Appl. Phys. A 72, 197–208 (2001) / Digital Object Identifier (DOI) 10.1007/s003390100773

Applied Physics A Materials Science & Processing

Diffusion of hydrogen in heterogeneous systems A. Herrmann, L. Schimmele, J. Mössinger∗ , M. Hirscher∗∗ , H. Kronmüller Max-Planck-Institut für Metallforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany Received: 19 September 2000/Accepted: 6 November 2000/Published online: 9 February 2001 –  Springer-Verlag 2001

Abstract. The effective long-range long-time tracer diffusivity Deff for interstitial diffusion of hydrogen through heterogeneous systems was studied theoretically for model systems consisting of isolated grains of material G embedded in a matrix of material M. Different solubilities of hydrogen in these two materials as well as different diffusivities are allowed for. Additionally, modified diffusion barriers at the phase boundaries were included in the diffusion model. The effect of different sizes, arrangements, and forms of the grains was also considered. Deff was determined by Monte Carlo (MC) simulations on simple lattice models of the systems described above. An equilibrium distribution of hydrogen atoms among the two constituent materials was assumed. Our main interest was focused on whether and how Deff may be related to mesoscopic or macroscopic quantities characterizing the heterogeneous system and its constituent materials, such as the volume fractions of the two materials, the fraction of lattice sites in the immediate vicinity of the phase boundary, the hydrogen concentrations c G and cM in the grains and in the matrix and the respective hydrogen diffusivities DG (cG ) and DM (cM ). In order to obtain good estimates for these relations in terms of analytic formulas, we attempted to model a heterogeneous system by a network of diffusion elements connected in series and in parallel, in analogy to an electric network. The properties of the basic connections, in parallel and in series, were studied on layered structures, for which analytic expressions for Deff could be derived. The network formulas for different grain–matrix systems were tested by comparing with results of MC simulations. In general, the network formulas describe the corresponding MC results for Deff fairly well. It was found that differences in the hydrogen solubilities in the two phases as well as modified energy barriers at the phase boundaries may have dramatic effects on Deff . PACS: 66.30 Dn; 66.30 Jt ∗ Present

address: Robert Bosch GmbH, Stuttgart, Germany author. (Fax: +49-711/689-1912, E-mail: [email protected])

∗∗ Corresponding

When using alternative energy sources, such as solar radiation, wind force, etc., a regular question to answer is the one about storage of energy. Chemical energy in hydrogen produced by decomposition of water is one possible form of stored energy, and metal hydrides used in hydrogen storage systems or batteries can be important components in the technical application of this energy form [1]. The usage of nanocrystalline or certain heterogeneous multiphase alloys as hydrogen storage materials is very promising due to their good absorption and desorption kinetics and to their good mechanical stability during cyclic operation. Hydrogen diffusion through such heterogeneous materials is an important issue, as the diffusivity crucially enters into the charging and discharging time of storage systems or batteries. The diffusion of hydrogen in heterogeneous alloys is strongly affected by their microstructure [2–13]. Therefore, the aim of the present paper is to provide, in terms of analytic formulas, good estimates for the effective long-range diffusivity Deff through a heterogeneous material composed of isolated grains of material G embedded in a matrix of material M. The investigations are focused on the dependence of Deff on quantities such as the diffusivities of hydrogen in the constituent phases, the volume fractions of the phases, solubility differences between the phases, and additional energy barriers at the phase boundaries. Furthermore the influence of the grain form and arrangement was investigated. Most of the following discussion is restricted to systems with isolated grains and to statistically isotropic systems. It is important to notice that the diffusivities DG and DM , which characterize the hydrogen diffusion within a grain and within the matrix phase, respectively, depend not only on temperature but also on the hydrogen concentrations c G and cM in the respective constituent materials. These concentrations have to be determined for the actual heterogeneous material under study and the chosen experimental conditions. In the following we always assume that hydrogen is distributed between the two materials according to thermal equilibrium (see Sect. 3). The concentration dependence of the diffusivity is particularly pronounced if the material is disordered, for instance if the matrix is amorphous.

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Different methods were used to tackle the problem (see Sect. 2). Firstly, Monte Carlo simulations were performed on a simple periodic lattice model, described in Sect. 1, which includes all the essential features of a grain–matrix system, mentioned above. Another approach used was to model the problem of diffusion through a grain–matrix system as closely as possible by a network of diffusion links connected in series and in parallel (as, for example, done in [14–16]). These models will be called network models in the following. The ‘addition’ rules for the two basic connections, i.e., the parallel and series connections, were studied on layered structures. These were modelled by lattice systems of stacked layers with two-dimensional periodicity, and the diffusivity Deff was determined by a third method, i.e., from the longtime, long wavelength solutions of master equations for the site occupancies. For these quasi-one-dimensional systems in the limit of infinite dilution and in the long-time limit, the master equations can be solved analytically for arbitrary stacking and these solutions provide us with the rules for connecting two diffusion links in parallel or in series, respectively. It turns out, as will be discussed in Sect. 4, that these rules are identical to those for resistor networks, if the electrical conductivity in material i is replaced by the product c i Di (here, i = G,M). By introducing c i Di instead of just Di different solubilities of hydrogen in the different constituent materials can be allowed for in the network models, in contrast to similar models considered in the literature (e.g., [15, 17]). Eventually, additional energy barriers at the transition of hydrogen from the matrix into a grain and vice versa, which is also generally not considered explicitly in the literature, may be accounted for by introducing additional diffusion links. Several network models which were constructed on the basis of the above considerations but otherwise merely intuitively, are described in detail in Sect. 5. The quality of these models was tested by comparison with Monte Carlo simulations of hydrogen diffusion in model systems. Of course the appropriate cG (T, c), cM (T, c), DG (T, cG ), and DM (T, cM ) for the model system have to be determined and plugged into the network formulas to be tested. The network formulas, which originally were developed for a special geometry of the system, have been generalized so as to be applicable to more general systems. We close this introduction with two remarks. Firstly, we expect that the rules for parallel and serial connections of diffusion links do not depend on the hydrogen concentration as long as the diffusion properties in each link (or at least in a typical ensemble of links) may be sensibly described by bulk or mesoscopic diffusivities. All complications due to blocking of sites, concentration dependent activation or site energies, complicated diffusion mechanisms, or quantum effects may be absorbed in the macroscopic (or mesoscopic) quantities like DG , DM , cG and cM . In particular, our analytic formulas describing Deff (see Sect. 5) as a function of DG , DM , cG , cM , etc. should be valid in systems with arbitrary hydrogen concentration. Therefore we concentrated our Monte Carlo investigations mainly on systems with low hydrogen concentration (c = 0.01) and carried out only a few calculations at higher concentrations up to c = 0.999 in order to test the validity of the analytic formulas over the whole concentration range. Secondly, Deff , as studied in the present paper, describes the long-time long-range tracer diffusivity for thermal equi-

