Published in In Solid State Physics, eds. H. Ehrenreich and F. Spaepen, Elsevier, Amsterdam (2004), Vol. 59, I.

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1 Published in In Slid State Physics, eds. H. Ehrenreich and F. Spaepen, Elsevier, Amsterdam (004), Vl. 59, I. Intrductin Slid slutins f hydrgen in cmplex materials REINER KIRCHHEIM Institut für Materialphysik, Gerg August Universität Göttingen Tammannstr. 1, D Göttingen, Germany II. Fundamental prperties f hydrgen in metals 1. Equilibrium pressure and slubility. Diffusivity 3. Partial mlar vlume and interactin with stress and strain fields III. Behavir f hydrgen in defective and disrdered metals 4. Density f site energies (DOSE) and Fermi-Dirac Statistics (FD-Statistics) a. Slubility b. Validity f Henry s Law 5. Diffusivity a. Tracer diffusin b. Diffusin in a cncentratin gradient c. General randm walk d. Lw cncentratin limit 6. H-H interactin IV. Interactin f hydrgen with defects 7. Interactin with ther slutes and vacancies 8. Interactin with dislcatins 9. Interactin with grain bundaries 10. Interactin with metal/xide bundaries 11. Defect frmatin energy 1. Interactin with crack tips and hydrgen embrittlement V. Hydrgen in disrdered and amrphus allys 13. Disrdered crystalline allys 14. Metal/nn-metal glasses 15. Early transitin/late-transitin metallic glasses 16. Bulk metallic glasses VI. Other interstitials in amrphus materials 17. Mdeling diffusin 18. Small mlecules in glassy plymers 19. Hydrgen in amrphus silicn and germanium 0. Ins in xidic glasses VII. Hydrgen in systems with reduced dimensins 1. Thin films. Multi layers 3. Clusters 1

2 I. Intrductin Hydrgen in metals has attracted cnsiderably attentin f physicists, chemists, material scientists and engineers fr many decades. Mst f the exciting prperties are related t the small size f the H atm which leads t a high mbility in materials. Namely in metals its diffusivity is very high at rm temperature and may reach values which are the same as fr ins in aqueus slutins. The physical reasns fr the high H-mbility are twfld. On the ne hand H-atms are disslved interstitially and migrate via a direct interstitial mechanism which at dilute cncentratins des nt require the frmatin f vacancies. On the ther hand a site exchange may ccur via quantum mechanical tunneling. The cnsequences f the high H-mbility are manifld: (i) Thermal equilibrium is established in rather shrt times at rm temperature between the H-disslved in the metal and either hydrgen gas r prtns in aqueus slutins. Thus thermdynamic prperties, especially the chemical ptential f hydrgen can be btained simply by measuring the partial pressure r the electrchemical ptential. (ii) Hydrgen strage in metals and its use as an energy carrier becmes pssible at rm temperature. (iii) Hydrgen can easily redistribute and segregate at defects prduced during plastic defrmatin, i.e. dislcatins, crack tips etc.. This interactin gives rise t hydrgen embrittlement. The small size f the H-atm allws a large packing density in thse metals which have a high affinity, i.e. large negative heat f slutin, fr hydrgen. In ther metals H-cncentratins at reasnable H -pressures may be very lw despite the same number density and size f interstitial sites in these metals. In metal hydrides the atmic density can be even larger than in liquid hydrgen. This prperty f the hydrides is advantageus when using hydrgen as a fuel althugh the energy strage per weight is still t lw fr many applicatins. At present cnventinal strage allys cntain wt.-% H whereas fr their use in vehicles figures abve 3 wt.-% are required. Hwever in rechargeable batteries metal hydrides are widely used nwadays. It will nt be the purpse f this review t discuss issues related t hydrgen ecnmy in general r the special rle f hydrides. Research results in this area are published elsewhere [1,, 3, 4, 5]. Besides hydrgen strage hydrgen embrittlement is a technlgical subject f majr cncern, where the negative rle f hydrgen is played at much lwer cncentratins, i.e. in irn base allys embrittlement effects are bserved at cntents as lw as a few ppm. Here the interactin f H-atms with lattice defects and its effect n plasticity has t be studied in detail. Again the high mbility f H-atms is crucial fr the phenmena, because hydrgen has t reach r t fllw mving dislcatins and r crack tips. It appears as if the quantum mechanical diffusin plays a rle as well. Especially fr adjacent tetrahedral sites in bdycentered cubic metals, which are nly a very shrt distance apart, this effect is strngly prnunced and leads t the result that ferritic steels (bcc-lattice) are mre susceptible t hydrgen embrittlement than austenitic steels (fcc-lattice). Tpics f embrittlement phenmena are treated in cnferences [6, 7, 8] which seldm verlap with the nes n hydrgen strage. Metal-hydrgen systems are ften used as mdel systems t study physical r chemical prperties and hw they change with cmpsitin. This is ften very easy because hydrgen can be dped in a cntrlled way by either measuring changes f the H -pressure in clsed systems r by electrchemical depsitin n a metal electrde applying Faraday s Law,. This

3 advantage f easy allying is supprted by the pssibility f btaining the chemical ptential f hydrgen by measuring partial pressures and/r electrde ptentials. Cases where metalhydrgen served as mdel systems are (i) slute-slute interactin measured and interpreted in the framewrk f a quasi chemical apprach fr the first time in the Pd-H system by Lacher [9], (ii) tunneling as a diffusin mechanism fr atms in slids was discvered and discussed fr hydrgen in metals [10, 11, 1], (iii) the behavir f hydrgen in systems with reduced dimensins can be studied nicely in metals [13], (vi) hydrgen interactin with defects in metals is representative fr ther slute/slvent systems and has been studied extensively [14]. After knwing the basic features f the interactin H-atms can be used as prbes fr the defects [15, 16]. By gradually increasing the H-cncentratin the sites f increasing energy within the defects are saturated successively and, therefre, a kind f spectrscpic methd will be available (examples will be discussed thrughut this study). Anther peculiarity f the lightest element is the large mass difference f its istpes which gives rise t prnunced istpe effects in mst f its prperties. The different scattering length fr neutrns which even changes sign frm H t D gave rise t an extensive use f neutrn scattering and diffractin techniques in the area f metal-hydrgen systems [17]. Hydrgen at high cncentratins may change the physical prperties f a material remarkably. Examples are changes f the magnetic cupling between ferrmagnetic layers [18, 19] r even mre exciting a metals-insulatr transitin in yttrium ging frm the dihydride t the trihydride [0]. In additin micrstructural changes have been bserved during allying a metal with hydrgen. Examples are the generatin f abundant vacancies [1] and dislcatins [, 3, 4], the decmpsitin f miscible allys [5, 6] and the imprvement f mechanical prperties f Ti-allys by decreasing the grain size [7] r by variatin f the α t β vlume fractin [8]. Thus hydrgen can be used as a temprary allying element, in rder t set up a desired micrstructure. In ther materials like semicnductrs hydrgen is used as a permanent allying additin fr the purpse f saturating deep impurity levels r dangling bnds at the Si/SiO interface as well as in amrphus silicn [9]. As the present study will fcus n slid slutins f hydrgen in metals, the prperties f hydrides will be mitted and the reader is referred t mngraphs including these subjects [30, 31, 3, 33]. The majr tpic will be the behavir f hydrgen in disrdered and amrphus systems including the interactin with defects. The experimental and theretical findings are relevant fr ther materials like plymers, xidic glasses r amrphus silicn which will be treated in a special chapter f this study. 3

4 II. Fundamental prperties f hydrgen 1. EQUILIBRIUM PRESSURE AND SOLUBILITY a) Slubility Hydrgen mlecules interacting with a metal are dissciating n the surface and disslved as atms within the metal accrding t the fllwing reactin H ( gas) H ( metal). (1.1) Evidence fr this reactin is prvided fr instance by measuring pressure cmpsitin istherms (pc-istherms) and shwing that Sieverts Law c H = K p H (1.) is valid, where c H is the H-cncentratin within the metal, p H is the partial pressure f hydrgen and K is Sieverts cnstant. In rubbery plymers fr instance the mlecules are nt dissciated during srptin and prprtinality between cncentratin and pressure arises. Eq. (1.) is derived by assuming thermdynamic equilibrium between gaseus and disslved hydrgen which requires that the chemical ptentials in the tw phases are the same. H-atms are ccupying interstitial sites in the metal lattice which are mstly tetrahedral r ctahedral sites. This can be prven directly experimentally by in channelling experiments using single crystals f the metal [34]. Then the cnfiguratinal entrpy s cf is given by NH N s H cf = kb ln = kb ln, (1.3) N βn i Me where k B is Bltzmann s cnstant, N H, N i, N Me are the numbers f H atms, interstices and metal atms respectively. Thus β is the rati f interstitial sites t metal atms being 6 fr tetrahedral sites in bcc lattices and 1 fr ctahedral sites in fcc lattices. Then the chemical ptential f ideally disslved H-atms is expressed as N µ µ H r µ H H = H + kbt ln H + kbt ln, (1.4) βn β Me where µ H is a standard value f the chemical ptential. As a measure f hydrgen cncentratin the rati r H f hydrgen and metal atms is used mstly. The chemical ptential f the mlecules in the gas phase is µ H = µ ln H + k B T p H. (1.5) Eq. (1.1) and equilibrium require 1 µ H = µ H. (1.6) Inserting Eqs. (1.4) and (1.5) int the last ne yields Sieverts Law µ H µ H r = p H β H exp, (1.7) kbt because the relatin between r H and the H-cncentratin c H which is usually defined as the number f mles f H per unit vlume is prprtinal. Often the H-cncentratin at a given hydrgen pressure and temperature is called slubility. This has t be distinguished frm the maximum r terminal slubility f hydrgen in a metal fr thse case where a hydride is frmed. The difference f the standard values f the chemical ptentials in Eq. (1.7) is equal t the Gibbs free energy f the reactin given in Eq. (1.1) r the Gibbs free energy f absrptin (disslutin), respectively, i.e. 4

5 H H G = H T S = µ µ. (1.8) The majr cntributin t the entrpy change f disslutin stems frm the lst degrees f freedm the free H mlecules have in the gas phase. Thus S is abut equal t the standard entrpy f gaseus hydrgen at rm temperature [35] S S95 =131 J / K / ml (1.9) Because f this rather high and psitive entrpic cntributin t the Gibbs free energy f disslutin hydrgen can be desrbed at high temperatures. The enthalpy f disslutin in metals is very negative at the left hand side f the peridic table and increases t very psitive values ging t the right via the transitin metals [3, 36]. An atmistic interpretatin is rather invlved as elastic and electrnic cntributin play an imprtant rle. As the metal expands during disslutin f hydrgen (see II.C) elastic energy has t be paid. Assuming n changes f the electrnic structure the additinal electrn f the hydrgen atm has t be placed in states abve the Fermi level which gives rise t a psitive cntributin t the enthalpy f disslutin. Hwever, it has been shwn [3, 37] that by the incrpratin f H-atms new energy levels belw the Fermi-level are generated yielding negative cntributins t H. The Gibbs free energies f hydride frmatin per H-atm r mlecule are defined as the free energy change during the fllwing reactin H + xme MexH G f. (1.10) Values f G f are very clse t the Gibbs free energy f disslutin [36] which has been defined via the reactin in Eq. (1.1). The similarity f the crrespnding entrpy changes arises frm the fact that in bth reactins the majr cntributin t it stems frm the lss f translatinal freedm f gaseus hydrgen. The similarity f the enthalpy changes means that cntributins f slute slute interactin in the hydride are small in cmparisn with the elastic and electrnic cntributins t the enthalpy f disslutin.. DIFFUSIVITY Diffusin f hydrgen in metals ccurs via the direct interstitial mechanism with a diffusin cefficient given by [38] r H l Γ D* = f (1 ) (.1) d β where Γ is the jump frequency, l the jump distance and d the dimensinality f the lattice (which is mstly 3 but fr diffusin alng grain bundaries we have d=). The quantities r H and β have been defined after Eqs. (1.3) and (1.4) and f is a crrelatin factr which is unity fr r H 0 but will be smaller than unity fr r H /β 1 where the interstitial lattice becmes filled [39] and the direct interstitial mechanism f diffusin changes gradually t a vacancy mechanism. Then blcking f sites cmes int play as well which is accunted fr by the factr in brackets. A tacit assumptin fr the validity f Eq. (.1) is that all sites visited during a randm walk f the H-atm have the same average jump frequency, i.e. the same site and saddle pint energy fr the case f thermally activated hpping r the same transitin prbability fr tunneling thrugh the ptential barrier. Very ften the chemical diffusin cefficient f hydrgen D H is measured via the flux f hydrgen J H in a cncentratin gradient accrding t Fick s First Law 5

6 C J D H H = H. (.) x Using the gradient f the chemical ptential f hydrgen as the driving frce fr diffusin it can be shwn [40] that C ln a ln D H µ H D* H γ = = D* = 1 H + D *, (.3) kbt CH ln CH ln CH where a H is the thermdynamic activity f H and γ H the activity cefficient (a H =γ H C H ). The experimental techniques f measuring diffusin cefficients f hydrgen in metals are manifld [30, 31, 3, 33]. The nes which were used t btain mst f the data presented in this study are described in the fllwing (i) Electrchemical techniques: The metal sample is immersed in an electrlyte and a current is passed frm a cunter electrde. Then the amunt f hydrgen depsited n the sample surface is simply calculated frm Faraday's Law. Whether this hydrgen prduced in statu nascendi, i.e. in the atmic frm is absrbed by the sample r whether it recmbines t H mlecules and escapes by disslutin in the electrlyte r as gas bubbles, depends n the H-slubility f the metal, the current density and the permeability f surface xides [14]. Natural xides can be remved by sputtering and replaced by a palladium layer as shwn fr nibium and tantalum [41]. Hydrgen can be als desrbed frm a sample by reversing the directin f the current. The H-activity n the surface r the chemical ptential, respectively can be btained by measuring the vltage between sample and a reference electrde. Depending n the bundary cnditins with respect t either current r electrchemical ptential, transient, changes f these parameters can be evaluated in rder t btain a value fr the chemical diffusin cefficient [4]. If the sample is munted in a duble cell as suggested in [43], the values f the diffusin cefficient are mre reliable [44]. As diffusin ccurs in a cncentratin gradient, values f the chemical diffusin cefficient are btained. The main advantages f electrchemical techniques are applicability at lw H-cncentratins (dwn t a few at-ppm), simplicity f equipment, ease f dping and the pssibility f getting values f the (electr-)chemical ptential. Drawbacks are a limited temperature range between freezing and biling pint f the electrlyte and nn-permeable surface barriers. Because f the latter mst measurements were made with palladium and its allys. (ii) Grsky-Effect [30]: Here the hydrgen cntaining sample is bend prducing bth expanded and cmpressed regins. This way a gradient f the chemical ptential f hydrgen is set up and hydrgen having a psitive mlar vlume migrates frm cmpressed t expanded regins until an equilibrium cncentratin prfile has established. The crrespnding strain in the sample remains in the sample after the bending stress is released, i.e. the sample is still bend. Then the cncentratin gradient is n lnger in equilibrium and vanishes by diffusin. The assciated strain gradient vanishes t and can be measured by mnitring the decreasing bending f the sample. In rder t get measurable strains the H-cncentratin has t be at least few tenth f ne at.-%. This cnditin sets a lwer limit fr the temperature range, because by lwering the temperature the cncmitantly decreasing terminal slubility will finally be higher than the H-cncentratin and hydride precipitatin ccurs. Cntrary t the electrchemical technique surface xides are advantageus fr Grsky-effect measurements, because they prevent desrptin f hydrgen and/r an equilibratin via the gas phase. At higher temperatures a limit is set by the increasing permeability f hydrgen thrugh the xide r the disslutin f xygen within the metal and the destructin f the barrier. 6

7 (iii) Permeatin: Here a pressure difference is set up acrss a metallic membrane separating tw clsed cmpartments and the hydrgen flux thrugh the membrane is measured, i.e. by mnitring pressure changes. Transient and steady state behavir yield the chemical diffusin cefficient and the permeability (prduct f diffusin cefficient and difference f H-cncentratins between entrance and exit surface f the membrane). The technique requires rather high permeabilities (high temperatures, diffusivities and slubilities) and may be dminated at the lwer temperatures by rate-cntrlling reactins at the interface. (vi) Internal frictin (mechanical spectrscpy) [45]: Damping f vibratins in a vibrating reed r in a trsinal pendulum is ften caused by jumping atms, if their strain field interacts with the externally applied stress field which excites the vibratins. If the frequency f vibratin and the jump frequency are highest, damping reaches its maximum. Thus measuring damping as a functin f sample frequency yields a damping peak which is called Snek peak [45] and which has its maximum at a value which is equal t the jump frequency. The frequencies f sample vibratins can be changed by changing the sample dimensins r by exciting varius mdes f vibratin. Experimentally it is easier t change the temperature and, therefre, changing the jump frequency f the atms while keeping the frequency f vibratin cnstant. Then the Snek peak ccurs in a damping versus temperature plt. H-atms incrprated in ctahedral sites f an fcc lattice (fr instance Pd r Ni) cause a strain in the lattice which has cubic symmetry and, therefre, des nt cause any damping. Hwever, H-atms in tetrahedral sites f a bcc lattice (fr instance Nb and Ta) give rise t a tetragnal distrtin and a Snek relaxatin is expected. It has nt been bserved s far which may be due t a relaxatin strength being smaller than the detectin limit f the technique r due t the very rapid tunneling f H-atms between the 4 adjacent tetrahedral sites may smear ut the tetragnal distrtin. Hwever, fr hydrgen in amrphus allys an internal frictin peak has been discvered [46, 47, 48, 49] and it was shwn that the jump frequency is in agreement with the ne calculated frm diffusin cefficients via Eq. (.1). In additin hydrgen atms being trapped in the neighbrhd f a freign slute atm, i.e. at a substitutinally disslved titanium atm in a nibium lattice [50] give rise t a Snek peak. The diffusin cefficients usually bey an Arrhenius Law when they are measured in a limited temperature range. Hwever, the values btained frm Grsky effect measurements ver a large range dwn t very lw temperatures revealed fr the Vb metals vanadium, nibium and tantalum a prnunced curvature when presented in an Arrhenius plt [11, 30]. The curvature crrespnds t a decreasing activatin energy f diffusin with decreasing temperature which is in accrdance with a quantum mechanical tunneling f the H-atm thrugh a ptential barrier between tw adjacent sites. Thus values as large as cm /s have been measured in α-fe and V at rm temperature which crrespnd t diffusin length f abut 1 cm in ne day. 3. PARTIAL MOLAR VOLUME AND INTERACTION WITH STRESS AND STRAIN FIELDS Partial mlar vlumes f hydrgen in metals are usually btained by measuring changes f the lattice parameter as a functin f H-cncentratin [51]. Because hydrgen is mstly disslved in interstitial sites with a smaller vlume, the lattice parameter increases with increasing H-cncentratin. Fr many metals and their allys the partial mlar vlume is.9 7

