concentration in the stress field of a stressed crack. In the AIDE model, external hydrogen is adsorbed on the crack surface where it induces hydrogen emission, which by the defactant concept would occur with a decreased formation energy. For the chemical potential of hydrogen determining the dislocation generation rate, it does not matter whether the chemical potential is determined by internal or external hydrogen as long as hydrogen transport is not rate controlling.

It has been shown *in situ* in an electron microscope that hydrogen enhances or triggers dislocation motion. This will be interpreted in the following in the framework of the defactant concept. On the atomic scale, dislocations move by generation of a kink pair where the two kinks move in opposite directions along the dislocation line until they reach a pinning point. Then the whole dislocation line has moved from one Peierls valley to an adjacent one. The time needed may be either the time for generating the kink pair or the time for moving to the pinning points. Kinks are discontinuities and their formation energy should be decreased in the presence of hydrogen segregating to places, where kink pairs are formed. Thus, a dislocation moves faster, if the rate controlling process is kink pair formation being accelerated by hydrogen. However, if the motion of the separated kinks to the pinning point is rate controlling, hydrogen segregating to the kinks will retard their motion and, therefore, slowing down the dislocation motion as a whole. On the macroscopic scale softening occurs by hydrogen, if kink pair formation is rate controlling, whereas hydrogen hardening is due to kink motion being rate controlling [44]. Again the chemical potential being a suitable controlling parameter in this case as in other cases presented in figures 8–10.

## Data accessibility

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## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Appendix A. Random walk in an energy landscape

We consider *N*_{H} particles in *N*_{o} sites of a lattice with a given density of site energies. The following derivation is rather general and, therefore, it is applicable to any system being modelled as a lattice gas. The equilibrium distribution of the particles according to FDS shall be stationary in space and one tagged particle shall do a random walk via the empty lattice sites. It might not be really necessary to assume a stationary distribution for the remaining particles. What might be the important effect only is that the moving particle will not be able to visit all lattice sites as the low-energy sites are preferentially occupied according to FDS. One might call this simplification the one-particle approximation resembling the one electron approximation of solid-state physics. Further on the walk is uncorrelated besides the few blocking events and, therefore, the mean distance *R* after *Z* jumps is given according to simple random walk theory [47] as
A 1
where *a* is a constant jump distance. Then a diffusion coefficient can be defined in [47]
A 2
where *t* is the total time required for the walk. The time *t* is the sum of all the times of residence *τ _{m}* in the various sites labelled

*m*which the particle visited during the walk. Here, it has been tacitly assumed that the time interval required for the site exchange is small compared with

*τ*. Thus, we get A 3 This way the jump rate is the reciprocal of the residence time and it has not to be described by a special mechanism like thermally activated hoping for instance or quantum mechanical tunnelling through an energy barrier.

_{m}Among the various sites, we combine those having the same energy *E _{i}* (

*i*being a subset of sites labelled

*m*) and average over the distribution of empty sites, i.e. sum over all

*N*−

*N*

_{H}sites. Thus, we obtain from equation (A.3) A 4 where

*t*is the time the particle spends in sites of energy

_{i}*E*. It is tacitly assumed that the number of jumps

_{i}*Z*and the number of lattice sites

*N*

_{o}are large enough in order for the laws of random walk and statistical mechanics to be valid. Then the

*Z*sites which have to belong to the empty category are as representative for empty sites as are the

*N*

_{o}−

*N*

_{H}ones. If

*N*are the number of sites having energy

_{i}*E*, the probability that they belong to the empty ones is given by FDS as A 5 where

_{i}*o*(

*E*) is the occupancy of sites of energy

_{i}*E*(cf. equation (2.3)). Thus, the number of empty sites of energy

_{i}*E*is

_{i}*N*=

_{ie}*N*[1−

_{i}*o*(

*E*)] and these can be visited by the tagged particle only.

_{i}According to the ergodic hypothesis, the fraction of time a particle spends in empty sites of type *i* is equal to the occupancy of particles in this type of site yielding
A 6
For a box-type distribution, the term on the very r.h.s. has a maximum for *μ* = *E _{i}*, i.e. particles with energies around the chemical potential or Fermi level, respectively, contribute to diffusion only. This is realized for particles migrating through amorphous materials (cf. [3,4,10]). For a bimodal distribution with a large fraction of normal lattice sites (

*E*=

_{i}*E*) and a remaining small fraction of traps, the maximum in equation (A6) will be at

_{o}*E*=

_{i}*E*

_{o}>

*μ*. Note that the tagged particle needs the sites of subset

*i*with

*μ*=

*E*to be connected, in order to walk through the lattice. For the traps at dislocations or in grain boundaries that may be the case leading to short circuit diffusion along these defects. The present analysis is applicable to these cases also by neglecting the bulk and considering diffusion in grain boundaries only (cf. [7] and figure 5). Strictly speaking, there may be also correlation effects for isolated and filled traps, because the tagged particle is not allowed to jump into the occupied trap but has to choose one of the remaining directions. Correlation effects are known for substitutional diffusion via the vacancy mechanism. However, these effects are small when compared with the exponential dependencies arising by FDS or thermally activated jump mechanisms.

_{i}For the dilute solution, most of the free sites have a low occupancy and equation (A5) gives
A 7
This way FDS is replaced for the empty sites by Boltzmann statistics. For convenience, we choose an arbitrary reference site having energy *E*^{o} and a low value of *o*(*E*^{o}) as in equation (A7). Note that *E*^{o} could be *E*_{o}. Then the following relation is derived from equations (A6) and (A7)
A 8
where *t*^{o} is the fraction of the total time *t* particles reside within sites of energy *E*^{o}. Inserting equation (A8) in equation (A4) [1 − *o*(*E*^{o})]≈1 and equation (A5) gives
A 9
or
A 10
where the last sum on the r.h.s. is nothing else than equation (2.3) replacing the continuous distribution *N*(*E*) by a discrete form and, therefore, it can be expressed by *N*_{H}. Then the following simple result is obtained
A 11
where *τ*_{r} is the mean residence time as defined by
A 12

Inserting equation (A11) into equation (A2) yields A 13

## Footnotes

One contribution of 24 to a discussion meeting issue ‘The challenges of hydrogen and metals’.

- Accepted March 19, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.