librium with respect to the hydrogen distribution between the two materials. Therefore, strictly speaking, Deff can only be applied to the study of charging or discharging of a sample with hydrogen if thermal equilibrium between the hydrogen concentrations in the constituent materials holds locally. In other words the local hydrogen concentrations in the grains and in the matrix should follow to a good approximation from the average concentration c(r), taken over volumes containing several grains, and the requirement of local thermodynamic equilibrium. 1 Model of the heterogeneous material The heterogeneous material is modelled by a periodically repeated three-dimensional unit cell containing one crystalline grain (phase G) which is embedded in a matrix of a different material (phase M). (Only when the effect of the arrangement of the grains is investigated do several grains have to be placed in one unit cell.) Hydrogen is assumed to diffuse on a periodic lattice of (interstitial) sites. The same lattice, a face centered cubic lattice with cubic lattice parameter a, has been chosen for the grains and for the matrix phase, and diffusion has been assumed to proceed in each √ case via nearest-neighbour jumps with the jump length d = 2a/2. It is further assumed that the hydrogen atom blocks the site on which it sits but no further sites. Sites belonging to either a grain or to the matrix are distinguished by their different probabilities to be occupied by hydrogen and/or by different characteristic hopping rates among them. The first of these differences has been introduced in order to model the different hydrogen solubilities in the two materials, the second to account for different diffusivities. The boundaries between grains and matrix have been assumed to be perfectly sharp, i.e., a site either belongs to a grain or to the matrix. The hopping rates across the boundaries are treated separately. A number of elementary cells of the diffusion lattice are combined to form a supercell, the unit cell of the periodic grain–matrix structure mentioned at the beginning of this section, which is then repeated periodically. In the present case a√rhombohedral supercell with edges √ √ n (110) 2/2, n (011) 2/2 and n (101) 2/2, i.e., parallel to the jump vectors, has been chosen. Typically the length n of the edge of the unit cell was chosen to be n = 51 d for a concentration c = 0.01. For higher concentrations n was chosen to be smaller (see Sect. 2). In most of our Monte Carlo studies one rhombohedral grain with an edge length l (l < n) has been placed in the supercell. The grain is completely surrounded by the matrix phase which has a width of b = n − l (a two-dimensional presentation of the system is shown in Fig. 1). For convenience of presentation in the following we always draw rectangular supercells and grains (Fig. 1b). A few systems with grain shapes other than a rhombohedral one or with different arrangements of the grains were also investigated for comparison. We focused our investigations on rhombohedral grains for several reasons. For such a system the jumps across the phase boundaries are especially important: For an atom sitting at the grain boundary, 4 of 12 possible jump vectors cross the phase

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dimensional cut of the potential surface shown in Fig. 2). Q ij together with E i parameterizes the distribution of the hopping rates from site i to its nearest neighbour j (if site j is empty) by assuming thermally activated jumps described by νij = ν0 exp[−(Q ij − E i )/kT ],

Fig. 1a,b. Unit cell of our model of a heterogeneous material. Shaded: grain phase G, white: matrix phase M. a Rhombohedral unit cell and rhombohedral grain (only one plane is shown); b schematic. l: edge length of a grain, n: edge length of the unit cell, b: width of the matrix phase

boundary, a fraction which is not achieved for any other grain form. Moreover for rhombohedral grains in a face centered cubic lattice zigzag diffusion paths of consecutive boundary jumps are possible, and this additional diffusion paths may simulate diffusion along a phase boundary. The model of hydrogen diffusion in the heterogeneous material described above may best be visualized and parametrized by introducing a schematic potential landscape, as shown in Fig. 2, in which the hydrogen atoms move. The energies of interest for the present purpose are first the site energies E i at a site i (energies at the potential minima), which determine the site-occupation probabilities, and secondly the energies Q ij at the saddle points between pairs of nearest neighbour sites i and j (energy maxima in the one-

Fig. 2. Schematic potential landscape for interstitial diffusion of hydrogen through a heterogeneous system

(1)

where k is the Boltzmann constant, and ν0 an attempt frequency chosen to be common to all nearest-neighbour hopping processes. In our description we take into account a possible random energetic disorder in the matrix phase which is modeled by a Gaussian distribution of site and saddle point energies. The distribution functions for the site and saddle point ener1 exp[−(E − E)2 /2σ 2 ], with gies take the form n(E) = √2πσ the appropriate average energies E and widths σ of the distributions (see also [18]). The distributions of site and saddle point energies have been taken to be uncorrelated, except for the restriction that the energy of a saddle point must not be lower than the site energies of the adjacent sites. Other correlations between site and saddle point energies which can possibly be observed in real materials (see, e.g., [19, 20]) are not considered. As shown in Fig. 2, we have introduced the notation E M  for the average site energy in the matrix, σ P for the corresponding width of the distribution, Q M for the average saddle point energy Q ij , and σS for the distribution width of the saddle point energies. In the grains no energetic disorder has been allowed for. The constant site energy in the grains is E G = E M  + ∆E, where ∆E can be positive or negative. The constant saddle point energy in the grains is given by Q G . As a possible phase-boundary effect we include a modification by Q B of the saddle point energy for jumps between a site in the grain and one in the matrix. Q B may have either a positive or a negative sign. Q B is defined as the difference between the saddle point energy for a jump over the phase boundary and the average saddle point energy in that phase in which the average site energy is higher (see Fig. 2). Furthermore, the system is characterized by the volume fraction vG = (l/n)3 of the grains and by the grain form. The hydrogen concentration c averaged over the whole system is an external parameter. (c in this work is defined as the ratio of the number of occupied sites to the total number of sites which could be occupied by hydrogen.) The model described above is rather special with respect to the assumed diffusion mechanism and the law (1) describing the temperature and concentration dependences of the hopping rates. However, we are interested here in the general aspects of the problem of diffusion through a heterogeneous material. The investigation of a special, but nevertheless characteristic model of diffusion and of a heterogeneous material, serves mainly as a means to develop and test the formulas representing Deff as a function of the properties of the constituent materials and of purely geometric quantities. However, at the end of the investigation any conceivable complications in the diffusion processes in the constituent materials can be included by plugging the correct diffusivities in these materials into the final formulas. In a sense, the model considered here is already more complex than absolutely necessary for our purposes, because it explicitly models the matrix as an energetically disordered phase, although this property could also be included at the

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end, as described above. However, by including this additional detail, it becomes possible to test whether it is always appropriate to describe diffusion through possibly narrow matrix channels by diffusivities characteristic of bulk materials, although it is conceivable that for diffusion through narrow channels percolation-like effects may play a rôle. The Monte Carlo simulations do not give any indication that this might be the case, and thus no detailed studies in this direction were undertaken. Furthermore, including an energetically disordered phase allows us to give a more general discussion of diffusion across the phase boundaries. 2 Methods To determine the diffusion of an ensemble of atoms in the heterogeneous system described above, we applied two methods. Monte Carlo simulations are a common method to determine diffusion constants. Atoms are placed on the available lattice sites at random, but according to their site occupancy probability, until the given average concentration c is reached. Then one atom is chosen at random to perform jumps statistically with given jump probabilities. At the end of the simulation the long time tracer diffusion coefficient (termed Deff in the present paper) Deff =