8 Å 3 per H-atm r 1.7 cm 3 /ml, respectively [3, 51, 5]. Details f the lattice distrtin in the neighbrhd f the H-atm can be revealed by Huang scattering [51]. Fr high H-cncentratins measurements f the density f the sample yield the partial mlar vlume, whereas at lw cncentratins in-situ dilatmetry is mre apprpriate. An example fr the latter case is shwn in Fig. 1 [53], where results btained by electrchemical dping f ne sample f plycrystalline palladium and ne f an amrphus Pd-Si ally are presented. Increasing the H-cncentratin step by step and measuring the length change f the ribbnshaped samples gave the same partial mlar vlume fr the crystalline Pd, whereas large variatins were bserved fr the amrphus ally. The vlume cntractin being bserved fr the first 40 atppm f hydrgen were explained by trapping f H-atms in vacancy-like defects [53]. At higher cncentratins the expected vlume change f abut 1.5 cm 3 /ml fr hydrgen being disslved in interstitial-like sites was measured. In this cncentratin range there is a tendency f increasing partial mlar vlume with increasing H-cncentratin which is indicative f filling larger interstices in the amrphus structure first [54]. In rder t prvide additinal experimental evidence fr the unusual negative vlume change, the partial mlar vlume was als measured by measuring changes f the chemical ptential under tensile stress as explained in the fllwing. Fr a slute atm which leads t an istrpic strain f insertin the chemical ptential changes if an arbitrary stress state σ ik is present accrding t the relatin [55, 56] ( ) ( 0) σ ik s µ σ = µ σ = V (3.1) 3 where V s is the partial mlar vlume f the slute. Fr uniaxial tensile stress (σ 11 =σ and σ ik =0 therwise), hydrgen as a slute and measuring changes f the electrmtive frce E the last equatin becmes σ V E = H (3.) 3F where F is Faraday s Cnstant. By measuring changes f the electrchemical ptential f metal-hydrgen systems as a functin f the applied stress the partial mlar vlume V H was determined via Eq. (3.) in an independent way when cmpared with the dilatmetric studies [53]. Namely, the negative changes shwn in Fig. 1 culd be reprduced. Besides allwing a determinatin f V H Eq. (3.1) is much mre imprtant in the cntext f H- embrittlement as discussed in sectin 1. 8

9 III. Behavir f hydrgen in defective and disrdered metals The behavir f hydrgen in single crystalline metals r plycrystalline metals with a small density f lattice defects has been studied fundamentally and extensively during the 1960ies 70ies and 80ies. In the 1990ies the metal hydride batteries replaced the nickel cadmium batteries gradually and the fcus f researchers was directed tward metal hydrides and their applicatin. In rder t extend the applicatin f hydrgen strage materials t vehicles driven directly with hydrgen r indirectly via a fuel cell the density f metallic materials r the strage capacity in terms f hydrgen per unit f mass is a matter f serius cncern. In additin, the theretical strage capacity f an ally is ften nt reached because f crystalline defects, namely dislcatins [57, 58]. They are present frm the very beginning, r even mre, they are frmed during lading and unlading f the ally. Besides effecting strage capacity grain bundaries play an imprtant rle, because they increase the kinetics f uptake and release f hydrgen. Thus nancrystalline allys are favrable strage allys [59]. The interactin f hydrgen with crystalline defects is even mre imprtant in the cntext f hydrgen embrittlement. There are varius mechanisms leading t hydrgen embrittlement but they are all related t hydrgen/defect interactin. The mst prminent ne is the direct actin f hydrgen at the crack tip by either a dechesin mechanism [60] r the frmatin f a brittle hydride [61]. In the first case a single hydrgen atm migrating with the prpagating crack tip will be sufficient whereas in the latter higher H-cncentratins are required. Fr ductility being part f the failure mechanism, the interactin f hydrgen with dislcatin becmes imprtant. In general, the strage f hydrgen in crystal defects and/r the trapping by the defects determines hw much f mbile hydrgen will be available t reach a critical value fr embrittlement. As this is a rather cmplex prblem, it has been treated qualitatively nly at the beginning f research in this area. The present study will prvide a quantitative result by defining a density f site energies (DOSE) r an energy landscape, respectively, and filling the varius sites accrding t Fermi Dirac Statistics (FD-Statistics). It will be shwn in chapter VI that the cncept f a DOSE and FD-Statistics is useful fr ther interstitials in ther materials besides metallic nes. It als prvides the basis fr a fundamental study f the interactin f slute atms with the micrstructure f a material. Hydrgen systems are especially suited fr these fundamental experiments because f the ease f dping and the ease f measuring its chemical ptential which is f central imprtance in FD-Statistics. As the system can be studied arund rm temperature the crystal defects will nt be annihilated, i.e. the micrstructure remains the same during experimentatin. 4. DENSITY OF SITE ENERGIES (DOSE) AND FERMI-DIRAC STATISTICS (FD-STATISTICS) The DOSE is defined as usual in slid state physics as the nrmalized number f sites n(e) in a given energy windw E, E+dE with n ( E) de = 1. (4.1) This definitin is equivalent with the density f states (DOS) functin used fr electrns r ther particles in quantum mechanics. Fr electrns in slids the distributin f energy states in reciprcal space is cnsidered, whereas here we are interested in sites in real space. Hwever, the DOSE des nt cntain any infrmatin abut the lcalizatin f the sites and spatial crrelatins between lw and/r high energy sites are nt accessible. Fr the energy 9

10 scale the standard state f gaseus hydrgen at ne bar and 98 K is used. In the fllwing a few relevant example f a DOSE f hydrgen in metals will be presented (cf. Fig. ). Because f the translatinal symmetry all the interstitial sites in a single crystal have the same energy and, therefre, the site energy fr hydrgen atms is the same. In ther wrds, the system is ttally degenerate. This simple pictures has t be mdified, because in a crystalline lattice different interstices are available amng which the tetrahedral and ctahedral sites are the mst prminent nes. The situatin becmes even mre cmplex, if the single crystal cntains different cnstituents, i.e. a slid slutin r an intermetallic cmpund. Then the degeneracy f the system is partly lifted and a discrete number f site energies have t be cnsidered. Nevertheless, fr dilute slutins f hydrgen and single cmpnent crystals the simplificatin f ne interstitial site is usually fulfilled because the site energies f ther interstices are t high t be ccupied. The mathematical representatin f the crrespnding DOSE will be a Dirac delta functin. Frm the ne level system f a single crystal we cme t a tw level system by intrducing islated pint defects like substitutinally disslved freign atms r vacancies, by restricting the interactin with hydrgen t the nearest interstices. As spatial crrelatin des nt play a rle all the freign atms may be cmbined t a layer in between the hst metal and by neglecting the interface this becmes a tw level system, t. Ging frm zer t ne and tw dimensinal defects f the lattice, it is mre cnvenient t intrduce cntinuus functins instead f discrete energy levels. This will be discussed in mre detail in the fllwing sectins dealing with dislcatins and grain bundaries. And finally in a perfect amrphus structure all sites may be cnsidered t be different frm each ther representing a system withut degeneracy. The varius sites f a DOSE cmpete fr the ccupancy with hydrgen. By ccupying the sites f lwest energy the system reduces its ttal energy whereas distributin amng sites being present in large numbers increases the cnfiguratinal entrpy bth effects reduce the Gibbs free energy. If we allw a site t be ccupied by ne H-atm nly, the crrespnding minimizatin can be treated in the framewrk f the Fermi-Dirac Statistics (FD-Statistics) [6]. Here the cnfiguratinal entrpy in each energy windw f the DOSE is calculated under the assumptin f single ccupancy. During this prcedure it des nt matter whether electrns are distributed amng energy states in reciprcal space r particles amng sites in real space. The minimum f Gibbs free energy fr all energy windws yields the fllwing result fr the thermal ccupancy f energy level E i r the crrespnding site, respectively 1 ( Ei ) = 1+ exp ( E µ ) / k T (4.) [ ] i B where µ is the derivative f Gibbs free energy with respect t particle cncentratin and, therefre, it is the chemical ptential f the particles despite being called Fermi energy in this cntext. Integratin ver the DOSE with the crrespndent thermal ccupancy yields the ttal fractin N/N f sites ccupied with particles, where N is the number f disslved particles and N is the ttal number f available sites: N N = n( E) de E µ 1+ exp kbt. (4.3) 10

11 The term particle instead f hydrgen atms has been used befre, in rder t stress the fact that the cncept f a DOSE and FD-Statistics is rather general and nt restricted t hydrgen in metals. It has been applied befre in the framewrk f hetergeneus adsrptin [6] and the interactin f slute atms with dislcatins [63, 64]. a. Slubility The relatin between hydrgen cncentratin in the metal expressed as the rati r H =N H /N Me and its partial pressure in the gas phase is btained by using Eq. (1.5) rh N = H n( E) de =. (4.4) β βn Me 1 E ( µ H / ) 1+ exp p k T B Fr a single crystal with n(e)=δ(e-e ) and dilute cncentratins (r H <<1) the well knwn result (cf. Eq. (1.7)) µ H E r = p H β exp (4.5) k T B is btained. A clsed slutin f the integral in Eq. (4.4) will be pssible fr simple frms f n(e) nly. In rder t understand the cmpetitin between energy decrease and entrpy gain the fllwing apprximatin based n the step r T=0 apprximatin f the Fermi-Dirac functin (i.e. (E) 1 fr E<µ and (E) exp[(µ-e)/kt] fr E>µ) will be applied µ n( E) de c = n( E) de+ c1 + c (4.6) exp ( µ [ E µ ) / kt ] where c will be used fr the cncentratin instead f r H, in rder t be cnsistent with previus publicatins f the authr and t refer t the general validity f the equatins. Thus c refers t the fractin f interstices ccupied by particles. The first term c 1 n the right hand side f the equatin is that part f the ttal cncentratin c arising frm particles belw the Fermi level and c crrespnds t sites abve µ. Occupying sites belw the Fermi level the system decreases its energy whereas fr the sites abve it increases its cnfiguratinal entrpy. Thus fr c 1 >c energy play a mre imprtant rle than entrpy and vice versa. Fr c <<c 1 c which is always fulfilled at lw temperatures because f the step-type behavir f the Fermi-Dirac functin, we get frm the last equatin c = n(µ). (4.7) µ The last equatin can be used t get the DOSE frm measured values f µ as a functin f c (fr instance by the electrchemical methd). b. Validity f Henry s Law Fr c<<1 r µ<<0, respectively, and c 1 <<c c we btain frm the last equatin n( E) de µ E c = exp n E de = aγ [ E µ kt ] kt ( ) exp exp ( ) / kt (4.8) 11

12 Then Henry's Law a=γc is fulfilled with the thermdynamic activity a=exp(µ/kt) being prprtinal t c. At lw cncentratins the cnditin c 1 <<c is nt fulfilled fr all site energy distributins. Fr a Gaussian distributin it was shwn that the cnditin is valid fr µ<<-σ /kt, where σ is the width f the distributin [14]. Then γ exp[σ /(kt) ] fr c 0. Hwever, if we cnsider a cntinuus expnential distributin, n(e) = σ -1 exp[e/σ] fr E 0 and n(e)=0 therwise, Eq. (4.8) yields exp( µ / kt ) exp( µ / σ ) c1 = exp( µ / σ ) and c = (4.9) 1 σ / kt where independent f µ (being negative) c is smaller than c 1 fr σ>>kt. In this limiting case the system decreases its free energy because it lwers its energy by ccupying sites f lwest energy nly. Thus we have c 1 c and the activity cefficient becmes µ µ σ / kt 1 γ = exp = c. (4.10) kt σ This pwer law has been derived in a different cntext as well [65] and it shws that γ appraches 0 fr c 0. The activity cefficient depends n c and, therefre, Henry's Law is nt fulfilled. Hwever, as all the samples have a finite number f sites the expnential DOSE has t have a cut-ff at the lw energy side and it can be shwn that this leads t the validity f Henry's Law again. Very ften a sample cntains a small fractin f f crystal defects nly. Then the majrity f sites is the same as in a single crystal and the DOSE can be written as n( E) = (1 f ) δ ( E E ) + fn ( E) with f << 1, (4.11) f If the average energy f n f (E), i.e. its first mment, is abve E the defects have a negligible effect and the H-atms are almst all in sites f the single crystalline regin. Fr the reverse case f the average energy being smaller than E integratin ver the first part f the DOSE in Eq. (4.11) yields the cncentratin in the single crystalline regin 1 f µ E E µ H c fr = exp p exp. (4.1) E µ + kbt k T B 1 exp kbt The subscript fr refers t these H-atms as being free and nt bund t crystalline defects. Then the activity f hydrgen r its partial pressure, respectively is slely determined by the free hydrgen. 5. DIFFUSIVITY a. Tracer diffusin Diffusin f tagged particles in a system with a DOSE and cnstant cncentratin can be btained simply frm averaging ver the jump frequencies Γ and calculating D* with the jump distance l via Eq.(.1) written as * l l Q E E 1 D exp + = Γ = Γ [ n( E1) ( E1)][1 n( E) ( E )] 6 6 k T (5.1) B 1

13 13 where Γ is the average jump rate. The jump rate fr particles hpping frm sites f energy E 1 int sites f energy E is written as the prduct f 1.) a cnstant prefactr Γ which is equal fr all sites,.) a Bltzmann factr exp[-(q +E -E 1 )/ k B T] because thermally activated jumps are cnsidered (cf. Fig. 3), 3.) the partial cncentratin in sites f energy E 1 [= n(e 1 )(E 1 )] and 4.) the availability f empty sites f energy E [= n(e ){1-(E )}]. Nte that a cnstant saddle pint energy has been assumed, in rder t allw fr an uncrrelated randm walk. The effect a distributin f saddle pint energies has n the diffusin cefficient will be discussed in chapter V. By averaging ver E 1 and E ne btains: exp 1 ) ( ) ( exp 1 ) ( exp 1 de de T k E E n E n T k E E n T k E E Q c B B B Γ = Γ µ µ (5.) Thus Q is the activatin energy f diffusin in the reference material. Integratin f the last term in brackets immediately yields ne factr (1-c). By using the identity + = + T k E E n E n T k T k E T k E E n B B B B µ µ µ exp 1 ) ( ) ( exp exp 1 ) exp ( (5.3) integratin f Eq. (5.) with respect t E 1 gives ) (1 exp exp c T k E T k Q c B B Γ = Γ µ. (5.4) Inserting in Eq. (5.1) yields a simple result B c D c T k Q l D γ γ ) (1 ) (1 exp 6 * = Γ = (5.5) with D being a tracer diffusin cefficient in a material which cntains sites f energy E nly and γ being an activity cefficient with respect t this material as a reference state. One f the factrs (1-c) is due t blcking f sites whereas the secnd factr takes care f the fact that γ appraches (1-c) -1 fr c 1 in a single crystal [66]. The temperature dependence f the tracer diffusin cefficient is slely determined by the expnential dependence f the average jump frequency accrding t Eqs. (5.4) and (5.5). Thus at a given cncentratin the effective activatin energy is Q +E -µ, which is the energy difference between saddle pint energy and Fermi energy. The Fermi energy µ will nt change very much with temperature fr brad DOSE and, therefre, D* beys an Arrhenius Law despite a distributin f site energies r jump rates, respectively. This is a cnsequence f the step behavir f the FD-functin, where the ccupancy at lw temperatures and a brad DOSE is smeared ut arund the Fermi energy withut changing the latter very much [67]. The cncentratin dependence f the effective activatin energy Q +E -µ is very prnunced as µ increases with increasing cncentratin because f the secnd derivative f Gibbs free

14 14 energy being always psitive in equilibrium. Thus an increasing cncentratin always leads t a decreasing activatin energy and an increasing diffusivity, if we neglect the effect f c in the denminatr f Eq. (5.4) (cf. sectin 5.4). Again the atmistic interpretatin is very simple. With increasing cncentratin sites f higher energy have t be ccupied and, therefre, the Fermi energy rises and cmes clser t the saddle pint energy. Thus an increasing number f particles experience a smaller activatin barrier. b. Diffusin in a cncentratin gradient In rder t derive an expressin fr the chemical r intrinsic diffusin cefficient D we cnsider tw adjacent lattice planes f distance l (= jump distance). The x-axis is parallel t the nrmal f the planes and the average cncentratins c 1 and c within the planes are different. Thus we have a cncentratin gradient and D is defined by Fick's First Law: Ω = Ω = β β l c c D x c D J 1, (5.6) where J is the flux f particles frm plane 1 at x t plane at x+l, Ω is the atmic vlume f the metal and β is the number f interstices per metal atm. The rati β/ω n the right hand side f Eq. (5.6) has t be included because c was defined as the fractin f ccupied interstices whereas in Fick's Law cncentratin is defined as particles per vlume (cf. Eq..). The flux J is als btained by averaging ver the jumps in between the planes frm sites f energy E 1 in plane 1 t sites f energy E in plane and vice versa ) ( exp 1 ) ( ) ( ) ( exp 1 ) ( ) ( 6 de de T k l x E E n l x T k x E E n x l J B B Γ + Γ Ω = µ µ β (5.7) where the jumps ccur in all directins and nly 1/6 f them t the plane under cnsideratin. The first term in brackets crrespnds t the jumps f particles in sites f energy E 1 ut f plane 1 at x and the secnd term accunts fr particles in sites f energy E in plane at x+l, where the jump frequencies Γ i have t be weighted with the fractin f ccupied sites f a given energy n(e i )(E i ). Jump frequencies are calculated as befre fr thermally activated prcesses ver saddle pints f cnstant energy E +Q (cf. Fig. 3) including blcking f sites by a factr f n(e i )[1-(E i, µ)] [ ] + + = Γ Γ T k E E Q l x E E n B 1 1 exp )) (, ( ) 1 ( µ (5.8) [ ] + = Γ Γ T k E E Q x E E n B 1 1 exp )) (, ( ) 1 ( µ (5.9) In Ref. [68] similar calculatin were presented fr the first time and the fact that the ccupancy depends n the chemical ptential (cf. Eq. (4.)) which depends n psitin x was nt taken int accunt prperly leading t a slightly different expressin fr D. Inserting Eqs. (5.8) and (5.9) in Eq. (5.7) and partly integrating yields: Ω Γ = T k l x E de E n T k E T k x E de E n T k E T k E Q c l J B B B B B ) ( exp 1 ) ( exp ) ( exp 1 ) ( exp exp 6 ) ( µ µ β (5.10) Using Eq. (5.3) and expanding µ(x+l) in a Taylr series gives:

15 βlγ (1 c) Q µ E l µ J = exp exp (1 c) (1 c)(1 + ) (5.11) 6Ω kbt kbt kbt x and βd (1 c) µ E 1 µ J exp =. (5.1) Ω kbt kbt x This way we have derived an expressin which is in agreement with irreversible thermdynamics stating that the gradient f the chemical ptential is the driving frce fr diffusin. Using the expressin derived fr the tracer diffusin cefficient D* (Eq. (5.4) and (5.5)) gives D* C µ a J = = D (1 c), (5.13) kbt x x where C is the cncentratin as usually defined, i.e. mles per unit vlume. Thus the factr in frnt f the gradient term is the mbility f the particles as expected frm the Einstein-Stkes relatin [38]. Thus a self cnsistent derivatin is prvided fr bth diffusin cefficients D* and D. In rder t apply the cncept t diffusin and permeatin f hydrgen thrugh materials, the material and its defects have t be characterized first and described by a DOSE. Then Eq. (4.3) is used t calculate µ(c) numerically r as a clsed slutin. This functin allws t calculate D* and D via Eqs. (5.5) and (.3). With apprpriate bundary cnditins Fick's Secnd Law has t be slved with a cncentratin dependent diffusivity D. Instead f using calculated values f µ measured nes can be inserted in Eqs. (.3) and (5.5) as well. c. General randm walk It will be shwn in the fllwing that the assumptin f cnstant saddle pint energies can be replaced by less restrictive cnditins. This has been dne fr thermally activated hpping in a previus study [14] including a Gaussian distributin f saddle pint energies. Hwever, ne may argue that thermally activated hpping is nt apprpriate fr H-atms, because they may migrate via quantum mechanical tunneling (cf. sectin ). Therefre we apprach the prblem f diffusin in an energy landscape frm a different perspective. We cnsider P particles in N sites f a lattice with a given DOSE. The equilibrium distributin f the particles accrding t FD-Statistics shall be statinary in space and ne tagged particle shall d a randm walk n the empty lattice. It might nt be really necessary t assume a statinary distributin fr the remaining particles. What might be the imprtant effect nly is that the mving particle will nt be able t ccupy all lattice sites as they are differently blcked accrding t Eq. (4.). One might call this simplificatin the ne particle apprximatin resembling the ne electrn apprximatin f slid state physics. Further n the walk is uncrrelated besides the few blcking events and, therefre, the mean distance R after Z jumps is given accrding t simple randm walk thery [38, 40] R = Zl, (5.14) where l is the jump distance. Then a diffusin cefficient can be defined by [38, 40] R l Z D* lim = lim, (5.15) t 6t 6 t t where t is the ttal time required fr the walk. The time t is the sum f all the times f residence τ m the particle stayed in the varius sites visited during the walk, if the time interval required fr the site exchange is small cmpared t τ m. Thus we get 15