Ri2  1 lim 6 i→∞ iτ

(2)

is determined averaging over all hydrogen atoms and over several simulations with different distributions of the hydrogen atoms at the beginning [2, 18]. Ri is the distance of an atom from its starting point after i jumps, τ is the mean time between two consecutive jumps of an atom. The movement of about 1300 atoms (which for an overall concentration of c = 0.01 corresponds to a unit cell length of n = 51d; for higher concentrations the unit cell of the periodic system was chosen to be correspondingly smaller) for about 10 9 jumps per atom was simulated and four such simulations per parameter set were performed. The statistical error of the results is estimated to be less than 3%. The second method, applicable to systems of low hydrogen concentration, involves the writing of the master equations for the site occupancy probabilities ci (R, t) = c(R + ri , t) as a function of time t, ri giving the position of site i in the unit cell with respect to the origin of the cell at position R. The master equations read n n

dci = c j ν ji − ci νij , dt j=1 j=1

(3)

with the jump rates νij for a jump from site i to site j. Performing a Fourier transformation in space and a Laplace transformation in time, one obtains for periodic systems a finite linear equation system. In the long-time, longwavelength limit (s → 0 and k → 0) the Fourier and Laplace transformation G i (k, s) of ci (r, t) adopts the form Gi ∼

s+

Dx k 2x

1 + D y k 2y + Dz k 2z

(4)

with the diffusion coefficients D x , D y and Dz parallel to the corresponding coordinate axes.

3 Properties of the constituent phases, diffusion across phase boundaries The equilibrium hydrogen concentrations c G and cM in the two phases at a given average hydrogen concentration c enter the formulas for Deff , which will be obtained from the network models (see Sect. 5), in two ways. Firstly, they come into play implicitly via the concentration dependences of the diffusivities DG (cG ) and DM (cM ), and secondly, explicitly in terms of weight factors. In order to obtain accurate values for cG and cM for specific model systems, e.g., Fermi statistics can be used (the so-called step approximation [21], however, is generally not sufficient) together with the density of states for the site energies which adequately describes the heterogeneous system. Alternatively, values obtained from Monte Carlo simulations may be used or the stationary solutions of rate equations, which also include blocking of occupied sites, may be determined (see, e.g., [22]). In the particular case of a crystalline matrix and crystalline grains, i.e., σP = 0, one obtains  1 c 1 cM = + (1 − vG ) + xvG 1 − vG 2 2(1 − x)   ± [vG x + (1 − vG) + c(1 − x)]2 − 4(1 − x)c(1 − vG) (5) with the volume fraction vG of the grains and x = exp [(E M − E G)/kT ] = exp(−∆E/kT ). The + sign applies if 2 vG (1 + x) + c (1 − x) − 2 > 0; otherwise, the − sign applies. cG follows from the conservation law c = vG cG +(1 − vG)cM . If σP = 0, only numerical solutions exist. The tracer diffusivities DG (cG ) and DM (cM ) were determined from Monte Carlo simulations for diffusion in the constituent model materials of the heterogeneous model systems. In any case the time-independent long-time limit of the diffusivity was used. For the sake of working always on the same basis, we chose to use our own Monte Carlo data (see also [18]), although analytical results (for crystalline materials and low c) and approximate analytical expressions or Monte Carlo data, comparable to ours, exist in the literature. See, for example, [23–25] for crystalline materials and [26–30] for energetically disordered systems. In addition to knowledge of the diffusion properties of the single phases, a characterization of atomic transport across the phase boundaries is required. For this purpose two ‘diffusivities’, DMG for a description of the movement of hydrogen from the matrix phase into a grain and D GM for the reverse process, are introduced. These are formally defined by DMG/GM = d 2 (Z/6) νMG/GM ,

(6)

where Z is the number of possible jump directions (Z = 12 for an fcc diffusion lattice) and νMG is an appropriate average over the distribution of jump rates ν ij from a site i in the matrix and at a grain boundary to a neighbouring site j in the grain. νGM is defined correspondingly. Correlation effects between multiple jumps over the boundary are not included in (6). The definition of two diffusivities for the characterization of diffusion across phase boundaries seems to be superfluous, and indeed they always appear symmetrically as

201 −1

−1

(cM DMG ) + (cG DGM ) in our final expressions for Deff . However, the chosen definition fits best into the general framework developed in the next section. The weighted average over νij was calculated according to νMG/GM = ci νij /cM/G ,

(7)

where ci is the occupation probability of site i (in the matrix if νMG is considered or in the grain otherwise). Experimentally the required combination of D MG and DGM might be deducible from investigations on layered structures. In order to calculate νMG/GM and thus the ‘diffusivities’ across the boundary for the specific model system studied by the Monte Carlo simulations, we start from available expressions for the average jump rates within a single phase. For thermally activated jumps [see (1)] one must distinguish two cases in an energetically disordered phase. If all the sites are energetically equivalent (i.e., σP = 0, E i = E) the average jump rate in any given direction is given (if E is chosen to be the energy zero) by   ν = ν0 (1 − c) e−Q/kT , (8) with the attempt frequency ν0 and an average . . .  which runs over a possible spectrum of saddle point energies Q. To treat the case with site energy disorder (σP = 0) we generalized a formula for ν originally derived by Kirchheim and Stolz [29] for a model with constant saddle point energies. We write (1 − c)2 µ/kT  −Q/kT  e e , (9) c where the energetic disorder of the site energies is characterized by the chemical potential µ, which itself is concentration dependent. The average site energy was defined to be zero. Equation (9) was confirmed by Monte Carlo simulations. The appropriate νMG and νGM are now determined from (8) and (9) by adapting these to the respective situations. Since the saddle point energies for the hops across the grain boundaries have no energy distribution in the chosen model, the averages (. . . ) in (8) and (9) can be omitted. ν MG/GM are then given as follows: To determine νGM ,use (8) with the substitutions ν = ν0

c → cM , Q → Q G + Q B , if ∆E > 0, Q → Q M + Q B + |∆E| , if ∆E < 0.

(10a) (10b) (10c)

To determine νMG , use (8) if σP = 0 (no distribution of site energies) and the substitutions c → cG , Q → Q G + Q B + ∆E, if ∆E > 0, Q → Q M + Q B , if ∆E < 0.

(11a) (11b) (11c)

If σP = 0, use ((9)) with the substitutions (1 − c2)/c → (1 − cG)(1 − cM)/cM , Q → Q G + Q B + ∆E, if ∆E > 0, Q → Q M + Q B , if ∆E < 0, µ → µ(cM ),

(12a) (12b) (12c) (12d)

where µ(cM ) is the chemical potential at the given concentration cM in the matrix phase. 4 Diffusion in layered systems As the simplest example of a heterogeneous system we first examined the diffusion of hydrogen through layered structures. Layered structures were modelled by lattice systems of stacked layers with two-dimensional periodicity. For these quasi-one-dimensional systems in the limit of infinite dilution and in the long-time limit, the master equations can be solved analytically for arbitrary stacking. These solutions provide us with the rules for connecting two diffusion links in series or in parallel, respectively (the diffusivity D x with x perpendicular to the layers represents the series connection; D y with y in the layer plane represents a parallel connection). It turns out that these rules are identical to those known for resistor networks if the electrical conductivity in material i is replaced by the product ci Di (here, i = G, M), where Di is the diffusivity in material i and ci the equilibrium concentration of hydrogen in that material. This may be demonstrated as follows: The resistivity of a piece of conductor of length l and cross-sectional area A is given by R = l/(A σ), with the material-specific conductivity σ. Replacing σ by ci Di , an analogous quantity Wi characteristic of a ‘resistance’ to diffusion through a diffusion link i of length l i and cross-sectional area A i may be defined by Wi = li /(Ai ci Di ).

(13)

The quantity W characterizing diffusion through a whole network of diffusion links can be calculated from the W i of its individual elements by the same rules that apply to the calculation of the total resistance of a network of resistors. For example, in a series connection we may write W = i Wi . If we apply this rule to hydrogen diffusion through a stack of alternating layers of materials 1 and 2 and in the direction x perpendicular to the layers, we observe first that the crosssectional areas perpendicular to the current direction are all equal, i.e., A i = A, with A being the area of the front face of the stack. We may thus write Wi = li /(A ci Di ) with the layer thicknesses li . If for simplicity we assume that all layers of material 1 have a thickness l 1 , and those of material 2 a thickness l2 (i.e., we have double layers repeated say, n times), we then get ⊥ W =: l n/(A c Deff ) = n[l1 /(A c1 D1 ) + l2 /(A c2 D2 )]

(14)

or ⊥ l/(c Deff ) = l1 /(c1 D1 ) + l2/(c2 D2 ),

(15)

where l = l1 + l2 , c is the average hydrogen concentration and ⊥ Deff is the effective diffusivity in direction x on a length scale much larger than l 1 + l2 . Solutions of master equations for such systems give exactly the expression (15) for the long⊥ time long-range diffusivity Deff in a direction x perpendicular to the layers. Monte Carlo simulations confirmed that this result holds for finite hydrogen concentration as well. we may write for a parallel connection W −1 = Similarly −1 i Wi . Again we apply this formula to the stacked layer

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system described previously, but now to diffusion in direction y parallel to the layers. We obtain 

(A1 + A2) n c Deff /L = n[(A1 c1 D1 )/L + (A2 c2 D2 )/L], (16) where L is an edge length of the front face of a layer, defining the common length of the diffusion links, and A 1 and A 2 are the areas of the side faces of the two different types of layers. Rewritten in terms of volume fractions v1 and v2 , (16) reads 

c Deff = v1 c1 D1 + v2 c2 D2 .

(17)

vi ci /c also gives the fraction of hydrogen atoms moving  through material i; thus, in order to determine D eff , Di have to be weighted by these fractions. The product ci Di of the concentration ci and the diffusivity Di appearing in (13)–(17) instead of just D i accounts for the different solubilities of hydrogen in the different constituent materials. Such solubility differences are an additional feature in the problem of diffusion in heterogeneous materials compared to the related problem of determining the effective electric conductivity in heterogeneous alloys (see, e.g., Fan [15] or McLachlan [17]). A comparison of our results with those of the works of Ash et al. [31], Barrie et al. [32], Song and Pyun [33] and Schmitz et al. [34], who investigated the diffusion in layered structures, is only meaningful if it is restricted to stationary diffusion in the limit of a large (infinite) number of layers, since our formulas only apply to such systems. Taking these limits, the earlier results agree with ours. The diffusion process across the boundary between the matrix and a grain, which is generally not explicitly considered in the literature, may be accounted for in our treatment by introducing additional diffusion links with diffusivities Dij as given by (10a)–(12d). As a side remark, the diffusivities Di in the solutions of the master equations mentioned above are formally introduced as in (6) via the hop rates among the sites modelling phase i. ‘Diffusivities’ for crossing the phase boundary can be introduced in the same way. The appearance of D i always in the combination ci Di (resulting from ci νij ) in our formulas, together with the law of parallel connection (16), strongly suggests that averaging ci νij , as done in (7), is the correct choice if one or both materials at the boundary are disordered. We add that the effective diffusivity in the series connection is dominated by the material with the lowest diffusivity Di . Thus in a system with a high energy barrier for jumps from one phase into the other the diffusivity perpendicular to the layers is governed by these jumps across the phase boundary.

Deff . The quality of those expressions are tested afterwards by comparison with mainly Monte Carlo simulations. We developed and tested several models, the best three of which we present in this section. In contrast to many similar models (see, e.g., [15, 17]) ours take account of the possibly different solubility of hydrogen in the two phases in terms of weighting factors for the diffusivities (compare to [35, 36]) and of additional resistances which simulate modifications of the energy barriers at the transition between the two phases. In the network model I, termed the model of pure parallel connection, we strongly emphasize the fact that in a grain– matrix system with isolated grains there are always diffusion paths penetrating the whole sample, which stay entirely within the matrix phase. Diffusion paths entering the isolated grains, however, must cross the matrix and the grain phase alternately. The paths through the matrix, independent of whether the matrix forms straight or winding ‘channels’, is modelled now by a diffusion link made up entirely of the matrix phase. This link is connected in parallel to a second link, which again is built up of alternating layers of the grain and matrix phases. In order to come to a quantitative formulation, the particular model described in Sect. 1, of a periodic grain–matrix structure with a single grain in the supercell is viewed along a direction parallel to an edge of a supercell. For the sake of simplicity, only grains having the same shape as the supercell are considered. (The normals of the layers are not parallel to the column axes; but this does not effect any of the following derivations.) Looked upon in this way, the grains and the intermediate matrix phase form a system of columns with constant cross-section, with the axes of the columns being parallel to the chosen particular view direction. The columns are completely surrounded by the matrix phase and the columns themselves are layered structures of alternating grain and matrix phases. The length of each grain layer in column direction is defined to be l − d and that of the matrix layer to be n − l − d, with the edge length l of a grain, the edge length n of a supercell, and the extension b = n − l of the matrix phase (see Fig. 1). The length d representing the thickness of the grain–matrix interface (chosen to be the jump length d, because in the model system this interface is assumed to be perfectly sharp on an atomic scale) is subtracted in the above definitions because two additional diffusion links each of length d are introduced in the columns between the grains and the matrix. These additional links describe diffusion from the grain into the matrix and vice versa in terms of the ‘diffusivities’ DGM and DMG as defined by (10a)–(12d). The described network together with a schematic drawing of the grain–matrix system is shown in Fig. 3. The addition rules for diffusivities derived in the last section may now be directly applied to this network. The effective diffusivity D layer through the link describing the layered columns follows from the law of series connection (15) to be