16 t = Z m= 1 τ. (5.16) m This way the jump rate as the reciprcal f the residence time has nt t be described by a special mechanism like thermally activated hping fr instance. Amng the varius sites we cmbine thse having the same energy E i and average ver the distributin f empty sites, i.e. sum ver all N-P sites. Hereby it is assumed that the Z sites which have t belng t the empty categry are as representative fr empty sites as are the N-P nes. Thus we btain t = Z m= 1 N P Z τ = τ t, (5.17) m i m i N P Em = Ei i= 1 where t i is the time the particle spends in sites f energy E i. It is tacitly assumed that the number f jumps Z and the number f lattice sites N are large enugh in rder t apply the laws f randm walk and statistical mechanics. Accrding t the ergdic hypthesis the fractin f time a particle spends in sites f type i is equal t the fractin f particles in this type f site yielding ( E ) n [ 1 ( E ] ( E ) ti ni i i i ) i = =, (5.18) t P P where n' i is the number f free sites f energy E i which is btained frm the site energy distributin by multiplying with [1-(E i )]. Fr the dilute slutin mst f the free sites have a lw ccupancy [(E i )<<1] and Eq. (4.) gives µ Ei ( Ei ) exp. (5.19) kbt This way FD-Statistics is replaced fr the empty sites by Bltzmann statistics. Fr cnvenience we chse an arbitrary reference site having energy E and a lw value f (E ). Then the fllwing relatin is derived frm Eqs. (5.18) and (5.19) ] [ ] ti ni 1 ( Ei ) E Ei = exp t n E, (5.0) 1 ( ) kbt where t is the fractin f the ttal time t particles reside within sites f energy E. Inserting Eq. (5.0) in Eq. (5.17) with [1-(E )] 1 and Eq. (4.) gives N [ ] P Zt E Ei exp ( Ei µ ) / kbt t = ni exp (5.1) ( N P) n [ ] = k + i BT 1 exp ( Ei µ ) / kbt 1 r Zt t == exp ( [ E µ ) / k T ] N(1 c) n B N P ni 1+ exp ( i= 1 i / [ E µ ) k T ] B (5.) where the last sum n the right hand side is nthing else than the discrete frm f Eq. (4.3) and, therefre, it can be expressed by the cncentratin c. Then the fllwing simple result is btained [ E µ ) / k T ] czt exp ( Zt Zτ t (1 c) n γ (1 c) n (1 c) γ where τ is the mean residence time as defined by B = = = (5.3) n 1 τ = s fr Es = E n τ (5.4) s=1 Inserting Eq. (5.3) int Eq. (5.15) yields 16

17 D* [ µ E ) / k T ] l (1 c) l (1 c) exp ( B = = γ (5.5) 6τ c 6τ If we cnsider the hypthetical material with sites f energy E nly the residence time τ as defined by Eq. 5.4 is enlarged cmpared t the empty lattice due t blcking f sites. The dilute residence time is then τ =τ(1-c) and the diffusin cefficient in the dilute regime f the reference lattice is given by l l D = =. (5.6) 6τ 6τ (1 c) By cmbining the last tw equatins the same result as in Eq. (5.5) is btained and, therefre, the result is independent f thermally activated hpping. Hwever, the cncentratin dependence f the diffusin cefficient despite cntaining the same expnential term exp(µ/k B T) has t be interpreted differently. Independent f the atmistic mechanism f diffusin the ccupancy f sites with higher energy at high cncentratin is accmpanied with a decrease f the time f residence (ergdic hypthesis) and a cncmitant increase f mbility. d. Lw cncentratin limit The previus discussin f slubility at lw cncentratins is relevant fr the diffusin cefficient, t. Fr thse cases where Henry s Law is beyed the activity cefficient is independent f cncentratin and s are bth the tracer and the chemical diffusin cefficient, if in the dilute regime the (1-c) term is neglected. Hwever, fr the academic case f an expnential DOSE discussed in sectin 4. the tracer diffusin cefficient D* will always decrease with decreasing cncentratin. Mre realistic cases f a DOSE like the Gaussian ne and experimental results shw that Henry s Law is fulfilled and, therefre, D* becmes independent f c at c 0. The bvius cntradictin t the discussin at the end f the last sectin is vercme by realizing that at lw cncentratins the step behavir f the FD functin n lnger hlds. The step behavir is a cnsequence f energy minimizatin, but a very lw cncentratins maximizatin f entrpy is mre imprtant. Thus mst f the particles g in sites abve the Fermi level withut filling these up t saturatin and, therefre, gain cnfiguratinal entrpy. Fr a Gaussian distributin it has been shwn that the average energy f the ccupied sites becmes independent f cncentratin at c 0 and, therefre, the average time f residence and D* are independent f c. This delicate balance between energy and entrpy is discussed in mre detail in Ref. [14, 69]. 6. H-H INTERACTION Althugh in the framewrk f this study dilute H-systems are cnsidered nly, H-H interactin has t be taken int accunt fr thse cases, where segregatin at extended defects takes place and, therefre, lcally high cncentratins ccur. Then the chemical ptential may be written in a first rder apprximatin as [70] µ = µ id + Wc lc, (6.1) where W is an interactin parameter. Again it is reasnable t assume that we have a bimdal DOSE like the ne in Eq. (4.11) yielding [70] n E de f ( ) µ E c = lc = f and c fr exp. (6.) E µ Wc + kbt lc 1 exp kbt 17

18 With the trivial equatin c=c fr +c lc we can slve, analytically r numerically depending n the frm f n f (E), Eq. (6.) t get the relatin µ(c). 18

19 IV. Interactin f hydrgen with defects 7. INTERACTION WITH OTHER SOLUTES AND VACANCIES An apprpriate DOSE fr the interactin f hydrgen with pint defects is a tw-level system n( E) = n ( E) + nt ( E) = (1 ct ) δ ( E E ) + ctδ ( E Et ) (7.1) which is a special frm f the DOSE in Eq. (4.11). The traps prvided by the defects are present with a cncentratin c t and have all the same binding energy E t -E with respect t the "nrmal" sites. Hydrgen residing in nrmal sites will be called free and integratin f n (E) accrding t Eq. (4.3) yields the cncentratin f free hydrgen: (1 ct ) c f c f = r µ = E + k BT ln (7.) E µ 1 ct c f 1+ exp kbt Fr the dilute case c t <<1 and c f <<1 Eq. (5.13) gives the same result as derived by Oriani [60] βd c f J = (7.3) Ω x stating that the flux f hydrgen is prprtinal t the cncentratin gradient f free particles. This assumptin is valid beynd the tw-level case as shwn in the fllwing. The derivatin f the last tw equatins is independent f the functinal frm f n t (E) as can be seen by cmparing Eq. (4.1) and (7.). The interactin f hydrgen with freign atms bth substitutinal and interstitial is rather weak with a binding energy f abut 10 kj/ml [9, 71]. This has been determined by a large variety f experimental techniques measuring permeatin, internal frictin, resistivity and neutrn scattering. Because f the small value f the interactin energy it is difficult t decide hw much f it is elastic r electrnic interactin. Hwever, interactin with vacancies is very strng with measured binding energies between 30 and 100 kj/ml. They are in gd agreement with calculated values using the effective medium thery [9]. The hydrgen atm is cnsidered t be disslved within the vacancy althugh slightly ff centre. The metal hydrgen distances are s large that the interactin ptential is in the attractive regin giving rise t a vlume cntractin. This is in agreement with experimental findings in heavily defrmed Pd, where negative vlume changes ccurring at lw cncentratins have been attributed t vacancies frmed during plastic defrmatins [53]. 8. INTERACTION WITH DISLOCATIONS Theretical mdels describing the interactin f slute atms with dislcatins [7] are based n an elastic interactin between the stress field arund dislcatins and the strain caused by a slute atm (cf. II.C). The interactin is strngest fr edge dislcatins because f the hydrstatic stress field. Thus slute atms changing the vlume during disslutin, i.e. cntaining a nn-vanishing trace in their strain tensr experience a strng elastic interactin with edge dislcatins. Hydrgen in metals causes vlume expansin and, therefre, interacts strngly with the stress field f edge type dislcatins. As stresses f dislcatins are 19

20 calculated by applying cntinuum thery they are less reliable within the dislcatin cre and, therefre, trapping f hydrgen in dislcatin cres has t be treated separately. In the fllwing the DOSE is calculated based n these cnsideratins fr edge dislcatins. The hydrstatic part f the stress field f an edge dislcatin is given by [7] σ ii Gb(1 + ν ) sinθ p = =, (8.1) 3 3π (1 ν ) r where G is the shear mdulus, ν Pissn's rati, b the magnitude f the Burger s vectr, θ and r are cylindrical crdinates as defined in Fig. 4 with the z-axis alng the dislcatin line. Then the interactin energy with H-atms n a circle f cnstant pressure is btained frm Gb(1 + ν ) AVH pvh = VH, (8.) 6π (1 ν ) R R where R is the radius f the cylinder and A is defined by the last equatin. Fr H in Pd the number f ctahedral sites which are chsen by hydrgen is the same as the number f Pd-atms. We assume that like in the β-phase f Pd nly the fractin α (being ca. 0.6 at rm temperature) is ccupied. Then in a material cntaining ρ dislcatins per unit area the number f sites, n, in a cylinder f radius R and unit length is ρπr and the DOSE becmes dn dn dr AVH αρπavh n( E) = αρ = αρ = αρπr =. (8.3) 3 de dr de E E Inserting this in Eq. (4.3), using the step apprximatin, i.e. the first part n the right hand side f Eq. (4.6) and slving fr µ gives αρπav µ = H. (8.4) c The last equatin is checked by pltting in Fig. 5 measured values f µ fr H in cld rlled Pd [73] versus the reciprcal square rt f cncentratin. Different t the theretical predictin the interactin energy (difference f chemical ptentials between defrmed and annealed sample) becmes cnstant at very lw cncentratins which is explained in terms f a direct interactin with the dislcatin cre which was nt included in Eq. (8.). The value f abut -50 kj/ml-h has been als determined fr H in Fe [74]. The linear dependence n 1/ c in Fig. 5 yields a slpe which crrespnds t reasnable dislcatin densities fr heavily defrmed metals ( cm - ). Cntrary t Eq. (8.4), the straight line crrespnding t the interactin with the lng range stress field f the dislcatin des nt intercept the rdinate at 0 but at a value f abut -0 kj/ml-h. This is attributed t a direct H-H interactin. It can be calculated frm the values btained fr this interactin in well annealed Pd at high H-cncentratins where the parameter, W, defined in sectin 6. was measured t be -30 kj/ml-h [75]. With a maximum lcal cncentratin f α=0.6 the cntributin t µ frm H-H interactin becmes -18 kj/ml-h in gd agreement with the intercept in Fig. 5. During the electrchemical measurements f µ the standard value was defined such that µ = k T ln, (8.5) B c f where accrding t Eq. (4.1) E =0. This is als the chemical ptential fr the single crystalline metal, where the cncentratin f free hydrgen c f is equal t the ttal cncentratin. Besides measuring µ fr a metal with a high dislcatin density in cmparisn 0

21 with a single crystal, the cncentratin f free hydrgen can be determined als frm measurement f the electrical resistance. It has been shwn fr Pd [76] that hydrgen trapped as a hydride at the dislcatin lines cntributes t the resistivity t a negligible part nly when cmpared t the same number f H-atms distributed hmgeneusly. Thus the resisitivty increment ρ H caused per unit f H-cncentratin divided by the same increment in a single crystal ρ H yields the fractin f H-atm being free. The same is true fr a dislcated metal and its tracer diffusin cefficient D*=D γ (cf. Eq. (5.5) with c<<1) where D has a well defined peratinal meaning being the tracer diffusin cefficient in the single crystal. Therefre, we have µ c f ρh D * γ = exp = = = kbt (8.6) c ρh D It is shwn in Fig. 6 that this simple relatin hlds fr hydrgen in strngly defrmed palladium. At very lw H-cncentratins the fractin f free hydrgen is negligible and, therefre, all the hydrgen is trapped at dislcatins. At intermediate cncentratins hydrgen atms are partitined between sites far away frm dislcatins and thse clse t them. But sites at dislcatins never becme saturated because f bth the lng range elastic interactin and an attractive H-H interactin. Thus the crrespnding cylinder being enriched in hydrgen is steadily grwing. The cncentratin dependence f the chemical diffusin cefficient f H in defrmed crystalline Pd is shwn in mre detail fr very lw cncentratins in Fig. 7. Using measured values f the chemical ptential fr the same defrmed Pd and a diffusin cefficient btained fr a single crystalline Pd the lines in Fig. 7 were calculated by using Eqs. (.3) and (5.5). Thus withut a fitting parameter the cncentratin dependence f D is btained in very gd agreement with experimental results. Deviatins ccurring at very lw cncentratins are either due t an enhancement f diffusin alng the dislcatin cre r they arise because f difficulties measuring chemical ptentials at very lw H-cncentratins. The prnunced changes f H-activity, diffusivity and resistivity in crystalline Pd which are caused by plastic defrmatin and arise frm the presence f dislcatins d nt ccur after cld rlling f amrphus PdSi-allys [77]. The absence f a detectable H-trapping in defrmed amrphus allys is cnsidered t reveal the absence f edge dislcatin-like defects. By measuring vlume changes caused by disslved hydrgen in severely defrmed crystalline palladium (99% reductin in crss sectin by cld rlling) it was shwn [53] that samples cntracted fr the first 50 t 100 atppm f H. This was attributed t trapping in vacancies (cf. sectin 3) which suppsedly frm during cld-rlling. After saturating the vacancies hydrgen was trapped in dislcatin. In this cncentratin range the mlar vlume f hydrgen was slightly smaller than in single crystalline Pd. This behavir is in accrdance with the assumptin f hydrgen being disslved in the expanded regin belw the glide plane f edge dislcatins. With increasing cncentratin the dislcatins became saturated and additinal hydrgen was predminantly disslved in nrmal ctahedral sites far away frm dislcatins. As a cnsequence the hydrgen partial mlar vlume apprached that f the single crystal. S far the segregatin f hydrgen at edge dislcatin and the cncmitant frmatin f hydride cylinders belw the glide plane have been prven in an indirect way nly. Direct evidence can be prvided by small angle neutrn scattering (SANS). Scattering by randmly riented cylinders has t be described by the fllwing macrscpic crss sectin [78] 1

22 3 4 d Σ π ρr0 g 1 = exp Q R 0 dω Q 4, (8.7) where ρ is the dislcatin density, R the radius f the cylinders, g is the difference f scattering length densities and Q is the magnitude f the scattering vectr. By pltting the lgarithm f the prduct f Q and measured values f the macrscpic crss sectin (after apprpriate subtractin f backgrund and incherent scattering) versus Q straight lines are expected accrding t Eq. (8.7). This is in agreement with experimental findings as shwn in Fig. 8. The slpe f the straight lines yields the radius f the cylinders and the intercept with the rdinate yields the dislcatin density. Similar plts have been evaluated fr different H- cncentratins and the results are presented in Fig. 9. Hwever, the natural chice fr SANS is deuterium instead f hydrgen because the frmer has a larger crss sectin fr cherent scattering. In additin, hydrgen gives rise t prnunced incherent scattering, i.e. raises the backgrund. In agreement with the larger cherent scattering measurements with deuterium lead t larger macrscpic crss sectin as shwn in Fig. 10. Hwever, they are larger by a factr f 1.5 nly, whereas Eq. (8.7) predicts a factr f 3. crrespnding t the squared rati f scattering length densities. This discrepancy is vercme by taking int accunt that there are tw cntributins t the scattering cntrast. The first ne is due t the H- r D-atms having a higher cncentratin at the dislcatin lines. The secnd ne is a cnsequence f this segregatin, because bth istpes expand the Pd-lattice and, therefre, reduce the scattering cntrast f Pd with respect t the matrix far away frm dislcatins. H-atms have a negative scattering length fr neutrns and the crrespnding negative cntrast (difference f scattering length density) is exaggerated by the lattice expansin. Fr deuterium with a psitive scattering length the ppsite is true. Thus the peculiar scattering behavir f the hydrgen istpes yields additinal insight int the segregatin at dislcatins. The high lcal cncentratins f hydrgen at the dislcatins may be treated as suggested in sectin 6. Hwever, it is simpler t apply a thermdynamic mdel treating the prnunced segregatin as a hydride frmatin in a hydrstatic stress field. Withut stresses (p=0) and fr equilibrium between the hydride and saturated slid slutin, the chemical ptentials in the tw phases have t be the same, i.e. 0 µ = kt ln c (8.8) ( hydride, p= 0) µ + ts where c ts is the terminal slubility f hydrgen in Pd in equilibrium with the β-phase (f cmpsitin PdH α ). As the terminal slubility f H is 0.01 H/Pd at rm temperature, the ideal slutin apprach fr the cnfiguratinal entrpy (lgarithmic term in Eq. (8.8)) is justified. At the brder between the cylindrical hydride and the slid slutin a cnstant hydrstatic pressure p (cf. Eq. (8.1)) is present, and the chemical ptential is changed t 0 µ = µ + kt ln c ts + pv H (8.9) Far away frm the dislcatin where hydrgen is free and where it has a lcal cncentratin f c f, the chemical ptential is given by Eq. (8.5) (here with the standard value µ ο ). Then Eqs. (8.), (8.5) and (8.9) yield Gb(1 + ν ) VH C c f = cts exp = cts exp (8.10) 6π (1 ν ) ktr R where C is 1.0 nm fr edge dislcatins f b=0.75 nm in Pd. Besides c f, the hydrgen trapped as a cylindrical hydride f cmpsitin α=0.6, radius R and length ρ d cntributes t the ttal cncentratin c tt. Thus we have in terms f H/Pd