5 Network models Diffusion in a three-dimensional grain–matrix system cannot be mapped exactly onto a primitive network composed of simply a number of parallel and series connections of diffusion links. Nevertheless, in the following several such networks are considered that capture some essential features of a grain–matrix system (see also Sect. 2). The rules of parallel and series connections developed in the preceding section are applied in order to deduce the corresponding expressions for

(clayer Dlayer )−1 =

l−d (cG DG )−1 n n −l −d d + (cM DM )−1 + (cG DGM )−1 n n d + (cM DMG )−1 , (18) n

with the average hydrogen concentration c layer in the layered columns, which in general differs from the average concentra-

203

tion c in the total system because the volume fractions of the grains and the matrix in the total system and in the subsystem formed by the columns are different. I For model I, the effective diffusivity D eff through the total network is given by the parallel connection of a layered column with the link characterizing the surrounding matrix channel. Using (16), one obtains c

I Deff

= (Alayer / A) clayer Dlayer + [(A − Alayer) / A] cM DM (19)

where A layer / A = (l/n)2 is the fraction of the area A of a face of a supercell that is cut out by a layered column. The considered grain–matrix system is cubic and diffusion is thus isotropic and describable by a single effective I diffusivity Deff . Although in the derivation of D eff artificial cuts in the system in special directions were introduced, we I gives a very good representation of the assume that Deff true Deff . Comparison with Monte Carlo results presented in Sect. 6.1 to 6.4, show that this indeed is the case. I A disadvantage of the representation of D eff by (18) and (19) is the usage of the geometric quantities n and l, which have a clear meaning only for the very special periodic grain– matrix system used in most Monte Carlo simulations discussed in the present paper. Our aim, however, is to give a reasonable description of Deff for the whole class of cubic or statistically isotropic grain–matrix systems with grains completely embedded in a matrix but with non-periodic grain arrangements in general and arbitrary shape and size distributions of the grains. Even with a presumption that diffusion within the grains and the matrix is isotropic, it is still required that the arrangement of the grains does not reduce the symmetry, e.g., by a particular layering of the grains or, if the grain shape is of lower-than-cubic symmetry, by certain preferences in the orientations of the grains. In order to recast Deff as given by (18) and (19) in a form containing only quantities which have a well-defined meaning for all the grain–matrix systems envisaged, we write 1/3 2/3 l/n = vG and A layer / A = (l/n)2 = vG , which are identities for the special system considered above, using the volume fraction vG of the grains. Instead of d/n (i.e., the ratio between the thickness of the grain boundary layer, the one jump

distance in the present case, and the distance between the centers of the nearest-neighbour grains), we introduce d 1/l GG , where d is the jump distance and 1/l GG  is the average reciprocal distance between the centers of nearest-neighbour grains in the system. I Using these quantities Deff may be re-expressed as 2/3

2/3

I = vG clayer Dlayer + (1 − vG ) cM DM c Deff

(20)

with (clayer Dlayer )−1 = (vG − d 1/l GG ) (cG DG )−1 1/3

+ (1 − vG − d 1/l GG ) (cM DM )−1 1/3

+ d 1/l GG  [(cG DGM )−1 + (cM DMG )−1 ] . (21) In our model of pure series connection (model II), we assume that one hydrogen stream crosses four heterogeneous resistances in a series as shown in Fig. 4. Three of these again describe a parallel connection of two hydrogen streams. Model II differs from the preceding model I formally by ‘cutting’ the system in layers perpendicular to the considered diffusion direction and not in columns parallel to it. In model II different independent particle streams are considered within each heterogeneous layer, these streams, however, mix and exchange atoms after each such layer. Using the addition rules for the diffusivities of the network links and replacing the model-specific lengths by the more general quantities as before, we obtain the effective diffusivity in this model

 1 1/3 II −1 ) = 1 − vG − d 1/l GG  (cDeff cM DM + d 1/l GG  (cM Dpar−MG )−1

 1/3 + vG − d 1/l GG  cpar−G Dpar−G + d 1/l GG  cpar−GM Dpar−GM ,

(22)

with the following notations for the effective diffusivities through the different layers, as indicated in Fig. 4, 2/3

2/3

cM Dpar−MG = vG cM DMG + (1 − vG ) cM DM , 2/3

2/3

cpar−G Dpar−G = vG cG DG + (1 − vG ) cM DM and 2/3

2/3

cpar−GM Dpar−GM = vG cG DGM + (1 − vG ) cM DM . In our third network model (model III), parallel and series connections are combined as shown in Fig. 5. After having crossed a matrix-phase layer the hydrogen stream splits up into three: one flowing through the matrix phase, one in a series connection across the phase boundaries and in the grain phase and the third one models diffusion along a phase boundary. Then, in the following matrix-phase layer, these three streams unite again and mix. Applying again our addition rules for the diffusivities, the effective diffusivity for this model III can be written as

1/3 III −1 (cDeff ) = 1 − vG − d 1/l GG  (cM DM )−1 Fig. 3. a Model of a heterogeneous system. Shaded: grain phase, white: matrix phase, black: phase contact. b Network model I: parallel connection

+ (vG + d 1/l GG )(cpar Dpar )−1 , 1/3

(23)

204

Fig. 4. a Model of a heterogeneous system. Shaded: grain phase, white: matrix phase, black: phase contact. b Network model II: series connection

with cpar Dpar describing the parallel connection of three types of resistances (see Fig. 5). It is given by 1/3

2 cpar Dpar = vG − d 1/lGG  cS DS

 1/3 + 1 − (vG + d 1/l GG )2 cM DM 1/3

+ 4 vG d 1/l GG cGB DGB .

(24)

The first term describes the series connection characterized by cS DS of the diffusion through the grain and over two interfaces, the second term the diffusion in the matrix phase, and the third one the phase boundary diffusion. c S DS is given by (vG + d 1/lGG ) (cS DS )−1 = d 1/l GG  (cM DMG )−1

Fig. 5. Network model III for a heterogeneous system: a combination of parallel and series connections. Shaded: grain phase, white: matrix phase, black: phase boundary III to Deff in terms of detail-independent variables, may indeed be usefully applied to a wide class of heterogeneous systems. I We would like to briefly note that Deff exactly reproduces analytical results obtained for models describing simple, thermally activated hopping in systems with constant saddle point energies. In such systems, jumps from any site occur with a jump frequency that is independent of the jump direction. Hence, at low hydrogen concentration atomic jumps are uncorrelated. In general, however, the situation is much more complicated, as discussed in the following subsections.