23 = + = C ctt αρ dπr c f αρ dπr + cts exp (8.11) R If we use this implicit functin f R(c tt ) and cmpare it in Fig. 9 with measured values, a gd agreement is btained by using a dislcatin density f cm -. This is cnsidered t be additinal evidence fr an extended segregatin f hydrgen, which requires taking int accunt bth elastic and slute/slute interactin. The experimental results presented in this sectin are in qualitative agreement with a variety f studies by ther grups [4, 44, 79, 80]. 9. INTERACTION WITH GRAIN BOUNDARIES Segregatin f slute atms at grain bundaries is ften studied by breaking a sample in a UHV-chamber, where in the case f intercrystalline fracture the crack runs alng the grain bundaries and the slute atms are expsed t surface analytical techniques such as AES, XPS and SIMS. Fr hydrgen this is difficult t achieve as nly the latter methd is able t detect H. In additin the high H-mbility at rm temperature allws surface segregatin t be established n the frmer grain bundary befre the measurement f the riginal grain bundary cverage takes place. Again the cncept f a site energy distributin is useful t study H-segregatin at grain bundaries. Similar t dislcatins the number f traps prvided by the grain bundaries is rather small and, in rder t study them by gradually filling, we need a large density f them, i.e. a small grain size. This will be the case fr nancrystalline metals. Electrchemical measurements [81] f the chemical ptential µ which were cnverted int partial pressures are presented in Fig. 11 fr a nancrystalline sample and a single crystal f Pd. In the latter case Sieverts Law is fulfilled in the slid slutin range. The behavir f the nancrystalline sample is described by a Gaussian distributin f site energies fr sites within the bundaries and a single level fr sites within the grains (cf. Fig. 1). With respect t the large variety f grain bundaries in a nancrystalline material and a variety f structural units within a certain grain bundary the cncept f a cntinuus distributin f segregatin energies appears t be mre reasnable. Fr the sake f simplicity the sites within the grains are assumed t have the same energy as sites in a single crystal. This is still a gd apprximatin fr higher cncentratins, where the interfacial stress affects the site energy within the grains [8]. Thus at a given chemical ptential r partial pressure, respectively the cncentratin in the grains has t be the same as in the single crystal and, therefre, its cntributin t the ttal cncentratin can be subtracted yielding the amunt segregated at the bundaries. Fitting Eq. (4.3) t the experimental results presented in Fig. 11 yields values fr E seg and σ. A mre detailed descriptin f the prcedure and the results is given in Refs. [14] and [81]. Fr H in nancrystalline Ni experimental value are nt available ver the same large range f H-cncentratin but they can be described within the same framewrk f a distributin f segregatin energies as well [83, 84]. Althugh tw parameters (E seg and σ) are available, experimental results at large H- cncentratins cannt be fitted (dashed curve in Fig. 11). The discrepancy arises frm neglecting H-H interactin. This can be included via a quasichemical apprach withut intrducing a new fitting parameter (cf. sectins 6 and 8), because the interactin parameter W btained frm pressure cmpsitin istherms f carse grained Pd was used. Then the slid curve in Fig. 11 is btained in excellent agreement with experimental data. It is 3

24 interesting t nte that the width f site energies fr the grain bundaries σ = 15 kj/ml H is in between the nes btained frm fitting data f liquid quenched and sputtered amrphus Pd-Si allys, where the width is 11.5 r 17.5 kj/ml H, respectively (cf. sectin 14). H-Diffsuin in nancrystalline Pd was measured via a time-lag methd [14, 81], where the pellet shaped samples were electrchemically charged with hydrgen frm ne side and the delayed respnse f the electrchemical ptential at the adjacent side was mnitred (cf. sectin ). The cncentratin was raised in small steps, in rder t minimize errrs arising frm the cncentratin dependence f the diffusin cefficient. As the transprt f H thrugh the sample is a mixture f grain bundary and bulk diffusin, the numbers evaluated frm the time lag were called effective diffusin cefficients (see discussin belw). The results are presented in Fig. 13. Fr atms diffusing alng grain bundaries being perpendicular t the surface three limiting cases are discussed [85] as shwn in Fig. 14. Case (A) crrespnds t the cnditin D g t>>d, where D g is the diffusin cefficient within the grains, d is the distance f grain bundaries and t is the time. Fr the results presented in Fig. 13 D g = cm /s, d 10 nm and t is between a few secnds and a few minutes and, therefre, the cnditin fr type A diffusin is always fulfilled. Then the effective diffusin cefficient is given by [85]: D = fd + ( 1 f ) D (9.1) eff gb g where f is the vlume fractin f grain bundaries which (fr the case shwn in Fig. 14 with additinal bundaries running parallel t the drawing plane) is given by: f δ = d + δ (9.) The case f slute segregatin described by a factr S=c gb /c g can be frmally included by replacing δ by Sδ [85]. Then the rati f the tw cntributins t the effective diffusin cefficient becmes: fdgb SδDgb = (9.3) (1 f ) D dd g g The cntributin f the grain bundaries can be dminant r determine the transprt, respectively fr tw reasns: A1: Grain bundary diffusin is much faster than bulk diffusin (D gb >>D g ) but the grains are fed frm the grain bundaries because f the shrt diffusin length d<< D g t and A: Grain bundary diffusin is slwer than bulk diffusin (D gb <D g ) but the segregatin factr S is much larger than unity. The cncentratin frnt is mving ahead within the grains but the bundaries act as sinks retarding the transprt thrugh the grains. The latter case A applies t the lw cncentratin results presented in Fig. 13, because frm the data f Fig. 11 the segregatin factr S can be btained frm the trivial relatin c = fc + ( 1 f ) c = fsc + (1 f ) c (9.4) tt gb g g g and a grain bundary thickness δ f abut 0.5 nm. The cncentratin within the grains c g is equal t the values f the single crystalline sample at a given chemical ptential, because they are assumed t have the same site energies. Thus the rati S in Eq. (9.4) is much larger than unity and, therefre, the measured H-transprt thrugh the nancrystalline membrane is determined by grain bundary diffusin and D eff = D gb. Case A is nt discussed in textbks n diffusin because grain bundary diffusin is always cnsidered t be faster than diffusin thrugh the grains. This judgment is based n the decreased density f atms in the bundaries which gives rise t a lwer frmatin energy f 4

25 vacancies r vacancy like defects as prerequisites fr substitutinal diffusin. Hwever, interstitial diffusin des nt require vacancies and it is slwed dwn in the presence f a distributin f site energies as the interstitials are trapped in lw energy sites at lw cncentratins. At higher cncentratins the traps are gradually saturated and interstitial diffusin takes advantage f the mre pen structure f the bundary as well. The cncentratin dependence f D gb can be calculated via Eq. (.3) and (5.5) neglecting the cntributin frm grains, i.e. cnsidering the Gaussian distributin in Fig. 1 nly. H- cncentratin has t be replaced by the grain bundary part c gb and γ is btained frm the measured chemical ptential. Thus nly ne fitting parameter, the reference diffusivity D, can be changed t btain agreement with experimental data. With the lgarithmic scale used in Fig. 13 the calculated curves are mved up and dwn in the directin f the rdinate withut changing their slpe and curvature. The steady increase f D with increasing cncentratin is equivalent t amrphus materials (cf. chapter VI) because a Gaussian distributin has been used there as well. Hwever, different t the glassy materials cncentratins in the bundaries becme s large that blcking f sites and H-H interactin has t be taken int accunt (withut additinal fitting parameters). The cncentratin in the bundaries is abut 0.5 H/Pd at c tt =0.01 fr δ=0.5 nm. As the H-H interactin is attractive (W=-30 kj/ml-h) the diffusivity is finally decreased at high H-cncentratins. In agreement with the generally accepted wisdm, the average r reference diffusin cefficient D f the grain bundaries is larger than the bulk value due t a lwer activatin energy fr H-atms arising frm the lwer metal density in grain bundaries. The cncept f interstitial diffusin in grain bundaries used in this study and the interpretatin f the cncentratin dependence have been embedded in a general cntext [86]. The increased H-slubility in the α-phase f nancrystalline Pd has been cnfirmed by ther experimental methds, i.e. dping samples frm the gas phase and measuring pressure drps [87] and r lattice parameters [8]. Like in Ref. [87] pressure-cmpsitin istherms r lattice parameters were als measured at high hydrgen cncentratins using an electrchemical technique [88]. In all studies [8, 87, 88] it was shwn that the miscibility gap between α- and β-phase is remarkably reduced in nancrystalline Pd. The crrespnding increase f the terminal slubility in the α-phase is generally accepted t be due t segregatin at the grain bundaries, whereas the reductin f the lwer limit f H-slubility in the β-phase is interpreted differently. In Ref. [88] it is argued that the cmpsitin f the grain bundaries d nt change within the miscibility gap, because the chemical ptential f hydrgen has t be cnstant in this regin. Then all f the grains are transfrmed int β-phase whereas the H- ccupancy f the grain bundaries des nt change, i.e. remains at a value which is larger than in the α-phase (segregatin) but smaller than in the β-phase. Thus an verall decrease f the ttal H-cncentratin ccurs at the (α+β)/β-bundary f the miscibility gap. Fllwing this reasning an average grain bundary thickness f 0.7 t 1 nm can be calculated withut data fitting [88]. Hwever, ne has t take int accunt [8] that the inhmgeneus distributin f hydrgen between grains and grain bundaries gives rise t a crrespnding inhmgeneus distributin f mechanical stresses which affect the chemical ptential. Thus segregatin in grain bundaries leads t cmpressive stresses in these regins whereas the grains g int tensin. Hydrgen atms are redistributed and crrespnding changes f the lattice parameter ccur. At the (α+β)/β-bundary f the miscibility gap the stress distributin is reversed because f the higher H-cncentratins in grains. These effects d nt change the data evaluatin presented in this study because at the dilute regin the crrespnding mechanical stresses are t lw t change the chemical ptential t a measurable extend. 5

26 10. INTERACTION WITH METAL/OXIDE BOUNDARIES Hydrgen trapping at metal/ceramic interfaces has been studied extensively [89, 90, 91] because f its relevance in the area f hydrgen embrittlement f high strength steels, where hydrgen is trapped at the metal/carbide interface [9]. Due t the large H-slubility and the ease f measuring chemical ptentials palladium allys are used as mdel allys again. Fr sme metal/xide interfaces high reslutin electrn micrscpy and analytical field in micrscpy [93, 94, 95, 96] revealed that mst f the terminating layers are dense packed xygen planes (cf. Fig. 15). As a cnsequence small xide precipitates have an excess f xygen when cmpared with the stichimetry resulting frm charge neutrality. Thus the negative charge f the excess xygen at the interface has t be delivered by the surrunding metal as it was shwn recently by electrn energy lss spectrscpy [97, 98]. In additin, it has been shwn [89] that the excess xygen at the Ag/MgO interface is bund there with an energy which crrespnds t the frmatin energy f Ag O in agreement with the structural mdel f Fig. 15 and electrn energy lss spectrscpy fr the same [98] and fr a similar bundary Cu/MgO [97]. The cncept f varying sichimetry is in agreement with experimental findings, where dependent n the xygen activity during sample annealing treatments irreversible and reversible trapping f hydrgen was bserved [90, 91]. The term irreversible trapping was used because the crrespnding part f trapped hydrgen culd nt be remved by prlnged andic plarizatin f the sample, i.e. the binding energy t the traps was s high that the crrespnding reductin f H-mbility did nt allw a remval f the trapped hydrgen. Raising the temperature abve 300 C finally leads t a depletin f these traps. The amunt f irreversibly trapped H crrespnds t abut ne mnlayer at the phase bundary and can be ascribed t the frmatin f O-H bnds at the interface accrding t the fllwing relatin: H (in Pd) + MgO + PdO Mg(OH) + Pd, (10.1) where the xygen f the PdO crrespnds t the excess xygen within the structural vacancies f the terminating O - -layer f the precipitate (cf. Fig. 15 fr the analgus case f Ag/MgO). Calculating Gibbs free energy f the reactin and the vlume change yields data fr the trapping energy and partial mlar vlume f H which are in agreement with experimental findings [90, 91]. Thus the frmatin f a Mg(OH) layer at the interface gives rise t a remarkable vlume change which is abut tw times as much as fr H-atms disslved in ctahedral sites f Pd. Althugh a higher elastic energy fr lattice distrtin has t be paid by segregatin at the xide/metal interface, it takes place, because the gain f chemical energy by frming the OH bnds is much larger than the crrespnding elastic energy. It is interesting t nte that the crude apprximatin f the chemistry at the interface by bulk behavir gives reasnable results. Because f the higher hydrgen mbility in silver it was pssible t internally xidize an Ag- 1at.-% Mg at rather lw temperatures leading t very small precipitates f MgO (1.6 t 5 nm in diameter). These were analyzed using a tmgraphic atm prbe [99]. The results cnfirmed an excess f xygen at the interface with the metal. Hwever, an analysis f hydrgen turned ut t be difficult because f the residual hydrgen in the vacuum chamber. Therefre, small angle neutrn scattering (SANS) was applied, in rder t get additinal infrmatin abut segregatin f bth excess xygen and hydrgen r deuterium, respectively. Again the Ag/MgO samples were advantageus because f the smallness f their precipitates. 6

27 As the slubility f hydrgen in silver is very lw, a temperature f 400 C was chsen, in rder t have a sufficient flux f hydrgen which is necessary t fill all the traps at the xide/metal interfaces. This way it was pssible t d SANS experiments with 3 types f samples: (i) samples after internal xidatin, (ii) internally xidized plus dping with hydrgen and (iii) internally xidized plus dping with deuterium [100]. The macrscpic crss sectin dσ/dω btained fr these samples is presented in Fig. 16. After the backgrund and the incherent scattering were subtracted, three different regimes can be distinguished fr all samples. At very lw values f the scattering vectr Q large MgO precipitates at the grain bundaries give rise t a steep decrease f dσ/dω. Then a plateau regin fllws which describes the scattering f the small precipitates within the grains at lw Q-values. The plateau is fllwed fr Q>R -1 by a Guinier regime which allws the evaluatin f the average radius R f the xide particles assuming a spherical shape [100]. Bth H- and D-dping change dσ/dω cnsiderably. MgO has a higher scattering length density than Ag and, therefre, segregated hydrgen (deuterium) having a negative (psitive) scattering length decreases (increases) the macrscpic crss sectin. The bservatin that hydrgen has a much mre prnunced effect than deuterium cannt be explained by the difference f the scattering lengths but is due t Ag-atms being repelled frm the interface. The crrespnding decrease f scattering length density is exaggerated by H with its negative scattering length, whereas the psitive cntributin f deuterium is cmpensated this way. This is in accrdance with the bservatin fr dislcatins, where the cntrast variatin fr H and D were smaller than expected because f a decrease f the packing density f Pd-atms. At first sight the changes presented in Fig. 16 are astnishing, because they are assumed t be caused by the segregatin f ne mnlayer f H-atms at the interface nly. Hwever, ne has t take int accunt that the number f H-atms n the surface f an xide particle as small as 3 nm crrespnds t abut 50% f all the ins in the xide. In additin, the decreasing macrscpic crss sectin in the Guinier regime is determined by the radius f gyratin with respect t scattering cntrast and, therefre, any changes f cntrast in the periphery f an xide particle is exaggerated. Besides the changes bserved fr H and D segregatin it is als pssible t detect desrptin f the excess xygen with SANS [101]. A quantitative analysis f the SANS data was perfrmed fr a set f samples with different average radii and different cverages at the interface [100, 101]. The results yield particle radii which are in agreement with measurements made with the tmgraphic atm prbe [99]. The cverage f the interface with excess xygen is half f the cverage with hydrgen r deuterium, respectively. This experimental finding supprts the mdel prpsed in Fig. 15 r the crrespnding reactin described by Eq. (10.1). Hwever, the ttal values f cverage are abut half f what the mdel predicts fr a (111) surface f the MgO. These crystallgraphic planes f the xide are adjacent t the (111)-planes f Ag [93, 95] and, therefre, the xide precipitates shuld have ctahedral shape. Hwever, as a cnsequence f minimizatin f interfacial area the actual shapes are truncated ctahedra with (100) planes. Fr these planes the rck salt structure f MgO predicts 50% Mg- and 50% O-ins, i.e. n structural vacancies n the xygen sublattice. Thus the fractin f (111) planes is reduced and s is the average cverage with excess xygen r hydrgen, respectively. Fr mre results and a detailed analysis see Ref. [101]. 11. DEFECT FORMATION ENERGY 7

28 It is well knwn fr surfaces and grain bundaries that their energy γ can be reduced by slute segregatin leading t an excess cncentratin Γ A at the interface. The crrespnding change f energy is expressed by the Gibbs Adsrptin Equatin [10] γ µ A T, µ B = Γ A, (11.1) where µ A is the chemical ptential f slute A and µ B the ne fr slvent B. In a related study Carl Wagner [103] defined the excess Γ A by the amunt f slute dn A ne has t add r remve frm a system were the interfacial area changed by da with a cnstant chemical ptential f A and a cnstant number f mles f the slvent B. Thus we have n Γ A A =. (11.) a T, P, V, n B Accrding t the system being partly pen (fr A) and partly clsed (fr B) a new characteristic functin was intrduced [103] F n A µ A (11.3) using the free energy F. The differential f the new functin is d ( F naµ A) = SdT PdV + γda + µ BdnB nadµ A. (11.4) Upn differentiating F-n A µ A nce with respect t a, nce with respect t µ A ne btains ( F naµ A) = a µ A n a γ A = T,, V, µ A, n µ B A T, V, nb, µ A which yields the classical Gibbs Adsrptin Equatin via Eq. (11.)., (11.5) In the present study this prcedure is generalized, in rder t include ther defects as well. Fr the sake f the same frmalism we intrduce an apprpriate defect density ρ, i.e. grain bundary area, dislcatin length r number f vacancies per vlume etc. and the specific energy γ nw being the energy f frmatin f the defect per area, length r number, respectively. Then Eq. (11.4) becmes d ( F naµ A) = SdT PdV + γd( ρv ) + µ BdnB nadµ A (11.6) The equivalent definitin f slute excess at the defect is defined in analgy t Eq. (11.) as ( ρ) 1 n Γ A A =. (11.7) ρ V T, P, V, n B Since n A is prprtinal t ρ, it fllws frm the last equatin that ( ρ) 1 n Γ A A =, (11.8) V ρ where n A is the ttal excess due t all the defects in vlume V. A generalized adsrptin equatin is derived by fllwing the same lines f derivatin as befre γ µ A T, µ B = Γ ( ρ) A. (11.9) With measurements f the chemical ptential f hydrgen and measurements f the excess amunt f hydrgen at dislcatin and grain bundaries as presented in sectins 8 and 9 the change f the frmatin energy f these tw types f defects can be calculated. Fig. 17 shws the changes f cncentratin as a functin f the hydrgen pressure fr nancrystalline Pd at rm temperature. The fllwing relatins have been used t btain the apprpriate quantities n c = n H Pd nan n n H Pd single aγ = n H Pd aγh Ω = V Pd 3ΓH Ω = g Pd (11.10) 8

29 with the last equatin being valid fr spherically shaped grains and g being the grain size and Ω Pd the atmic vlume f Pd. Numerical integratin f the Gibbs Adsrptin Equatin fr the data in Fig. 17 yields 3.3k T lg p γ = ΓH d B = 0.89 J/m. (11.11) 3 This value is f the rder f the energy f the grain bundaries meaning that segregatin may finally lead t zer frmatin energy. Then n driving frce fr grain grwth wuld be present. This phenmenn f zer grain bundary energies is discussed in mre detail in Ref. [104]. In rder t estimate the effect hydrgen has n the reductin f the line energy f dislcatins the simple relatin f Eq. (8.11) derived in sectin 8 fr the hydride cylinder frmed in the expanded regin belw the glide plane f an edge dislcatin will be used. ( ρ) ργ Pd c r H Ω = αρ 4π =. (11.1) V Then integratin f Eq. (11.9) yields γ = = r 0 µ µ 1 α 4πr Ω α 4π k Ω Pd B Pd dµ H = c c1 α 4πr Ω Pd α 4πk T (1.1nm ) dr = Ω k Td ln c B Pd B f = Tr (1.1nm ) r r1 α 4πr Ω Pd 1.1nm kbtd r (11.13) Fr a radius f r=1 nm as determined experimentally (cf. sectin 8 and Fig. 9) the last equatin gives γ= J/m. This is abut three times the line energy f a dislcatin, if we calculate this value frm the empirical relatin γb 1eV (b=burger s vectr=0.75 nm fr Pd) [105]. The reductin f the line energy by hydrgen segregatin may be verestimated because f the varius apprximatins made during the derivatin f Eq. (11.9). Nevertheless, a reductin f the energy f dislcatin frmatin is expected and it may be the reasn why higher dislcatin densities can be prduced by cycling palladium between the α and β phase when cmpared with severe cld rlling [106]. Crssing the α/β phase bundary dislcatin rings are punched ut, in rder t accmmdate the misfit between β and α phase [, 3]. This ccurs at a high chemical ptential, i.e. a high excess cncentratin and, therefre, a reduced energy f dislcatin frmatin. Fr a hydride the situatin is different, because the lcal cncentratin in the expanded regin arund an edge dislcatin is saturated and, therefre, it is the same as far away frm a dislcatin. The frmatin f an edge dislcatin leads t a negative H-excess because hydrgen is nw repelled frm the cmpressed regin abve the glide plane. Therefre, the energy f dislcatin frmatin is increased leading t a lss f ductility. This may explain the extreme brittleness f metal hydrides. The cmmnly accepted thery that slute drag decreases dislcatin mbility may nt apply t hydrgen at rm temperature being highly mbile even in hydride phases. Finally we discuss the effect hydrgen has n the frmatin energy f vacancies. In this case the defect density is the number f vacancies per unit f vlume. Then the excess defined by Eq. (11.7) has the simple meaning f number f H-atms trapped arund ne vacancy N HV. Assuming that this number is cnstant ver a certain pressure range Eq. (11.9) yields 9