1/3

+ (vG − d 1/lGG ) (cG DG )−1 + d 1/l GG  (cG DGM )−1 (25) 1/3

and the phase-boundary diffusion described by c GB DGB is considered to be a series connection of links for the diffusion from the grain into the matrix and vice versa and is therefore taken to be 2 (cGB DGB )−1 = (cM DMG )−1 + (cG DGM )−1 .

(26)

6 Monte Carlo simulations, tests of the network models 6.1 General remarks How well the effective diffusivity Deff in a heterogeneous material is described by the previously discussed network models was tested against Monte Carlo results as well as against rigorous results obtained by solving master equations. The aim of these investigations is twofold. Firstly, it is studied whether the network models I to III give good descriptions of Deff in those systems for which these models were originally devised, i.e., periodic systems in which the grains have a particular shape and arrangement (see Sect. 1). Secondly, it is tested how much Deff is influenced by variations in the arrangement and of the shape of the grains, provided other parameters, judged to be important in the previous section, such as the volume fraction vG of the grains, are kept constant. Only if the effects caused by the variations in shape and arrangement of the grains are small can I it be claimed that our expressions (20) to (26) giving D eff

6.2 Equal solubility in both phases In systems with equal solubility of hydrogen in both phases, for example, if ∆E = 0 and σP = 0, we can isolate the influence of diffusivity differences in the two phases on the effective diffusivity. A study of the exact diffusivities obtained by solving master equation systems for systems with I up to twelve sites shows that Deff (20), (21) is systematically II (see (22)) is sysbelow the exact value for Deff , whereas Deff III tematically too high and Deff (23)–(26) is even higher. Monte Carlo simulations for much larger systems give the same relations (see Fig. 6). I That Deff apparently is a lower bound to D eff seems to be plausible at least in cases in which the diffusivity in the matrix phase is higher than in the grains Fig. 6. Obviously, in model I, particles once having entered the diffusion link that represents the stack of grain–matrix double layers cannot avoid passing through the grain phase, which has the lower diffusivity in the considered case. In the real grain–matrix system, however, easy paths circumventing the grains and enhancing the effective diffusivity exist for all particles. By contrast, models II and III apparently overestimate the weight of the easy paths. The decrease of Deff with increasing volume fraction vG in the example shown in Fig. 6 can be easily underI stood if we look at Deff (20), (21) for instance. Since there are no solubility differences in the two phases, the hydrogen concentrations drop out of the equations and we obtain 2/3 2/3 I Deff = vG Dlayer + (1 − vG )DM . Dlayer is dominated by the smaller diffusivity, i.e., DG in the case shown in Fig. 6. In par−1/3 ticular, if DG  DM , one roughly obtains D layer ≈ vG DG (provided the volume fraction of the grains is not too low) and 1/3 2/3 I therefore in total Deff ≈ vG DG + (1 − vG )DM .

205

6.3 Different solubilities in the two phases Figure 7 shows Deff as a function of the energy difference ∆E that characterizes the solubility differences of hydrogen in the two phases. Roughly speaking, the solubility is higher in the matrix than in the grain phase if ∆E > 0 and higher in the grains if ∆E < 0. The behaviour shown in Fig. 7 is typical for grain–matrix systems with higher hydrogen diffusivity in the III matrix (DM > DG ). The agreement between Deff (23) and the II Monte Carlo data is very good, as Fig. 7 shows. Deff and in I particular Deff give very reasonable descriptions too. Two prominent features in the dependence of D eff on ∆E can be observed in Fig. 7. Firstly, for positive ∆E, D eff becomes independent of ∆E, i.e., it reaches a plateau at sufficiently high ∆E. Secondly, for sufficiently negative ∆E, Deff becomes proportional to exp(−|∆E|/kT ) as the logarithmic presentation in Fig. 7b shows. Generally, as long as the width of the site-energy distribution σP in the matrix is smaller than |∆E|, this exponential dependence of D eff on ∆E (for negative ∆E) is found as long as the grains do not become saturated. If c < vK the grains do not become saturated in the whole ∆E range. A simple understanding of the described features can be obtained by inspection of our network formulas, for inI stance for Deff . If ∆E > 0, and ∆E is increased further and further, eventually the hydrogen solubility in the grains becomes so small that the relation cG DG  cM DM holds. Then −1/3 one obtains from (21) c layer Dlayer ≈ vG cG DG (the links describing diffusion across phase boundaries can be neglected in a first approximation). Clearly, clayer Dlayer does not conI as given by (20) because of tribute substantially to Deff the inequality mentioned above, and thus we finally obtain 2/3 I Deff ≈ (1 − vG )(cM /c)DM . The plateau value of Deff is thus proportional to DM but reduced by a factor depending on the volume fraction of the grains. This factor is generally above 1/2 for vG below 80%. The interpretation of these findings is simply that a fraction of the diffusion paths is effectively interrupted by the grains, in which essentially no hydrogen dissolves. If DM < DG , the plateau in Deff (∆E) is reached at

Fig. 6. Deff as function of the volume fraction of the grains vG . Equal solI , (20)], ubility in the two phases. : Monte Carlo results, ◦: model I [Deff II (22)], : model III [DIII (23)]. Both phases crystalline, : model II [Deff • eff ∆E = 0, average concentration c = concentration in the grain rains cG = concentration in the matrix cM = 0.01. Saddle point energies QG = 0.5 eV, Q M = 0.4 eV, QB = 0. Diffusivities in the grain DG (cG = 0.01) = 4.13 × 10−11 D0 and in the matrix DM (cM = 0.01) = 4.29 × 10−9 D0 , with D0 = 2a2 ν0 , a = lattice constant of the fcc diffusion lattice, ν0 = attempt frequency, and T = 250 K

3

Fig. 7a,b. Deff as function of the energy difference ∆E (see Fig. 2). a Linear Deff axes, b logarithmic. : Monte Carlo results, ◦: model I , (20)], II , (22)], III , (23)]. n = I [Deff : model II [Deff •: model III [Deff 51 d, l = 41 d, vG = 0.520, average concentration c = 0.01, both phases crystalline, QG = 0.5 eV(23.2 kT), QM = 0.4 eV (18.6 kT), T = 250 K, DG (cG = 0.01) = 4.13 × 10−11 D0 , DM (cM = 0.01) = 4.29 × 10−9 D0 , Q B = 0, D0 = 2a2 ν0 , a = lattice constant of the fcc diffusion lattice, and ν0 = attempt frequency. DG (cG ) and DM (cM ) depend only slightly on ∆E as can be seen in Table 1a

3

higher ∆E, and furthermore it is approached from above and not as shown in Fig. 7 for the opposite case (DM > DG ) from below. In the case that hydrogen dissolves preferentially in the grains (∆E < 0), the limit cM DM  cG DG is eventually reached as |∆E| increases. Neglecting again spec2/3 1/3 I ial grain-boundary effects, one obtains D eff ≈ [vG /(1 − vG ) 2/3 +(1 − vG )](cM /c)DM , where now the factor cM varies as exp(−|∆E|/kT ) as long as the grains are not saturated. This ∆E dependence is not altered if diffusion processes across the phase boundaries are included explicitly, because these contribute with the same ∆E dependence. The exponential reduction of Deff with increasing ∆E can be understood in terms of repeated trapping and detrapping of hydrogen in and from the grains. 6.4 Phase-boundary barriers Modifications of the saddle point energies by an amount Q B (see Fig. 2) for hopping from a grain into the matrix and