30 kbt γ = N HV µ H = N HV ln ph. (11.14) As the interactin energies f hydrgen with vacancies in metals are rather high (ca. 1 ev [14]) hydrgen atms may be all trapped at lw cncentratins (i.e. 10 atppm). Up t the terminal slubility in Pd at rm temperature (ca ) three rders f magnitude in cncentratin are cvered which crrespnds t 6 rders with respect t pressure (cf. Eq. (.)). With an excess f 1 H-atm per vacancy (N HV =1) Eq. (11.14) leads t γ=-0.18 ev. By increasing temperature and pressure and a pssible higher excess, N HV, it is easily cncluded that the frmatin energy becmes zer. This is in agreement with high pressure experiments by Fukai et al. [1], where abundant vacancies have been detected in Pd. By discussing the effect slute atms have n the frmatin energy f a defect it is demnstrated again hw useful the knwledge f the chemical ptential is. And again hydrgen metal systems are mdel systems because it is rather easy t measure µ fr them. 1. INTERACTION WITH CRACK TIPS AND HYDROGEN EMBRITTLEMENT First f all a crack tip in a sample which is under external stress accrding t mde I, i.e. tensile lad perpendicular t the crack surface, attracts H-atms. The hydrstatic stress in frnt f the crack tip is enhanced and, therefre, H-atms can lwer their chemical ptential accrding t Eq. (3.1). A quantitative treatment f this elastic interactin in terms f a DOSE is presented in Ref. [107]. Hwever, the strength f the interactin is rather weak and the crrespnding cncentratin enhancement is small. Nevertheless, the presence f hydrgen at the crack tip gives rise t the fllwing effects: (i) Fr atms right at the crack tip a small fractin f H-atms is sufficient t ccupy and weaken the stretched metal-metal bnds at the tip. After rupture f these bnds hydrgen migrates t adjacent bnds and cntinues t weaken these bnds, t. Under these circumstances the fracture will be a brittle ne. This scenari f hydrgen embrittlement is called dechesin [60] and may apply t thse metals which have a lw H-slubility such as irn. (ii) Due t H-H interactin the segregatin f hydrgen will be enhanced near the crack tip because f the cncentratin enhancement stemming frm the elastic interactin. When the terminal slubility is reached a hydride frms at the crack tip and the crack can advance thrugh the brittle hydride. Again H-atms easily redistribute and fllw the prpagating tip. This mechanism f hydrgen embrittlement is expected t be relevant fr metals with a high H-slubility such as grup Vb and IVb transitin metals. Hydride frmatin in frnt f a crack has been bserved in-situ in an electrn micrscpe [61]. (iii) Opening a crack gives rise t the frmatin f fresh surfaces and, therefre, the wrk required fr crack grwth includes a surface energy term. In the presence f hydrgen the surface energy can be reduced by surface segregatin f H-atms [108] like the reductin f grain bundary energy discussed in sectin 11. This way the energy fr prpagating a crack tip is reduced. (iv) Fr thse cases were ductility plays an imprtant rle during fracture the interactin f hydrgen with dislcatin has t be taken int accunt as well. During TEM bservatins f dislcatins it was bserved that the mtin f these defects was accelerated in the presence f hydrgen [109, 110]. This enhanced lcal plasticity increases the grwth rate f a crack. Therefre, n a macrscpic scale it appears t be an embrittlement phenmenn. The reasn fr the increased dislcatin velcity as suggested by Birnbaum et al. [109, 110] is an elastic interactin between dislcatins and the strain field f slute atms such as carbn. In the light f the decreased line energy f a dislcatin by segregated H-atms as discussed in 30

31 sectin 11 it may als be that the rate f generatin f dislcatins is increased in the presence f hydrgen because their frmatin energy is decreased. Besides the direct interactin f hydrgen with the crack tip there are ther hydrgen effects which severely alter the mechanical behavir, t. A recent review is given in Ref. [111] 31

32 V. Hydrgen in disrdered and amrphus allys 13. DISORDERED CRYSTALLINE ALLOYS Fr a crystalline ally A 1-x,B x, where hydrgen ccupies tetrahedral sites and A and B atms are distributed randmly, an apprpriate DOSE is 4 4 i 4 i n( E) = f x (1 x) δ ( E Ei ), (13.1) i i= 1 where the factr f is equal t the number f tetrahedral sites per metal atm which can be ccupied by hydrgen. Hwever, due t the repulsive interactin between nearest hydrgen atms sme f the sites remain empty and, therefre, actual values f f are less than the ttal number f tetrahedral sites per metal atm. Five different types f tetrahedral sites (A 4, A 3 B, A B, AB 3 and B 4 ) have t be distinguished accrding t the ccupancy f their crners with either A r B atms. Fr a randm distributin f A and B these five types are present in 4 i 4 i cncentratins f x (1 x). Their site energies fr hydrgen are labeled E i where i=0, i 1,.., 4 is the number f B atms distributed amng the crners f the tetrahedrn. Fr an ally dilute in B the DOSE f Eq. (13.1) reduces t the ne given by Eq. (7.1), because the number f tetrahedra having mre than ne B atm becmes negligible. The DOSE prpsed fr a cncentrated ally has been used by C. Wagner [11] in rder t mdel the thermdynamic activity f interstitial xygen in liquid irn allys. Measurements f hydrgen slubility were perfrmed by Feenstra et al. [113] in nibium-vanadium allys ver a wide range f ally cmpsitin and hydrgen cncentratin. Because f the large data set they were able t shw unambiguusly that the fractin f the varius tetrahedra fllws the binminal distributin f a randm ally. Hydrgen prefers tetrahedra having a higher number f V-atms at their crners in agreement with the vanadium hydride being a strnger hydride frmer when cmpared with nibium, i.e. E 4 <E 3 <...<E 0 where the subscript refers t the number f V-atms. Besides the expected behavir the site energies with respect t i they als depend n ally cmpsitin, x. This is explained [113] by an verall change f the size f tetrahedral sites fllwing the changes f the lattice parameter. Accrding t the psitive partial mlar vlume f hydrgen the site energy is lwered, when the site vlume increases. Therefre, a V 4 -site has a lwer site energy in a nibium rich ally in cmparisn with a vanadium rich ally. All f these results are cmpiled in Fig. 18 as the DOSE f the Nb-V ally. 14. METAL/NON-METAL GLASSES A large number f metal/nn-metal glasses have a cncentratin f abut 0 at.-% nn-metal. Amng these amrphus allys the palladium-silicn allys were mstly used t measure hydrgen slubility and diffusivity [46, 114, 115, 116]. In sme studies it was cnsidered t be apprpriate t use fr the DOSE f the amrphus allys a tw-level system [114, 116] whereas thers preferred a Gaussian distributin 0 1 E E n ( E) = exp (14.1) σ π σ 3

33 where σ is the width and E the average value f this functin. Inserting this DOSE in Eq. (4.3) and applicatin f the step apprximatin yields 1 E µ c = erfc. (14.) σ Slving the last equatin fr µ gives 0 µ = E σ erf 1 (1 c ). (14.3) where the inverse errr functin erf -1 was used. By measuring the chemical ptential at lw cncentratins by an electrchemical technique and at high cncentratins with a high pressure equipment it was pssible t cver a range f hydrgen pressures extending ver 18 rders f magnitude [115] as shwn in Fig. 19. In this case the cncentratin was defined as the rati H/Pd which in crystalline Pd is equivalent with the rati f the number f H-atms and the number f sites (ctahedral nes in fcc-pd). Over the range f pressures shwn in Fig. 19 it will be impssible t fit a tw level system t the data pints. Besides the gd agreement between experimental results and a Gaussian DOSE there is als a slid theretical fundatin fr this DOSE. In a first rder apprximatin [14] the DOSE f amrphus PdSi allys was derived frm the distributin f atmic distances which is given by the first peak in the radial distributin functin. This distributin f atmic distances may be cnsidered as a distributin f strain. By multiplying the latter with a factr f 3 yields a distributin f the vlume f interstices. Its prduct with the bulk mdulus and the partial mlar vlume f hydrgen results int a distributin f site energies. As the first peak f the metal/nn-metal glasses has a Gaussian shape the frging simple cnsideratin will lead t a Gaussian DOSE. In a mre rigrus treatment P. Richards [117] shwed that the assumptin f an interactin ptential between hydrgen and metal f radial symmetry is sufficient t cme t the same cnclusin. Then the width σ f the DOSE can be calculated frm measurable and knwn quantities yielding a value which is nly abut 30% larger than the experimental ne. The pressure cmpsitin istherms shwn in Fig. 19 d nt shw a pressure plateau which is present in crystalline allys stemming frm the cnstant chemical ptential within a miscibility gap r a tw phase regin f a slid slutin and a hydride. The absence f a miscibility gap r the missing hydride frmatin in amrphus allys is treated by R. Griessen [70] shwing that there is an interesting analgy with ferrmagnetism and the Stner criterin. Thus, if the H-H interactin energy is smaller than the width f the Gaussian DOSE, n hydride frmatin ccurs, i.e. the H-atms prefer t be distributed amng the lw energy sites instead f being frced t ccupy high energy sites in a cncentrated phase, in rder t prfit frm a small H-H interactin energy. Examples f chemical diffusin cefficients in amrphus Pd 80 Si 0 measured by an electrchemical technique are presented in Fig. 0. Depending n the methds f preparatin f the Pd-Si ally the values differ by up t tw rders f magnitude, althugh the radial distributin functin as measured with X-rays revealed n differences. The chemical diffusin cefficient D f hydrgen in amrphus matrices can be calculated by using Eq. (5.5) and (.3) in cmbinatin with Eq. (14.1). Experimental values are always in gd agreement with the calculated nes [14, 46, 118] which is shwn fr an example in Fig. 0. The different results in Fig. 0 can be explained partly by different distributins f site energies but they are als affected by the distributin f saddle-pint energies as discussed in sectin 17. The latter effect changes the reference diffusivity D. Hwever, the cncentratin dependence, i.e. the slpe and curvature f the calculated curves in Fig. 0 is slely determined by the measured 33

34 chemical ptentials. The nly free fitting parameter D will mve the curves up and dwn, because a lgarithmic rdinate has been used in Fig. 0 and because D is a factr in Eq. (5.5). In Fig. 0 the curve with the steepest slpe crrespnds t Pd 80 Si 0 which was prepared by sputtering and had the bradest DOSE as btained frm fitting measured values f the chemical ptential. Fr a brad distributin an incremental change f the cncentratin dc give rise t larger changes f the Fermi energy dµ when cmpared with a narrwer DOSE (cf. Eq. (4.7) with n(µ) being smaller fr a brad DOSE). Fr the extreme case f zer width r a crystalline ally, respectively D is independent f c as shwn in Fig. 0 fr a crystallized Pd 80 Si 0 -ally [14, 119]. 15. EARLY TRANSITION/LATE-TRANSITION METALLIC GLASSES Fr an amrphus ally A 1-x,B x the same cnsideratins as in sectin 13 have t be applied, in rder t btain the fractins f the varius A i B 4-i -tetrahedra. Hwever, different t the crystalline case each type f tetrahedral site has a brad distributin f site energies. Assuming a Gaussian ne fr each f them with an average energy E i and a width σ i Eq. (13.1) becmes 4 f 4 i 4 i ( E E = i ) n( E) x (1 x) exp. (15.1) σ i π i= 1 i σ i Often binary metallic glasses are allys f an early and a late-transitin metal. The affinity f these metals may be btained frm their hydride frmatin energies. A cmpilatin f these values [36] shws that they in almst all cases increase within the transitin series frm left t right (Pd being an exceptin). Thus early transitin metals have a high affinity t H whereas the late nes have a much lwer ne. Then the average site energy E i fr the A i B 4-i - tetrahedrn increases with increasing i fr A being the late transitin cmpnent. This expectatin is in agreement with experimental findings [10, 11]. By using the step apprximatin f the FD-Statistics, i.e. Eq. (4.7) the DOSE is btained directly frm measurements f µ(c) as shwn in Fig. 1. The DOSE fr the amrphus Ti-Ni and Zr-Ni allys shws a structure which arises frm the varying cmpsitin f the next neighbr atms arund the H-atm. Fr the sake f simplicity the same width σ was assumed fr each type f tetrahedrn. The psitin and the area f the varius Gaussian functins yield values fr f and E i. In a first rder apprximatin the average energies are linear cmbinatins f the site energies fr A 4 and B 4 and the site energies fr the latter are abut the same as the hydride frmatin energies f the crystalline metals A and B. The rati f the areas f the varius Gaussian distributins is independent f f and is determined in agreement with Eq. (15.1) by the ally cmpsitin nly. This agreement is cnsidered t be a piece f evidence fr a randm distributin f Ni and Ti r Zr-atms, respectively. The situatin is ttally different fr a Mg 50 Ni 50 ally, where the pressure cmpsitin istherms resemble thse f a crystalline metal, i.e. have rather flat prtins like in the nes in the miscibility gap r the tw phase regin [1]. Thus the majrity f sites has t be very similar and accrding t the cmpsitin they are mst prbably Mg Ni - tetrahedra. The chemical diffusin cefficient f hydrgen in early transitin/late-transitin metallic glasses is increasing with increasing cncentratin [10]. Hwever, in the ascending curves f 34

35 D vs. c plts there is a dip at thse cncentratins, where a sub-set f tetrahedra f the type A i B 4-i is filled and the energetically less favrable type A i+1 B 3-i has t be ccupied. Again this feature prvides sme infrmatin abut the structure f a metallic glass where H-atms act as prbes. In additin, it is an indicatin f a crrelated randm walk thrugh the amrphus structure and, therefre, the cncentratin dependence f D calculated frm measured chemical ptentials deviates remarkably frm measured data [14]. 16. BULK METALLIC GLASSES Bulk metallic glasses can be quenched int the glassy state with rather lw cling rates [13, 14]. This is achieved - besides ther means by increasing the number f cmpnents. In additin a larger variety f atmic radii appears t be favrable fr glass frmatin. There are a few studies n the behavir f hydrgen in bulk metallic glasses nly [15, 16, 17]. This may be due t a lw H-diffusivity (see belw) and the extreme brittleness f hydrgenated samples. Thus electrchemical techniques, were tw electrchemical cells are separated by the sample acting as a membrane, fail, because cracks frm and hydrgen is permeating quickly alng these cracks pretending high diffusivities. Recently a new technique f measuring H-diffusin has been invented [18], in rder t circumvent these prblems. Thin film preparatin methds were applied in rder t prduce a multilayer f Pd/bulk-glass/Pd n a substrate (cf. Fig. ). The cvering Pd layer is very thin and serves the purpse t facilitate hydrgen entry. After passing thrugh the bulk metallic glass hydrgen is disslved in the Pd-layer belw. Because f the higher electrical resistance f bulk metallic glass when cmpared t Pd and because f the thickness f the varius layers, the bttm Pd-layer determines the verall resistivity f the package. Then changes f the resistivity as a functin f time can be used t calculate the diffusin cefficient f hydrgen, because the rate determining step is diffusin thrugh the bulk metallic glass. Experimental results f D in a Zr 66.8 Al 17.4 Ni 7. Cu 8.6 glass and the equilibrium pressures are presented in Figs. 3 and 4 as a functin f H-cncentratin. The equilibrium pressures were btained by electrchemical measurements as well but withut the bttm Pd-layer shwn in Fig.. In agreement with the behavir in ther metallic glasses the diffusin cefficient increases with increasing hydrgen cntent and the increase can be calculated frm the measured values f the chemical ptential [18]. Hwever, mdelling the behavir f the chemical ptential r the equilibrium pressures, respectively turned ut t be rather cmplicated. Because f the increased number f cmpnents the number f different tetrahedra increases frm 5 fr a binary ally t 15 fr a ternary and 35 fr a quarterny ally. Assigning different widths and average energies t each f the tetrahedra wuld result in a crrespnding large number f fitting parameters. Thus the measured istherms were mdeled with the smallest reasnable number f parameters by assuming that all Gaussians had the same width σ i =8 kj/ml-h and an average site energy fr a tetrahedrn btained frm the participating crner atms and their hydride frmatin energies E f as E k n = ν k1 E f 1 + ν k E f + + ν kne fn and ν kj = 4, (16.1) j= 1 where ν kj is the number f j-atms sitting at the crners f a tetrahedrn labeled k and n is the ttal number f cmpnents. The DOSE is given by the sum f Gaussians with these parameters multiplied with the frequency f their ccurrence which depends n cncentratin. 35

36 Then the nly fitting parameter left is the ttal number f tetrahedra, f (cf. Eq. (15.1)) which mves the calculated pressure cmpsitin istherms in a duble lgarithmic plt parallel in the directin f the lg c axis but des nt affect slpe and curvature. Calculating istherms this way resulted in gd agreement with experimental results fr the Zr 66.8 Al 17.4 Ni 7. Cu 8.6 ally as shwn in Fig. 3. Hwever, the agreement was nt achieved by this prcedure fr the Zr 46.8 Ti 8. Cu 7.5 Ni 10 Be 7.5 (Vitrally 1) ally. This failure f the simple cncept may arise because f tw reasns: (i) the atms f the varius cmpnents are nt distributed randmly but a prnunced shrt range rder exists, and (ii) the distances between a H-atm and the atm f ne f the cmpnents is very different frm the nes in the pure cmpnent (Hatms may actually ccupy nn-tetrahedral sites). The latter affect then has t be treated in a similar way as in the crystalline Nb-V allys (cf. sectin 13). 36