206

Fig. 9. Deff as function of the surface to volume ratio of the grains, in units of the lattice constant a of the fcc diffusion lattice, at constant volume fraction (rhombohedral grains). Monte Carlo simulations. : ∆E = −0.05 eV, Q B = −0.050 eV, : ∆E = −0.05 eV, QB = 0.025 eV, ∗: ∆E = −0.2 eV, Q B = −0.050 eV, ◦: ∆E = −0.2 eV, QB = 0.025 eV. Volume fraction of the grains vG = 0.73, average hydrogen concentration c = 0.01, both phases crystalline, QG = 0.5 eV, QM = 0.4 eV, DG (cG = 0.01) = 4.13 × 10−11 D0 , DM (cM = 0.01) = 4.29 × 10−9 D0 , T = 250 K, D0 = 2a2 ν0 , and ν0 = attempt frequency. cG , cM , DG (cG ), DM (cM ) as a function of ∆E are given in Table 1a

Fig. 8a,b. Deff as function of the additional energy barrier QB at the phase boundary: comparison of the Monte Carlo results (symbols) with Deff [model III, (23)] (lines). a ∆E = − 0.3 eV(), ∆E = −0.2 eV (), ∆E = −0.1 eV ( ), ∆E = 0 (), b ∆E = 0.1 eV (◦), and ∆E = 0.3 eV (•). The dashed horizontal lines mark DG (cG = 0.01) = 4.08 × 10−11 D0 and DM (cM = 0.01) = 1.35 × 10−9 D0 . Average concentration c = 0.01, vG = 0.52, T = 250 K, QG = 0.5 eV, QM = 0.4 eV, σP = σs = 0.05 eV, D0 = 2a2 ν0 , a = lattice constant of the fcc diffusion lattice, and ν0 = attempt frequency. cG , cM , DG (cG ), DM (cM ) as a function of ∆E are given in Table 1b

3

vice versa influence Deff more or less strongly, depending on the sign and size of ∆E. Figure 8 shows Monte Carlo data for Deff as a function of Q B and for different parameters ∆E. As Fig. 8b shows, there is no effect on Deff due

to variations in Q B within the investigated range for large and positive ∆E, i.e., when hydrogen dissolves preferentially in the matrix phase. For smaller positive and negative ∆E (higher hydrogen solubilities in the grains) always a stepwise increase of Deff towards smaller and negative Q B is observed (see Fig. 8a). For small |∆E| and negative Q B , Deff may be higher than the diffusivity D M (cM ) in the matrix phase due to the opening of channels with higher diffusivity along the phase boundaries (phase-boundary diffusion). (At the given low average hydrogen concentration c = 0.01, DM (cM ) does not vary much as a function of ∆E, although cM does [see Table 1]. DM and DG at the concentration 0.01 are shown as dashed lines in Fig. 8 for comparison.) The stepwise behaviour of Deff as a function of Q B is III well described by Deff . The other two network models, which do not account for phase-boundary diffusion, cannot describe this behaviour.

Table 1. Concentrations and diffusion coefficients in the single phases as a function of ∆E. (a) Values corresponding to Figs. 7 and 9. The parameters are: average concentration c = 0.01, vG = 0.520, both phases crystalline, T = 250 K. (b) Values corresponding to Fig. 8. The parameters are: average concentration c = 0.01, vG = 0.52, σp = 0.05 eV, T = 250 K. Factor D0 = 2a2 ν0 , with a = lattice constant of the fcc diffusion lattice and ν0 = attempt frequency (a) ∆E (eV) DG [D0 ] DM [D0 ]

−0.3 4.04 ×10−11 4.29 ×10−9

−0.2 4.04 ×10−11 4.29 ×10−9

−0.1 4.04 ×10−11 4.29 ×10−9

0 4.08 ×10−11 4.24 ×10−9

0.1 4.13 ×10−11 4.20 ×10−9

0.3 4.13 ×10−11 4.19 ×10−9

(b) ∆E (eV) cG DG [D0 ] cM DM [D0 ]

−0.3 0.0193 4.04 ×10−11 2.59 ×10−7 1.24 ×10−9

−0.2 0.0192 4.05 ×10−11 2.68 ×10−5 1.24 ×10−9

−0.1 0.0172 4.05 ×10−11 0.00218 1.24 ×10−9

0 0.0021 4.12 ×10−11 0.0186 1.63 ×10−9

0.1 2.30 ×10−5 4.13 ×10−11 0.0208 1.70 ×10−9

0.3 2.13 ×10−9 4.13 ×10−11 0.0208 1.70 ×10−9

207

6.5 Surface-to-volume ratio, shape and arrangement of the grains The surface-to-volume ratio of the grains characterizes the fraction of lattice sites next to the phase boundary relative to all lattice sites in a grain. It can be varied at constant volume fraction of the grains by scaling the linear dimensions of grains and super cells by a common factor. The surfaceto-volume ratio is also related to the (average) reciprocal distance 1/lGG  between the grains. Deff as a function of the surface-to-volume ratio was studied by Monte Carlo simulations on our standard model system (Sect. 1) for two different values of ∆E (both negative, i.e., higher hydrogen solubility in the grains, since in that case the effect of grain boundaries is more pronounced) and in both cases for positive and for negative Q B . The results are shown in Fig. 9. For positive Q B , diffusion along phase boundaries is hindered in comparison to diffusion in the matrix and thus Deff decreases with increasing surfaceto-volume ratio, i.e., as the phase boundaries become more and more important. For negative Q B , phase boundaries provide faster diffusion channels, and thus D eff increases as the surface-to-volume ratio rises. The dependences just menIII ) but not by the tioned are well described by model III (D eff other two models (I and II). The influence of phase boundaries was also investigated for a number of other sets of model parameters. Generally their influence is less pronounced than shown in Fig. 9. Whether details of the texture of a heterogeneous material do influence Deff was tested by Monte Carlo investigations on systems having the same general properties (i.e., the constituent phases are the same, the volume fractions of the grains or the fraction of lattice sites at the phase boundaries are identical, and so on) but which are different in some finer details, such as the precise grain shape or the arrangement of the grains. The influence of the grain shape has been investigated by studying rhombohedral, cubic, ‘spherical’ and star-like grains. The rôle of the arrangement of grains was investigated by arranging several grains in various different ways within the super cell (in rows or staggered). It was found that Deff is essentially independent of such finer details. We are therefore confident that the I III formulas Deff to Deff in the chosen general notation may be applied to essentially all statistically isotropic two-phase materials.