37 VI. Other interstitials in amrphus materials It has been stated several times in chapter III that the cncept f a DOSE and FD-Statistics is nt applicable t hydrgen alne but t all small particles disslved interstitially in a defective crystalline r an amrphus material. The term interstitially requires a new definitin fr an amrphus matrix and an attempt is made in sectin 18. As a cnsequence f the generalized treatment Eq. (14.3) can be used successfully fr ther amrphus materials, t, as shwn in Fig. 5. This way the slubility f interstitials in amrphus materials can be treated in a universal way. Hwever, in sme cases the slute particles change the DOSE like H in amrphus silicn (see sectin 19) r alkali ins in xidic glasses (see sectin 0) and Eq. (14.3) becmes inapplicable. Befre these special systems are cnsidered in mre detail a general treatment f interstitial diffusin in amrphus systems is treated first. 17. MODELING DIFFUSION During the derivatin f Eq. (5.5) it had t be assumed that diffusin ccurs via an uncrrelated randm walk. As Eq. (5.5) has been successfully used in many cases t describe the cncentratin dependence f measured chemical diffusin cefficients, crrelatin effects are apparently nt changing the cncentratin dependence in these cases. Hwever, they can change the magnitude f the D-values. Crrelatin effects can arise in extended defects where slvent particles migrate predminantly within the defect. This can be circumvented as demnstrated fr grain bundaries in sectin 9 by neglecting the material surrunding the defect and cnsidering the DOSE f the defect alne. If this diffusin within the defect is rate determining the verall effective diffusin cefficient is btained. Besides these spatial crrelatins there may be energetic crrelatins as well. It is easy t cmprehend that the saddle pint energy between tw sites f lw energy is reduced n the average when cmpared with the average saddle pint energy. If fr instance the site energy is reduced because the site vlume is large, then atmic distances f the neighbring atms are larger than the average value. Thus the crrespnding distances fr the saddle pint cnfiguratin are large, t, which gives rise t a lwer saddle pint energy. Besides attempts t btain analytical slutin [14, 65] fr a distributin f saddle pint energies, Mnte-Carl (MC) simulatin have been cnducted [14, 19, 130, 131]. The advantage f studying diffusin by MC-simulatins is the freedm f chsing varius kinds f DOSE and varius kinds f distributins f saddle pint energies (DOSPE). The dependence f the simulated tracer diffusin cefficient D* n cncentratin, c, and temperature, T, is summarized in a schematic way in Fig. 6. The fllwing fur cases are distinguished: 1) Delta functin fr bth DOSE and DOSPE (first clumn in Fig. 6) This simulates the behavir f an interstitial in an ideal single crystal. D* des nt depend upn c unless the lattice becmes filled. Fr c 1 (lg c = 0) blcking f sites and vacancy crrelatin effects (preferred jumps back int the site just left) leads t a decrease f D*. The temperature dependence fllws an Arrhenius Law. These dependencies are shwn fr cmparisn as dashed curves in Fig. 6 fr the fllwing cases. ) Delta functin fr DOSE and Gaussian functin fr DOSPE (secnd clumn in Fig. 6) Fr a Gaussian distributin f saddle pint energies having the same average value as the cnstant value in case 1) the tracer diffusin cefficient is increased, because the interstitials prefer jumps ver lwer barriers. In tw and three dimensins there will be always passes 37

38 thrugh the lattice having a lwer average activatin barrier than the mean value. Thus in agreement with MC-simulatins the effective activatin energy is reduced. At very lw temperatures and the cncmitant shrt diffusin lengths the lwest barriers allwing a perclated walk thrugh the lattice are preferred even mre giving rise t the cnvex curvature in an Arrhenius diagram when viewed frm the 1/T-axis. As there is n DOSE D* des nt depend n c unless c 1 and like in case 1) blcking and crrelatin effects decrease D*. 3) Delta functin fr DOSPE and Gaussian functin fr DOSE (third clumn in Fig. 6) As there is n distributin f saddle pint energies in this case, there is n crrelatin between successive jumps and the results derived in chapter III can be used. At very lw cncentratins D* is independent f c because Henry s Law is fulfilled and γ =cnst(c). Here the particle distributin amng the sites f the DOSE is cntrlled by maximizing entrpy and, therefre, sites abve the Fermi energy are filled predminantly. Then the mean energy f the particles is given by [69] E E exp[ ( E E ) / σ ] = E 1+ exp[( E µ ) / k T ] B σ > µ 4k T B fr c 0. (17.1) Then the activatin energy is the difference between the cnstant saddle pint energy and this average energy. Because f the temperature dependence f <E> given by Eq. (17.1) the D* values at lw cncentratins exhibit a cncave curvature in an Arrhenius plt when viewed frm the abscissa. By increasing c the Fermi energy µ increases and finally becmes larger than <E>. Then minimizing f energy cntrls ccupancy f sites and the activatin energy is the difference between the cnstant saddle pint energy and µ. This activatin barrier decreases with c as µ increases and, therefre, D* increases with c. This increase f D* is a cnsequence f FD-Statistics, because lw energy sites will be saturated and high energy nes have t be ccupied. This way trapping by lw energy sites vanishes and the activatin barrier is decreased. The c-dependence disappears, if the cnditin f single ccupancy f a site is nt taken int accunt during a MC-simulatin. Then FD-Statistics is replaced by Bltzmann Statistics. Finally fr c 1 D* starts t decrease because f the same reasns discussed befre in case 1) and ). Increasing temperature in the intermediate cncentratin range des nt change µ very much, despite an increasing spread between full and empty sites f the rder f k B T. Thus the activatin energy des nt depend n temperature and straight lines are btained fr the MC-results in an Arrhenius diagram. 4) Gaussian functins fr bth DOSE and DOSPE (furth clumn in Fig. 6) The case f having a Gaussian distributin in site and saddle pint energies appears t be the ne which is the mst apprpriate fr real amrphus materials. Evaluating the MC-results f this case shws that the effects f bth distributins can be superimpsed accrding t the relatin D * = f ( T ) f ( c, T ) D, (17.) sp st where the varius factrs are btained frm the limiting cases 1), ) and 3), i.e. D =D* in case 1), f sp =D*/D calculated frm the MC-data f case ) and f st =D*/D calculated frm the MCdata f case 3). There is als theretical supprt fr the validity f Eq. (17.) [65]. Thus the c- dependence f D* in case 4) is the same as in case 3) but the values are increased by the factr f sp >1. Despite the fact that bth activity and diffusivity depend strngly upn c the permeatin, P, being the prduct f bth shws a minr c-dependence in defective and amrphus materials nly. Fr the statinary flux thrugh a film f thickness d, a permeatin cefficient P is defined accrding t the fllwing equatin: 38

39 a J = P, (17.3) d where d is the membrane thickness and a is the activity difference (r partial pressure difference) between entrance and exit side f the film. Accrding t Eq. (5.13) steady state means a J = D ( 1 c). (17.4) d Thus the permeatin becmes P=D (1-c) which is independent f c at dilute slutin. It is astnishing t nte that this result, althugh it is derived fr a distributin f site energies, is equivalent with permeatin thrugh a matrix which cntains sites f energy E nly. This result is caused by a cmpensatin effect, i.e. any increase r decrease f D is accmpanied by a equivalent decrease r increase f the slubility and, therefre, bth changes have n effect n the cncentratin dependence f the permeatin cefficient P. An interesting example is a plymer fil which was prduced with and withut clusters f Pd-atms. Fr bth samples the permeability f hydrgen was nt changed very much althugh the diffusivity decreased by mre than tw rders f magnitude within the plymer cntaining the palladium clusters as trapping centres [13]. 18. SMALL MOLECULES IN GLASSY POLYMERS There are a variety f different applicatins where the slubility and diffusivity f small mlecules plays an imprtant rle like penetratin f CO thrugh PET beverage bttles, permeatin f xygen thrugh plastic fils used t wrap fd, drug release thrugh plymer catings, diffusin f dye mlecules int fibers, separatin f gases by plymeric membranes etc.. If mlecules are disslved in the intermlecular space between the macrmlecules f a plymer they interact mainly via Van-der-Waals r diple-diple frces with the matrix. Cntrary t metals the matrix is in the majrity f cases an amrphus ne. Fr plymers abve the glass transitin (rubbery r liquid state) the elastic cnstants are rders f magnitude smaller than in the glassy state and, therefre, any size misfit between the intermlecular site and the disslved mlecule is accmmdated by a negligible amunt f elastic stress which mst prbably relaxes. As a cnsequence the vlume change per disslved mlecule r its partial mlar vlume, respectively is equal t the vlume f the mlecule (in its liquid state). Then the DOSE is represented by a Dirac delta functin and fr small cncentratins Eq. (4.3) yields the well knwn relatin fr an ideal dilute slutin N µ = E + kbt ln = E + kbt lnc. (18.1) N Where N is the ttal number f sites which is usually nt knwn because f an unknwn structure (cf. the factr f in Eq. (15.1)). But N can be either estimated [133] r measured by psitrn annihilatin spectrscpy [134, 135]. Expressing µ by the partial pressure f the small mlecules in the gas phase via Eq. (1.5) yields the prprtinality between cncentratin and pressure. This is in agreement with experimental findings and it is called Henry s Law because the partial pressure is equivalent t the thermdynamic activity at lw pressures. Small mlecules disslved in plymers belw the glass transitin temperature behave very different. First f all the partial mlar vlume f the small mlecules is much smaller than the vlume f the mlecule and it increases with increasing cncentratin (cf. Fig. 7 and [136]). This is interpreted via a distributin f site vlumes belnging t the intermlecular vlume. 39

40 In accrdance with the nmenclature used in plymer science the sites are called hles in the fllwing. Because the plymer structure is frzen in belw the glass transitin temperature the free vlume ccupied by small mlecules is nt regenerated as in the case f rubbery r liquid plymers. Fr mlecules having larger sizes than the hles elastic energy has t be paid during disslutin and, therefre, larger hles are filled first. Increasing the number f disslved mlecules requires filling f the smaller hles as well leading t an increasing vlume change with increasing cncentratin. Assuming spherical shapes fr bth mlecule and hle allws the calculatin f the partial mlar vlume V p in the framewrk f cntinuum mechanics [137] V = γ ( V Vh ) (18.) p i where V i is the vlume f the interstially disslved small mlecule and V h the ne f the hle, γ is a factr f abut unity taking int accunt the different elastic cnstants f the spherical mlecule and the matrix [136]. Thus the smaller V h is the larger is V p. The elastic energy E el assciated with the incrpratin f the mlecule is [137] E µ s ( Vi Vh ) el =, (18.3) 3γVh where µ s is the shear mdulus f the plymer. Thus a brad distributin f the vlume f hles gives rise t a brad DOSE. The distributin f hle sizes is mdeled accrding t Bueche [138] as a Gaussian stemming frm vlume fluctuatins abve the glass transitin temperature 1 ( V V ktgv h h ) n( Vh ) = exp with σv =, (18.4) σ V π σ B V where B is the bulk mdulus at the glass transitin temperature T g. The riginal treatment by 138 is mdified in as far as the temperature, T, was replaced by T g assuming that the vlume fluctuatin are quenched in at the glass transitin. Fr the simplified case f a linear expansin f the elastic energies arund the average vlume V h the Bueche distributin leads t a Gaussian DOSE [133] with an average energy E and a width σ E given by E 0 h µ s ( Vi V ) µ s ( V V ) = Er + Eel = Er + and σ 0 E = σv 3γVh 3γV h i 0 h 0 h i 0 h fr V V, (18.5) where E r is that part f the disslutin energy stemming frm Van der Waals interactins. Thrughut this study site energies were cnsidered, althugh the frmula will nt change, if E is replaced by G, i.e. Gibbs free energies. Thus site entrpies can be taken int accunt. Experimental cncentratin-pressure istherms are shwn in Fig. 8 and cmpared with the predictins f a Gaussian DOSE. It can be shwn [139] that in a duble lgarithmic plt the shape f the calculated istherms is slely determined by the width σ E whereas the secnd fitting parameter E mve the curves in the directin f the abscissa. Straight lines with a slpe f unity crrespnd t Henry s Law which is apprpriate fr water mlecules in plycarbnate (see Fig. 8). This is a cnsequence f the small size f H O mlecules being smaller than the hles in plycarbnate. N elastic energy cmes int play and the DOSE degenerates t a Dirac-delta functin knwn t lead t Henry s Law. In text bks n plymers [140, 141] the slubility is described by the dual srptin mdel [14, 143, 144] were the DOSE is a tw-level system. It allws fitting f c-p istherms with three free parameters yielding an agreement with experimental data which is as gd as fitting with a Gaussian DOSE. Hwever, the parameters f the dual srptin mdel d nt have a 40

41 rigrus physical meaning and they are nt related analytically t ther physical quantities like the tw parameters f the Gaussian distributin as represented by Eqs. (18.4) and (18.5). Fr a mre detailed cmparisn f the tw appraches the reader is referred t Refs. [133, 139, 136, 145]. The quantities used in Eqs. (18.4) and (18.5) are cmpiled in Ref. [141] r btained as fllws. Fr the vlume f the disslved mlecule the partial mlar vlume in a rubbery plymer r the ne in the liquid state f the small mlecules was used. Frm a cmparisn f measured partial mlar vlumes with predicted nes the average hle vlume V h is btained. Once this value is knwn the vlume change f ther small mlecules can be calculated withut a fitting parameter (cf. Fig. 9). By knwing values f V h values f the width f the Gaussian DOSE, σ E, can be calculated via Eqs. (18.4) and (18.5). They are abut 30-50% larger than the nes btained frm fitting c-p istherms. This is cnsidered t be gd agreement in the light f the varius crude assumptin made during the derivatin f Eq. (18.5). The functinal relatin are fulfilled as well as shwn in Fig. 30 and 31, i.e. E pltted vs. E el yields a straight line f slpe 1. Pltting σ E vs. V i squared shuld give a straight line with an intercept (V h ) n the abscissa. Fr atms r mlecules like He and H O the width is zer because f V i <V h. Very large mlecules as ethene and acetne yield smaller values than predicted by the linear relatin. This deviatin at large V i may arise because the assumptin f an elastic incrpratin f the small mlecule is n lnger valid. A calculatin f the stresses within Eshelby s cntinuum apprach [137] yields values which exceed the flw stress f the plymer cnsiderably and, therefre, inelastic relaxatin f macrmlecules r plastic defrmatin, respectively will ccur. In ther wrds, the larger the mlecule gets the mre it is incrprated substitutinally. The analgue in a crystalline lattice wuld be an atm being t large fr an interstices which kicks ut a neighbring atm frm its lattice site and becmes a substitutinal slute. By generalizing these cnsideratins a slute atm is incrprated in a material interstitially, if it is straining the matrix elastically nly. Diffusin f small mlecules fllws the predictins fr a brad DOSE as discussed in sectin 6 and, therefre, the cncentratin dependence as btained frm measured c-p istherms (c-µ istherms, respectively) is in gd agreement with experimental data as shwn fr a few examples in Fig. 3. Again, nly the reference diffusin cefficient D has been used as a fitting parameter which in the presentatin f Fig. 3 mves the curves in the directin f the rdinate withut changing their shape. The cncentratin dependence is the larger the larger the size f the disslved mlecule is. Fig. 3 als shws that in cases where σ E is smaller the diffusin cefficient becmes independent f c fr c 0 in agreement with the expectatin fr a Gaussian DOSE. 19. HYDROGEN IN AMORPHOUS SILICON AND GERMANIUM Experiments n H and D slubility and interdiffusin f H and D in amrphus silicn and germanium were cnducted by using Secndary In Mass Spectrmetry (SIMS) [146, 147, 148, 149]. It was shwn that the activatin energy f diffusin decreased with increasing cncentratin. This was interpreted using the cncept f the chemical ptential and a site energy distributin as develped much earlier fr metallic glasses [14]. Besides striking similarities there are remarkable differences. Mst f the hydrgen atms are bund t dangling bnds f the silicn because the amrphus semicnductr was prepared by CVD frm SiH 4 at elevated temperature. H-cncentratin was varied by varying the preparatin 41

42 temperature. This way the site energy distributin r namely its fractin belnging t dangling bnds depends upn H-cncentratin. This is schematically shwn in Fig. 33. As the DOSE has small values arund µ any remarkable increase f the H-cncentratin requires a large change f µ. Thus the slubility seems t be cnstant r it is predetermined by the cncentratin f dangling bnds, respectively. Whereas in metallic glasses r glassy plymers the chemical ptential increased because f a filling f a pre-existing distributin f site energies, the situatin in amrphus semicnductrs is different. In the latter case an increase in cncentratin is accmpanied by changes f the distributin functin leading t an increase f the chemical ptential as well (cf. Fig. 33). A peculiar behaviur can be bserved, if a sample with a high cntent f H (r D) called sample B in the fllwing is brught in cntact with a sample f lw H (r D)-cntent which is called sample A. The ttal (H+D)-cncentratin will nt change in the tw samples althugh interdiffusin ccurs as bserved frm the istpe redistributin measured by SIMS depth prfiling. A step in the ttal cncentratin f H+D remains at the interface. In additin the diffusivity f H (r D) is nw much faster in sample A with the lw H-cntent when cmpared with an experiment withut a step in ttal cncentratin. This increase f H- diffusivity in sample A stems frm an increase f the chemical ptential as impsed by sample B (cf. Fig. 33). In rder t raise µ in sample A t abut the same level as in B, nly a negligible amunt f H has t be transferred frm B t A. Then the diffusin cefficient is abut the same in A and B in agreement with experiment [146, 147, 148, 149]. 0. IONS IN OXIDIC GLASSES Mixing an alkali xide with SiO leads t nn-bridging xygen atms in the netwrk f SiO 4 - tetrahedra [150, 151, 15]. These xygen atms are the centres f negative charge which are immbile far belw the glass transitin temperature, whereas the alkali catins are still mbile. They migrating via the hles (interstices) f the silicate netwrk (cf. Fig. 34). Due t the strng Culmb interactin amng catins and anins the sites next t the anins have the lwest energy. Because f charge neutrality the cncentratin f anins and catins has t be equal, in analgy t the equivalence f dangling bnds and H-atms in amrphus silicn. Therefre, the distributin f site energies is changing als in silicate glasses by changing the alkali cncentratin. Different t the materials discussed befre a decrease f the diffusin cefficient has been determined fr lw alkali cntents [153] which was explained [69] by the "weak electrlyte" mdel (cf. Fig. 35). Assuming that sites next t the anins have a much lwer energy cmpared t sites far away leads t a bimdal distributin f site energies where bth fractins vary with cncentratin as shwn in Fig. 36. With this distributin the weak electrlyte behaviur culd be mdelled fr this slid material [69]. It is wrth nting that a similar cncentratin dependence at lw c is expected fr H in amrphus silicn. The decrease f the diffusin cefficient with increasing cncentratin is in cntradictin with the simple minded interpretatin f Eq. (5.5) used s far, where an increase f µ by increasing c lwers the activatin energy f diffusin. This must nt be used in this cntext, because the DOSE changes with cncentratin, whereas it has been tacitly assumed in the derivatin f Eq. (5.5) that it will nt change. The decrease f D fr catins in glasses stems frm an entrpy effect [69]. As the number f anin and catin pairs increases the number f "free" catins increases as well. Hwever, the crrespnding enhancement f the effective mbility is mre than cmpensated by the increased trapping efficiency f the "naked" anins. 4

43 At high alkali cntents the diffusivity increases like in the case f the ther amrphus r glassy materials. Besides the interpretatin given befre either fr amrphus metals r silicn anther pssible cause exists fr a decrease f the activatin energy and the cncmitant increase f D*. At high alkali cncentratins a cnsiderable mdificatin f the netwrk ccurs accmpanied by a decrease f the O-atm density. Thus the netwrk becmes mre pen which culd lwer the activatin energy fr catin hpping. In additin it increases the mbility f neutral atms as He, t [154]. Besides changing the mesh size f the silicate netwrk with additins f alkali ins it can be als changed by externally applied hydrstatic pressure. The effect f pressure n catin diffusivity can be easily measured by mnitring changes f the electrical cnductivity yielding the activatin vlume f the catin diffusin cefficient. The average changes f the mesh size r strain, respectively by external pressure can be calculated frm cntinuum elastic thery and can be cmpared with the changes induced by the additin f alkali xides [154]. Thus a quantitative treatment f mesh size effects is achieved. If we accept this mesh size effect and include the effect f a DOSE and FD-Statistics, a new and quantitative explanatin f the mixed alkali effect can be ffered [155, 156]. The mixed alkali effect [151, 15, 153, 155] can be bserved as a decrease f in cnductivity r mbility, respectively, by several rders f magnitude when ne alkali in is substituted by anther ne withut changing the ttal alkali cntent (cf. Figs. 37 and 38). If we assume that all the alkali ins are distributed ver a brad DOSE and that the smaller nes ccupy the lwer levels (because they cme clser t the anins), the additin f smaller catins t a glass with larger nes reduce the mbility f the small nes as they are placed in the lwest energy levels. The mbility f the larger catins is reduced as well, because the mesh size f the netwrk is reduced accrding t an average reductin f the mean catin size. This behavir can be treated quantitatively, t, yielding agreement with experimental data by fitting ne free parameter, the width f the distributin, nly [155, 156]. 43