two-component materials of isolated grains embedded into a matrix. The models account for different hydrogen solubilities and diffusivities in the constituent materials, for phaseboundary diffusion and so on. These network models were tested against Monte Carlo simulations and rigorous results obtained on model systems. It was shown that the network models give a very good description of D eff for all tested model systems. Our model III has the widest range of applicability. In the tests of the network models against Monte Carlo simulations, only examples in which the average hydrogen concentration c is small (c = 0.01) were discussed explicitly in the present paper. This does not mean, however, that the network models do not work for high c. On the contrary, comparisons of the network models I to III with Monte Carlo data obtained at several higher hydrogen concentrations (especially at c = 0.1, c = 0.5 and c = 0.9) showed that for high c the network models work equally well as for low c, if the correct concentrations cM (c), cG (c) and diffusivities DM (cM ), DG (cG ) are plugged into the network formulas. (The variations of DM and DG due to changes of cM and cG caused by variations of the model parameters are stronger, however, for larger c.) Furthermore, in the examples shown, the diffusivity in the matrix is always higher than in the grains. In most cases the network models, however, also give reasonable descriptions in the opposite case. Significant discrepancies between the network formulas and the Monte Carlo results only exist if hydrogen is strongly trapped in the grains, i.e., if either much more hydrogen is dissolved in the grains than in the matrix or the additional barrier for hopping between the grains and the matrix phase is very I III to Deff might be conhigh (large Q B ). In those cases, Deff siderably smaller than the Deff values obtained from the Monte Carlo simulations. We suspect, however, that the Monte Carlo simulations have to be blamed for the major part of these discrepancies, since Monte Carlo simulations in those cases are problematic in the sense that extremely long runs are necessary to reach the long-time limit of Deff [37]. Finally, the rules for constructing network models for diffusion in heterogeneous materials developed in the present paper may be easily applied to more general situations, for instance to heterogeneous materials containing oriented grains with a shape of lower-than-cubic symmetry. Acknowledgements. This work was funded by the Deutsche Forschungsgemeinschaft as a subprojekt in the Sonderforschungsbereich 270.

7 Summary So-called network models, constructed in analogy to electrical networks, were developed in order to describe the effective diffusivity Deff of hydrogen through a heterogeneous material. In a first step, approximate representations of the heterogeneous material by networks of diffusion links were searched for, based on heuristic arguments. It was shown, in a second step, that Deff for any of these networks may be calculated by applying the rules valid for electrical networks, with the substitution of ci Di for the electrical conductivities, if ci is the hydrogen concentration and D i the hydrogen diffusivity in the diffusion link i [37]. Three different network models were presented which are able to describe Deff for heterogeneous statistically isotropic,

References 1. S. Suda, G. Sandrock: Z. Phys. Chem. 183, 149 (1994) 2. M. Hirscher, J. Mössinger, H. Kronmüller: J. Alloys Compd. 231, 267 (1995) 3. C.U. Maier, H. Kronmüller: J. Less-Common Met. 172–174, 671 (1991) 4. M. Hirscher, S. Zimmer, H. Kronmüller: Z. Phys. Chem. 183, 51 (1994) 5. R. Kirchheim, T. Mütschele, W. Kieninger, H. Gleiter, R. Birringer, T.D. Koble: Mater. Sci. Eng. 99, 457 (1989) 6. N. Mommer, M. Hirscher, H. Kronmüller: J. Alloys Compd. 253–254, 390 (1997) 7. K. Yamamoto, Y. Tsushio, S. Tanioka, T. Shimizu, T. Morishita, S. Orimo, H. Fujii: J. Alloys Compd. 231, 689 (1995) 8. H. Iba, E. Akiba: J. Alloys Compd. 231, 508 (1995)

208 9. H. Uchida, H. Uchida, Y. Huang: J. Less-Common Met. 101, 459 (1984) 10. L. Zaluski, A. Zaluska, J.O. Ström-Olsen: J. Alloys Compd. 253–254, 70 (1997) 11. J. Huot, G. Liang, R. Schulz: Appl. Phys. A same issue, (2000) 12. A. Zaluska, L. Zaluski, J.O. Ström-Olsen: Appl. Phys. A same issue, (2000) 13. S. Orimo, H. Fujii: Appl. Phys. A same issue, (2000) 14. F. Lux: J. Mater. Sci. 28, 285 (1993) 15. Z. Fan: Philos. Mag. A 73, 1663 (1996) 16. S. Kirkpatrick: Solid State Commun. 12, 1279 (1973) 17. D.S. McLachlan: J. Phys. C: Solid State Phys. 19, 1339 (1986) 18. J. Mössinger, A. Hörner, H. Kronmüller: Phys. Status Solidi A 166, 513 (1998) 19. N. Eliaz, D. Fuks, D. Eliezer: Acta Mater. 47, 2981 (1999) 20. V.V. Kondratyev, A.V. Gapontsev, A.N. Voloshinskij, A.G. Obukhov, N.I. Timofeyev: Int. J. Hydrogen Energy 24, 819 (1999) 21. T. Mütschele, R. Kirchheim: Scr. Metall. 21, 135 (1987) 22. O.F. Sankey, P.A. Fedders: Phys. Rev. B 15, 3586 (1977) 23. D.A. Faux, D.K. Ross: J. Phys. C: Solid State Phys. 20, 1441 (1987)

24. A.D. LeClaire: In Physical Chemistry – An Advanced Treatise, Vol. X, Chapt. 5, ed. by H. Eyring, W. Hendersson, W. Jost (Academic Press, NY 1970) p. 261 25. P.A. Fedders, O.F. Sankey: Phys. Rev. B 15, 3580 (1977) 26. Kirchheim, Acta Metall. 30, 1069 (1982) 27. R. Kirchheim: J. Non-Cryst. Solids 55, 243 (1983) 28. R. Kirchheim, F. Sommer, G. Schluckebier: Acta Metall. 30, 1059 (1982) 29. R. Kirchheim, U. Stolz: J. Non-Cryst. Solids 70, 323 (1985) 30. R. Kirchheim: Prog. Mater. Sci. 32, 261 (1988) 31. R. Ash, R.M. Barrer, J.H. Petropoulous: Br. J. Appl. Phys. 14, 854 (1963) 32. J.A. Barrie, J.D. Levine, A.S. Michaels, P. Wong: Trans. Faraday Soc. 59, 869 (1963) 33. R.H. Song, S. Pyun: J. Electrochem. Soc. 137, 1051 (1990) 34. G. Schmitz, Ph. Kesten, R. Kirchheim, Q.M. Yang: Phys. Rev. B 58, 7333 (1998) 35. D.A.G. Bruggeman: Ann. Phys. (Leipzig) 24, 636 (1935) 36. A. Turnbull, R.B. Hutchings: Mater. Sci. Eng. A 177, 161 (1994) 37. A. Herrmann: Thesis, Universität Stuttgart (1999)