44 VII. Hydrgen in systems with reduced dimensins The behavir f hydrgen in samples with reduced dimensins is ne f the mre recent areas f research in metal-hydrgen systems [0, 158, 159, 160, 161, 16, 163, 164, 165, 166, 167, 168, 169, 170]. The prperties f these systems are affected by the prximity f surfaces r interfaces prviding additinal sites t the DOSE. In this case the surface f the sample r its interface with a different material is just a surce f new sites and reductin f dimensins leads t an increasing fractin f the new sites. Thus the interactin f hydrgen with free surfaces falls in the same categry (cf. sectin 9). In additin stresses and strains play an increasing rle. On a free surface the nrmal cmpnent f the stress is zer and this bundary cnditin is imprtant fr hydride frmatin in the elastic regime [171, 158]. Inelastic prcesses like dislcatin emissin are als affected by free surfaces as they act as sinks fr defects. Reduced dimensin inhibit dislcatin generatin and mtin [17] and, therefre, hinder that mechanism f hydride frmatin which requires dislcatin emissin. Samples adhering t a substrate r being embedded in a matrix are subject t bundary cnditins with respect t their strain at the interface. Again the related stresses may nt relax as easy as in the extended bulk sample. It has als been discussed (cf. sectin ) whether the prximity f a different material changes the electrnic structure in the neighbrhd f an interface [159]. 1. THIN FILMS First measurements f hydrgen disslved in thin films were cnducted by H. Zabel et al. [160]. They determined the lattice expansin f nibium films in a hydrgen gas atmsphere f varying partial pressure. As the Nb-film was adhering t a substrate, n expansin in plane can ccur and cmpressive stresses develp. During the disslutin f hydrgen expansin ut f plane takes place and it is accmpanied by the Pissn effect, i.e. additinal expansin due t the increasing cmpressive stresses. By using H-cncentratins frm bulk pc-istherms the lattice expansin per H-atm was fund t be much larger than in bulk nibium [160]. Hwever, it was shwn later n [166] that the lattice expansin per H-atm was the same as in bulk nibium and the larger values btained in Ref. [160] were mst prbably due t less reliable H-cncentratin values btained frm the istherms. Strains and stresses develping in thin Nb-films as a functin f H-cncentratin are presented in Figs. 39 and 40. Within the α-phase f Nb and Pd the ut f plane expansin f thin films fllws the laws f cntinuum mechanics. Hwever, the assciated cmpressive stresses are smaller by 10 t 30% when cmpared with calculated values [166]. This discrepancy may be attributed t sme stress relaxatin. A much mre prnunced stress relaxatin takes place when the terminal slubility f the α-phase is reached and a hydride r a high cncentratin phase is frmed. Then the in-plane distance f lattice planes is changing, t. In sme cases the prnunced stress relaxatin ccurs within the α-phase, i.e. befre the terminal slubility is reached, because the cmpressive stresses simply exceed the yield strength f the thin metal film. The terminal slubility increases as the film thickness decreases which may be caused by a lwering f the critical temperature in finite dimensins r a kinetic barrier fr the necessary vlume expansin. The plt f stress vs. H-cncentratin as shwn in Fig. 40 may be interpreted as a stress-strain curve, where the strain is internally impsed by H-atms instead by an external machine. Therefre, similar features knwn frm the defrmatin f bulk metals can be bserved, i.e. single crystalline Nb-films have a much lwer yield strength and a lwer wrk hardening rate 44

45 than plycrystalline films. After remval f the hydrgen a dislcatin netwrk remains in the film and at the subsequent secnd lading with hydrgen yield strength and wrk hardening are increased (cf. Figs. 41 t 43). Recently thin Y-films have attracted cnsiderable interest [173, 174, 175, 176], because a metal/inslatr transitin ccurs between H/Y= and H/Y=3, i.e. between the di- and trihydride. This transitin can be used t build a switchable mirrr, if the Y-film is depsited n a transparent substrate. The Y-H system is als peculiar with respect t vlume changes. Besides the usual lattice expansin which is bserved fr H/Y< a cntractin ccurs with increasing H-cncentratin fllwed by an expansin again. The crrespnding tensinal and cmpressive stresses cancel mstly and, therefre, the switching between di- and trihydride is nt accmpanied by large changes f film stresses [177]. Thus a large number f transitins between a reflecting and a transparent mirrr are pssible withut failure f the device. If the adhesin between film and substrate is weak, the films start t delaminate frm the substrate during hydrgen lading. This has been bserved fr Nb-films n mica [178] and Pd-films n plycarbnate [179].. MULTI LAYERS With the increasing interest in multilayers, the behavir f hydrgen in these systems has als attracted intensive research. In their pineering wrk, Miceli, Zabel, and Cunningham [13, 180] describe critical phenmena in Nb/Ta super-lattices. Hjörvarssn and c-wrkers [159] emphasized the effect f electrn transfer at M/V interfaces n the hydrgen slubility in V layers. The influence f hydrgen n the magnetism f Fe/Nb and Fe/Ce multilayers was investigated by Weidinger et al. [16] and Felsch et al. [181]. Actual measurements n strain relaxatin and phase separatin in Pd/Nb multilayers are presented in Ref. [165]. Transprt f hydrgen thrugh thin metallic films is essential fr the develpment f catings preventing bulk materials frm hydrgen uptake. The influence f thin films f Pd, Ni, and Cu n the hydrgen permeatin thrugh Fe was investigated by Sng and Pyun [18] and Takan et al. [183] Furthermre, t avid xidatin and t enable hydrgen charging, thin films are ften cvered with an additinal Pd surface, such that the specimens are multilayered structures (substrate, film, and Pd layer). The influence f Pd catings n the hydrgen permeatin was investigated by Züchner [184].. In the case f thin metallic films prepared n substrates, the hydrgen permeatin thrugh the substrate, as well as the influence f the interface and the prperty f the film has t be taken int accunt. Assuming defect-free layers, far away frm the interfaces, the hydrgen transprt shuld be well described by the knwn diffusin behavir f the bulk elements. Hwever, deviatins frm the ideal bulk diffusin can be expected at the interfaces, where pssibly high densities f lattice defects, lcalized misfit strains, r different hydrgen slubility are present. Therefre, it is mst imprtant t understand whether the hydrgen transprt thrugh a layered specimen is crrectly mdeled by a layered structure with bulk materials f different hydrgen slubilities r by taking int accunt additinal interface sites f different site energies. The permeatin methd, i.e., lading the sample at ne side with hydrgen and determining the time dependence f the cncentratin change at the ppsite side, can be used t measure hydrgen diffusivity thrugh multilayers. Depending n the experimental cnditins f hydrgen lading, the cncentratin change at the ppsite side appears with a certain time lag [184]. In a single layer the time dependence f the cncentratin change can be slved 45

46 analytically fr varius bundary cnditins [185], whereas in cmpsite systems nly the characteristic time lag t reach the steady state has been mdeled s far. Often fur different bundary cnditins are applied in steady-state permeatin experiments: (A) cnstant current density at the input and utput surface, (B) cnstant cncentratin at the input and utput surface, (C) cnstant current density at the input and cnstant cncentratin at the utput surface, and (D) cnstant cncentratin at the input and cnstant current density at the utput surface. In their theretical treatment f diffusin thrugh multi-layers, Ash and c-wrkers [186, 187] derived the time lag fr the permeatin using bundary cnditin B with vanishing cncentratin at the utput side. Their result was applied t interpret hydrgen diffusin in metallic bilayers and triple layers by Züchner [184] and Takan et al. [183] Sng and Pyun [18] mdified the hydrgen lading cnditins by using a cnstant hydrgen current density at the input side (cnditin C) and derived the crrespnding time lag fr a bilayer system. In a recent paper by G. Schmitz et al. [188] slutins fr the cnditins A and C were prvided fr the mst general case f multilayers including the substrate as ne f the layers. Fr a substrate f thickness s with N alternating layers f tw metals f the thickness a the time lag t L is fr N>>1: s 1 N a N a N a tl = , (.1) 6Ds 6 kd1 6D1 6D where k is the rati f H-slubilities in bth metals 1 and fr the same H partial pressure chsen in a way that k<1 and D are the crrespnding diffusin cefficients. The first term n the right hand side f Eq. (.1) is three times the time lag f the islated substrate and the third and furth term are the time lags f the islated layers f either metal 1 r. Fr nanstructured multilayers n a thick substrate usually Na<<s applies and the last equatin simplifies t s 1 N a tl = 3 + (.) 6Ds 6 kd1 The secnd term n the right hand side f the last equatin cannt be neglected because k may be very small. The term can be als used t define an effective diffusin cefficient D eff = (Na) /6t L =4kD 1 fr the multilayers. Fr a different thickness f the tw metals the effective diffusin cefficient fr a multilayer can be written independently f the bundary cnditins as [186, 187, 188, 189] D eff = kv v (.3) 1 D1 where ν 1 and ν are the vlume fractins f metal 1 and metal. In rder t cmprehend the physics behind Eq. (.3), the ptential trace a migrating H-atm will experience is shwn in Fig. 44 fr the case f Pd=metal 1 and Nb=metal. The site energies E Pd and E Nb crrespnd t the energies f hydrgen disslutin in Nb and Pd. In agreement with the definitin f k=exp[(e Nb E Pd )/kt]<1 the experimental values crrespnd t E Pd > E Nb. At rm temperature k is as small as abut 10-4 fr Nb/Pd [188] and, therefre, the effective diffusin cefficient is decreased by fur rders f magnitude which are in agreement with experimental findings [188]. It is bvius frm lking at Fig. 44 that the Nb-layers prvide traps fr the hydrgen diffusing in Pd whereas Pd-layers act as barriers fr H in Nb. Experimental results f time lag measurements f Pd/Nb-multilayers [188] are shwn in Fig. 45 as a functin f the partial hydrgen cncentratin in Nb r Pd, respectively. Due t the lwer Gibbs free energy f H disslutin in Nb nearly all f the hydrgen is disslved in the Nb-layers [36]. The ttal thickness f the layers was abut 0. µm and they were prduced n 46

47 a Pd substrate f 1.5 µm thickness. With the slubility rati k in Pd/Nb f abut 10-4 and a rm temperature diffusivity f cm /s in Pd the time-lag is calculated t be 0.75 s fr the pure substrate and 14 s fr the substrate plus multilayers. Fr H-cncentratins f 0.01<c<0.04 experimental data are f the same rder f magnitude as this estimated value. Unfrtunately, the cmparisn is apprximate nly because scatter f data n Gibbs free energy f slutin yield data n k which scatter by mre than ne rder f magnitude [188]. It is interesting t nte that the high diffusivity f H in Nb with D= cm /s des nt play a rle and that despite that value being higher than in Pd the Nb-layers retard the verall transprt f H. At very lw H-cncentratins f c<0.01 r fr the smallest duble layer thickness the interactin with defects r with the interphase Pd/Nb-bundaries may increase the time lag, i.e. the effective diffusivity. At a few at.-% f H in Nb the terminating slubility at rm temperature is reached and the Nb-layers cntain tw phases, a slid slutin f H in Nb and a hydride phase. Then the time lag can n lnger be calculated by Eq. (.) which was derived fr the single phase case. Experimental values in Fig. 45 shw that t L is reduced in the tw phase regin. Increasing the ttal hydrgen cntent further leads t a ttal cnversin f the Nb-layers int hydride layers, where the k-values are higher because f the higher Gibbs free energies f slutin in saturated Nb-hydride. Then the secnd term n the right hand side f Eq. (.) becmes negligible cmpared t the first ne and the time lag reduces t the value f the pure Pdsubstrate (0.75 s). Fr the Ni/Pd system [190] the Gibbs free energies f H-slutin are such that H is enriched in the Pd-layers accrding t k-values f abut 10-4 [36] and the diffusin cefficient f H in Ni is cm /s [83]. Inserting these values in Eq. (.) yields time lags f the rder f thusands f secnds. Hwever, experimental values [190] are abut the same as the time lag f the Pd-substrate, i.e. the Ni-layers d nt act as barriers fr H-diffusin. This discrepancy between experimental findings and thery is vercme by assuming that grain bundary diffusin in the Ni-layers is relevant. The Ni-layers f the samples used were plycrystalline and the cncept described in Ref. [188] has t be mdified by using an effective diffusin cefficient D gb δ/d, where D gb is the diffusin cefficient f H in grain bundaries f Ni, δ is the thickness f these bundaries and d is the diameter f the grains (abut the same as the Nilayer thickness). The rati k has t be multiplied by a segregatin factr S defined as the rati f H-cncentratins in grain bundaries and grains. Experimental values f S and D gb [83] are s large fr the H-cncentratins used in the experiment that the secnd term in Eq. (.) is smaller than the first ne and in agreement with experimental findings the effective diffusin cefficient is nt affected by the presence f the Ni/Pd-layers. Similar cnsideratins shuld apply fr the case f Pd/Nb-layers where the Pd barrier layers were nancrystalline, t. Hwever, in this case the prduct SD gb δ/d is smaller than the crrespnding value f D fr the grains. At the H-cncentratins expected in the experiments this is mainly caused by a lwer diffusivity f H in grain bundaries f Pd when cmpared with the grains (see chapter IV). The interface bundaries between different metals prduced by either sputtering r electrn evapratin are nt atmically sharp but an intermixing zne f abut 1 nm thickness has been detected by analyzing the multilayers with a tmgraphic atm prbe [191]. This intermixing zne decreases the width f bth adjacent layers and it des nt absrb hydrgen at a given partial pressure [19], i.e. it acts as a dead layer. This way a different explanatin is 47

48 ffered fr the dead layers bserved in Refs. [159, 193], where a transprt f electrns acrss the interface was assumed t be respnsible fr the dead layer. 3. CLUSTERS In rder t study the slubility f hydrgen in metal clusters, a large number f clusters are necessary fr the cmmnly applied techniques. Hwever, clusters tend t agglmerate because f the large chesin energy f metals. This tendency f frming a plycrystalline metal rather than remaining as islated clusters is circumvented by stabilizing cluster with a surfactant shell, embedding them in a slid matrix r depsiting them n a substrate. Again the Pd-H system is the ne first chsen by experimentalists as the nbility f the metal excludes the frmatin f an xide layer which therwise wuld be an appreciable fractin f the sample. First measurements f pc-istherms were cnducted by Flanagan et al. [194] with Pd-black having a surface area f abut 40 m /g which crrespnds t a diameter f abut 1 nm. Althugh changes are small when cmpared t bulk Pd, a narrwing f the miscibility gap was bserved. Griessen et al. [195] bserved a much mre prnunced narrwing fr Pd clusters depsited n an aluminum xide substrate. Hwever, in this case stresses will develp during hydrgen lading like in thin films and, therefre, the pc-istherms will be affected by these stresses. The interactin f hydrgen with free palladium clusters f t 6 nm diameter which were stabilized by a surfactant shell has been described in detail in recent studies [164, 168, 169]. The results are summarized as fllws. The preparatin f small Pd-clusters is achieved by electrchemical disslutin f Pd and its reductin n a cunter electrde t neutral Pd-atms. If a nn-aqueus electrlyte such as tetrahydrfurn with surfactant mlecules like ctylammnium brmide is used the Pd-atms agglmerate t clusters which are stabilized by the surfactant [170, 196]. At a certain cncentratin f clusters they precipitate as a black pwder. The diameter f the clusters and the width f its distributin can be varied by changing temperature, electrde distance and current density. Thus clusters are prduced with a rather narrw size distributin, i.e. with a variance f the diameter being 10 t 15% f the average diameter. Detailed investigatins f the atmic structure f these cluster by using mlecular dynamic simulatins, high reslutin transmissin electrn micrscpy and X-ray diffractin revealed that they have an icsahedral structure fr sizes belw 4 nm whereas the larger nes have an fcc structure. All the clusters absrb hydrgen frm the gas phase which can be easily shwn by mnitring the weight change in a hydrgen atmsphere. Hwever, befre hydrgen is reacting with palladium it frms water with xygen being absrbed n the surface f the clusters. Pumping ff the water and the residual hydrgen in the vacuum chamber leads t xygen and hydrgen free Pd-clusters. After this pre-treatment the clusters can be expsed t hydrgen with a given partial pressure and the absrbed amunt is btained frm weight changes r frm pressure drps. In the latter case the valves t the pumps and t the hydrgen inlet have t be clsed. The absrptin is cmpleted in between a few minutes and a few hurs. This way pressure cmpsitin istherms are btained. Sme f them are shwn in Fig. 46. At abut the same pressure where bulk-pd decmpses in α- and β-pd the clusters exhibit a slped plateau. Hwever, the maximum cncentratin in the α-phase is larger and the minimum ne in the β-phase is smaller, i.e. the width f the miscibility gap is reduced and the reductin is the larger the smaller the clusters are. The increased terminal slubility in the α- phase can be explained by subsurface sites having lwer site energies than bulk sites [168]. This is analgus t segregatin in grain bundary sites. The surface sites were mst prbably 48

49 ccupied during the pre-treatment step where the xygen was remved with hydrgen. Because f the large binding energies f the rder f 100 kj/ml-h [197] this hydrgen will nt be remved at rm temperature and high vacuum cnditins. Within the miscibility gap there is a hysteresis between absrptin and desrptin [168, 170] in analgy with bulk behaviur. Hwever, the hysteresis cannt be explained by the wrk required t punch ut dislcatins arund the new phases frmed [198], because the clusters are t small t allw the frmatin f dislcatins. Thus a cherent phase transfrmatin takes place which leads t a hysteresis as well [199, 00]. Additinal evidence fr a phase separatin in small Pd clusters is prvided by X-ray diffractin, where at partial pressure f hydrgen belnging t the plateau regin X-ray peaks split int tw (cf. Fig. 47). With increasing H-cncentratin the peaks shift t smaller angles indicating a vlume expansin as in bulk Pd. Calculating lattice parameters frm the peak psitins and pltting them versus the hydrgen pressure yields the results shwn in Fig. 48. Within the regin f the slped plateau there is a prnunced change f the lattice parameter accrding t the transitin frm the lw t the high cncentratin phase f the Pd-H system. The verall change f the lattice parameter acrss the miscibility gap decreases with decreasing cluster size in accrdance with the crrespnding decrease f the width f the miscibility gap. The change per H-atm is abut the same as in bulk Pd. As the changes f the lattice parameter scales with the H-cncentratin there is als a hysteresis f the lattice parameter between absrptin and desrptin within the miscibility gap. The hysteresis is nt a matter f sluggish kinetics because it is nt bserved within the ne-phase regin. Thus the hysteresis is an additinal piece f evidence fr a phase separatin in the Pd-clusters besides the slped plateau f the pc-istherms discussed befre. Whether a cluster itself cntains tw phases in cexistence r whether a cluster is either α- r β-phase thrughut its vlume remains an pen questin. A piece f evidence fr the first case is prvided by the hysteresis. If it is caused by the cherent phase transfrmatin as discussed in Refs. [199, 00], high and lw cncentratin pahse have t be present in the same cluster. These and ther prblems are subject f nging research, where namely MD-simulatin may becme a suitable tl t answer part f the remaining questins. Sme f the attractin f the cluster research actually stems frm the pssibility f cnducting MD-simulatins with samples f the same size as ne used by the experimentalist. Acknwledgments: The authr is thankful fr the cllabratin with many diplma students, PhD students and pst-dcs wrking n metal hydrgen systems. They are listed here in alphabetical rder: Deni Arantes, Jchen Bankmann, Martin Drnheim, Jürgen Gegner, Xianya Huang, Frank Jaggy, Philipp Kesten, Wlfgang Kieninger, Christian Kluthe, Uwe Laudhan, Ruth Lüke, Christf Marte, Michael Maxeln, Thmas Mütschele, Chenng Park, Astrid Pundt, Christian Sachs, Guid Schmitz, Ulrich Stlz, Mhamed Suleiman, Fang Wang, Quin-Min Yang. In additin the cperatin with many clleagues mstly leading t jint papers is gratefully acknwledged. Mst f the financial supprt stems frm the Deutsche Frschungsgemeinschaft via several prjects and via SFB 70, SFB 345 and SFB

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56 Figure captin: Fig. 1: Partial mlar vlume f hydrgen in plycrystalline palladium (clsed circles) and in a liquid quenched Pd-Si metallic glass. The negative vlume changes fr the first 40 atppm H are explained by trapping f H-atms in a crrespnding fractin f vacancies [53]. Fig. : Schematic presentatin f a distributin f site energies which starts with the mst degenerate ne, i.e. the single crystal in the tp line. In the fllwing lines increasing structural disrder leads t a decreasing degeneracy by increasing the dimensins f lattice defects. Fr diffusin ptential traces r energy landscapes, respectively including the defect are imprtant. Fr grain bundaries traces within the grain and within the grain bundary are shwn separately, where the latter are present with a vlume fractin f c t. The density f site energies (DOSE), n(e), defined as the number f sites within a given energy windw is presented bth graphically and analytically. The varius frms f n(e) are derived in chapters IV and V. Fig. 3: Ptential trace fr an interstitial jumping frm site 1 with energy E 1 t site with energy E. E is an average Energy f the DOSE and Q the activatin energy with respect t this reference. Thus particles in a material cntaining sites f energy E nly have a jump frequency f Γ=Γ exp[-q/k B T]. Fig. 4: Circles f cnstant hydrstatic stress at an edge dislcatin as calculated in cntinuum mechanics (cf. Eq. (8.1)). The elastic interactin energy with hydrgen is cnstant at these lines and the crrespnding DOSE is the number f sites within the tw circles. Fig. 5: Electrchemically measured chemical ptentials f hydrgen in heavily defrmed palladium pltted accrding t Eq. (8.4) versus 1/ c. The varius regins f the data pints represent regins f a predminant interactin mechanism. Fig. 6: Hydrgen resistivity increment, hydrgen diffusivity and the cncentratin f free hydrgen in heavily defrmed palladium divided by the crrespnding values in well annealed palladium. These ratis are pltted versus H-cncentratin and in agreement with Eq. (8.6) they are abut the same ver the whle range f cncentratins with in the α-phase f Pd. Fig. 7: Diffusin cefficient f hydrgen in heavily defrmed Pd (clsed circles) and well annealed Pd (pen circles). The line fllwing the values f defrmed Pd was calculated frm measured chemical ptentials withut a fitting parameter (cf. text). Fig. 8: A mdified Guinier Plt f the macrscpic scattering crss sectin dσ/dω (cf. Eq. (8.7)) measured by small angle neutrn scattering fr defrmed Pd with 1 at.-% H. The slpe f the linear part yields the radius f hydrgen enriched cylinders frmed by segregatin at the dislcatin line. 56

57 Fig. 9: Radii btained frm small angle neutrn scattering fr a defrmed Pd-sample at varius H- cncentratins. The line is calculated by assuming the frmatin f a hydride f cylindrical shape in the strain field f dislcatins. The dislcatin density (ρ= cm - ) was used as a fitting parameter t btain agreement with experimental data. Fig. 10: Macrscpic crss sectin fr a defrmed Pd sample cntaining 0.8 at.-% hydrgen (clsed sircles) r deuterium (pen circles). The lines are calculated by using Eq. (8.7). The difference f dσ/dω between the tw istpes is less than expected frm their scattering length which is evidence fr a vlume expansin during the hydride frmatin at the dislcatin lines (cf. text). Fig. 11: H -equilibrium pressure versus H-cncentratin fr single crystalline ( ) and nancrystalline Pd () [81] at 95 K. The dtted line has a slpe f within the α-phase accrding t Sieverts Law fr c<0.01 and it becmes a plateau within the α+β tw phase regin fr c> The slid and dashed lines fr the nancrystalline sample are calculated assuming a distributin f site energies (cf. Fig. 1) and including r excluding H-H interactin as explained in the text. Fig. 1: Site energy distributin fr H in nancrystalline Pd used in rder t calculate the line in Fig. 11. The bimdal distributin cvers sites within the grains with the same energy E=0 by definitin and sites with a Gaussian distributin within the grain bundaries. The Gaussian has the average site energy E seg and the width σ. Fig. 13: Effective diffusin cefficient f hydrgen at 95 K in nancrystalline Pd () and single crystalline Pd () as a functin f the ttal H-cncentratin. The hrizntal line thrugh the single crystalline data crrespnds t H-diffusin f nn-interacting H-atms. The curves thrugh the nancrystalline data are calculated using Eqs. (.3) and (5.5) and assuming that the effective diffusin cefficient crrespnds t the grain bundary diffusin cefficient (cf. text). H-H interactin is included by adding a term Wc gb t the chemical ptential, where W equals -30 kj/ml as in plycrystalline Pd [75] and c gb is the lcal cncentratin in the grain bundaries. Nte that grain bundary diffusin f interstitials at lw cncentratins is slwer than in single crystals. Fig. 14: Schematic presentatin f three limiting cases f grain bundary diffusin with bundaries perpendicular t the surface, where d is the distance f grain bundaries r the grain size, respectively. δ is the thickness f grain bundaries and c(x,y) is the cncentratin prfile. With diffusin cefficients D g and D gb fr grains and grain bundaries and t as the diffusin time the cnditins fr the three limiting cases are [85]: A1 and A: D g t>>d, B: 100δ< D g t<d/0 and C: D g t>δ/0 Fig. 15: Mdel f the (111) Ag/MgO interface at lw (a) and high (b) partial pressure f xygen. In a) structural vacancies are frmed in the terminating xygen layer, in rder t maintain stichimetry f MgO (cf. text). At high xygen pressures the vacancies are filled with excess xygen which gets its electrns frm Ag frming a kind f silver xide at the interface. In 57

58 the presence f excess xygen strng trapping f hydrgen ccurs at the metal/xide interface. Fig. 16: Macrscpic scattering crss sectin dσ/dω versus scattering vectr, Q, fr an internally xidized Ag-1 at.-% Mg ally (clsed circles) and its changes after expsing t hydrgen (pen circles) and t deuterium (triangles). The steep decrease at the lwest Q-values is due t large xide precipitates at grain bundaries. The fllwing plateau and the decrease at large Q-values stems frm small xide particles (average radius 1.6 nm) within the grains. The prnunced changes caused by hydrgen are caused by bth segregatin f hydrgen at the metal/xide interface and displacement f Ag-atms frm the interface (cf. text). Fig. 17: Difference f H-cncentratin between nancrystalline and single crystalline Pd, c, at a given partial pressure f hydrgen, p. The values are btained frm the data set in Fig. 11. Fig. 18: Density f site energies fr the varius Nb 1-x V x tetrahedra in crystalline Nb-V allys as a functin f cmpsitin [113]. Fig. 19: Pressure-cncentratin istherms fr hydrgen in tw amrphus Pd-Si allys btained by varius experimental techniques at 95 K [115]. The results are pltted in accrdance with Eq. (14.3) and the fugacity f hydrgen, f, is used at high chemical ptentials instead f the partial pressure. The slpe f the straight lines yields a value fr the width σ f the Gaussian DOSE. Fig. 0: Hydrgen diffusin cefficient as a functin f H-cncentratin in amrphus Pd 80 Si 0 allys prepared by the duble pistn technique (squares), melt spinning (pen circles) and sputtering (triangles). The cncentratin dependence vanishes after crystallizatin f the melt spun ally (clsed circles) [14, 119]. Fig. 1 a) Measured density f site energies n(e) c/ µ (cf. Eq. (4.7)) fr an amrphus Ni-Ti ally (filled circles) and the cntributins frm different tetrahedral sites (dashed Gaussian curves and their sum as a slid line). The Gaussian curves have abut the same width and the rati f their areas crrespnds t the binmial distributin (cf. Eq. (15.1)). Fig. 1 b) same as a) but fr a Ni-Zr ally Fig. Bulk metallic glass sandwiched in between tw Pd-layers. The tp layer allws permeatin f gaseus hydrgen int the glass and resistivity changes in the bttm layer are a measure f hydrgen transprt thrugh the glass. The transient behavir f this transprt yields H- diffusivity [18] Fig. 3: Pressure cmpsitin istherms fr H in tw bulk metallic glasses (Jhnsn glass: Zr 46.8 Ti 8. Cu 7.5 Ni 10 Be 7.5, squares fr a first run and triangles fr a secnd run and Inue glass: 58

59 Zr 66.8 Al 17.4 Ni 7. Cu 8.6, circles). The line was calculated fr the Inue glass as described in the text. Fig. 4 Diffusin cefficients fr hydrgen in the Inue glass (tw samples) as a functin f the hydrgen pressure. The straight line is an apprximate behavir as expected frm the measured chemical ptentials [18] Fig. 5: Chemical ptential r lgarithm f the thermdynamic activity, λ, f varius small slute mlecules r atms in varius amrphus matrices pltted in accrdance with Eq. (14.3) versus inverse errr functin f (1-c). Fig. 6: Schematic presentatin f results f Mnte-Carl Simulatins fr interstitial diffusin in a 3- dimensinal energy landscape with and withut Gaussian distributins f site and saddle pint energies. The cncentratin and temperature dependence f D in a perfect crystal (first clumn n the left) is shwn in the ther clumns as a dashed line fr the sake f cmparisn. The temperature dependence fr high cncentratins is shwn as a dtted line in the last tw clumns. The different dependencies are explained in the text. Fig. 7: Measured partial mlar vlumes f CO in tw glassy plycarbnates and in a glassy plyimide (Kaptn) as a functin f CO -cncentratin and calculated behavir (slid lines) [139]. The partial mlar vlume increases because with increasing cncentratin smaller sites with a higher elastic energy have t be ccupied. The partial mlar vlume in the liquid r rubbery state f the plymer as well as in many rganic liquids is abut 46 cm 3 /ml. This value is apprached at high cncentratins f CO (clsed circles) because the plycarbnate is swelling and finally transfrms int the liquid state due t the strain induced by the disslved CO. Fig. 8: Cncentratin pressure istherms fr varius small mlecules in plycarbnates [139]. The lines are calculated assuming a Gaussian DOSE and FD-Statistics. A slpe f unity crrespnds t ideal dilute behavir and the validity f Henry s Law which is the case fr water. The width f the Gaussian distributin determines slpe and curvature f the istherms and the smaller the slpe the brader the width. Changes f the average energy E result in a parallel mvement f the istherms in the directin f the lg c axis. Fig. 9: Relative vlume changes f plycarbnate fr three different slute mlecules (carbn dixide, ethylene, and acetne). The changes scale with the size f the mlecules and the lines are calculated with ne fitting parameter fr all three curves. Fig. 30: Average energy f the DOSE f CO fr different glassy plymers btained frm cncentratin-pressure istherms pltted versus the elastic energy assciated with the incrpratin f the mlecule int a smaller hle f the plymer and as btained frm Eq. (18.3). The linear relatin between the tw quantities used as an assumptin in Eq. (18.5) appeared t be fulfilled. 59

60 Fig. 31: Widths σ E f a Gaussian DOSE fr varius small mlecules in bisphenl-a plycarbnate (BPA-PC) pltted versus the squared mlar vlume f the small mlecules. The width was btained by fitting cncentratin-pressure istherms as shwn in Fig. 8 and the linear relatinship is predicted by Eq. (18.5). The line intercepts the abscissa at V h, i.e. the average site vlume in BPA-PC. Fr mlecules being smaller than V h n elastic energy has t be prvided during disslutin and the Gaussian degenerates t a Dirac-Delta Functin. Fig. 3: Cncentratin dependence f the diffusin cefficient (scaled with a factr κ, in rder t fit int the diagram) f different small mlecules in different plymers [145]. The lines were calculated using measured cncentratin-pressure istherms and Eqs. (.3) and (5.5) and ne fitting parameter D (cf. Eq. (5.5)) which like the scaling factr κ mves the curves parallel in the directin f the rdinate nly. Fig. 33: Schematic presentatin [ ] f the DOSE fr hydrgen in amrphus silicn (r germanium). The shwn part f the distributin stems mstly frm dangling bnds being saturated with H atms. Fr high cncentratins (sample B with the DOSE shwn left) the distributin extends further t the energy E m f "mbile" hydrgen and, therefre, the energy difference between µ Β and E m is smaller when cmpared with the case f lw H- cncentratins (sample A and distributin n the right). Hwever, if tw samples with the distributins shwn are in cntact, the chemical ptential has t be equal. This is achieved by a few H-atms mving frm B t A due t the lw density n(e) arund µ A. Thus the activatin energy fr diffusin in sample A is nw much smaller and similar t B. Fig. 34: a) Schematic silicate netwrk cntaining an alkali xide. b) Catin distributin in a regular lattice cntaining fixed anins as a mdel fr catin diffusin in xide glasses. Fig. 35: Schematic presentatin f the cncentratin dependence f the tracer diffusin cefficient f alkali ins in xide glasses. Decreasing D* in cncentratin range A and vice versa in B. Fig. 36: Bimdal distributin f site energies and ccupatin accrding t the shaded area. The ttal area f the first peak increases prprtinal with catin cncentratin. With this distributin the "weak electrlyte behavir" f catin mbility can be derived [69] Fig. 37: dc-cnductivity f a mixed Na-Cs silicate glass [157] with 5 ml-% alkali xide at 00, 300 and 400 C. The lines are calculated using a rectangular DOSE fr the exchange energy f Cs and Na [155, 156] Fig. 38: Diffusin cefficient f Na + (slid circles) and Cs + (pen circles) at 397 C in a 5 ml-% alkali xide silicate glass [157]. The straight lines are fits t the linear behavir fr A + ins at y 1 and R + ins at y 0, i.e. in regins where the crrespnding ins are the majrity cmpnent [155, 156]. 60

61 Fig. 39: Intensity, I, f X-rays in a θ-θ scan f a 100 nm thick Nibium film cvered with a 10 nm thick Palladium film. The latter prtects Nb frm xidatin and allws easy electrchemical H-dping. With increasing H-cncentratin (c=h/nb) the Nb (110) peak mves t smaller angles indicating the ut f plane expansin f Nb. The Pd (111) peak at abut 40.5 degrees des nt mve unless saturatin f the Nb layer (c>1) ccurs. Splitting f the Nb-peak int tw peaks reveals the decmpsitin in a lw cncentratin phase f α-nb and a hydride (βphase). The frmatin f the high cncentratin phase ccurs at a higher terminal slubility when cmpared t bulk Nb. Fig. 40: Cmpressive stresses in a 190 nm thick epitaxial Nb-film n MgO as a functin f H- cncentratin. Stresses are determined frm substrate curvature. At lw H-cncentratins the cmpressive stresses increase linearly with c. Deviatin frm this steep increase, i.e. yielding ccurs at the decmpsitin in the tw phases α and β. The stress f abut 400 MPa where the deflectin ccurs is much larger than the flw stress f bulk Nb. Fig. 41: Same as Fig. 40 but fr a 00 nm thick nancrystalline Nb-film depsited n Silicn by laser ablatin. Here stress relaxatin r yielding, respectively ccurs within the α-phase. The relaxatin is increased by entering the tw phase regin. Fig. 4: Same as Fig. 41 but fr a 00 nm thick nancrystalline Nb-film prepared by electrn beam evapratin. Here yielding ccurs at abut 0.6 GPa within the α-phase. Cmpared t the laser ablated film presented in the previus figure the grain size is larger fr the electrn depsited film. Fig. 43: Same sample as in Fig. 4 after the remval f hydrgen and a secnd lading with hydrgen. Nw stress relaxatin starts at a higher stress f abut 0.9 GPa. Fig. 44: Ptential trace fr hydrgen atms in a Nb/Pd multilayer. Site energies E Pd and E Nb are knwn frm slutin energies in bulk metals. H-Diffusin thrugh the layers is hindered by Nb-sites acting as traps r by Pd-layers acting as barriers, respectively. Fig. 45: Time lag versus H-cncentratin fr diffusin thrugh multilayers n a Pd-subtrate f 1.5 µm thickness. The multilayer cnsists f 4, 8 and 16 alternating duble layers f Pd and Nb f the same thickness and a ttal thickness as given in the inset. Fig. 46 Pressure cmpsitin istherms f Pd-clusters f 3 nm (circles) and 5. nm (triangles) in diameter cmpared with Pd-pwder (crsses) and Pd-bulk (slid line) Fig. 47: X-ray diffractin peaks f Pd-clusters f 6 nm in diameter at different partial pressures f hydrgen. At the pressure f Pa which crrespnds t the tw phase field f α and β bulk Pd (cf. Fig. 46) peaks split int tw r braden at least. 61

62 Fig. 48: Lattice parameter pressure istherms fr varius Pd-clusters. The change f the lattice parameter at abut 0 mbar is decreasing with decreasing cluster size in accrdance with the narrwing f the miscibility gap..0 partial mlar vlume (cm 3 /ml) Pd 80 Si 0 (amrphus) Pd (crystalline) cncentratin (H/Pd) Fig. 1 Partial mlar vlume f hydrgen in plycrystalline palladium (clsed circles) and in a liquid quenched Pd-Si metallic glass. The negative vlume changes fr the first 40 atppm H are explained by trapping f H-atms in a crrespnding fractin f vacancies [53]. 6

63 material structure ptential trace energy distributin energy distributin single crystal E E n(e) δ ( E E ) single crystal + pint defect E E t E n(e) (1 c ) δ ( E E c δ ( E E ) t t t ) + single crystal + dislcatin E E n(e) K ( E E 3 ) single crystals + grain bundary E E t E σ n(e) (1 ct ) δ ( E E ) + ct ( E Et ) exp σ π σ amrphus state E E σ n(e) 1 ( E E ) exp σ π σ Fig. : Schematic presentatin f a distributin f site energies which starts with the mst degenerate ne, i.e. the single crystal in the tp line. In the fllwing lines increasing structural disrder leads t a decreasing degeneracy by increasing the dimensins f lattice defects. Fr diffusin ptential traces r energy landscapes, respectively including the defect are imprtant. Fr grain bundaries traces within the grain and within the grain bundary are shwn separately, where the latter are present with a vlume fractin f c t. The density f site energies (DOSE), n(e), defined as the number f sites within a given energy windw is presented bth graphically and analytically. The varius frms f n(e) are derived in chapters IV and V. 63

64 Energy Q + E E E 1 1 Fig. 3: Ptential trace fr an interstitial jumping frm site 1 with energy E 1 t site with energy E. E is an average Energy f the DOSE and Q the activatin energy with respect t this reference. Thus particles in a material cntaining sites f energy E nly have a jump frequency f Γ=Γ exp[-q/k B T]. 64

65 θ R r σ = cnst. Fig. 4 Circles f cnstant hydrstatic stress at an edge dislcatin as calculated in cntinuum mechanics (cf. Eq. IV.4). The elastic interactin energy with hydrgen is cnstant at these lines and the crrespnding DOSE is the number f sites within the tw circles. 65

66 chemical ptential, µ, kj/ml H in Pd H-H interactin cre interactin strain field interactin dislc. density ρ reciprcal square rt f cncentratin, 10 (H/Pd) -0.5 Fig. 5: Electrchemically measured chemical ptentials f hydrgen in heavily defrmed palladium pltted accrding t Eq. IV.7 versus 1/ c. The varius regins f the data pints represent regins f a predminant interactin mechanism. 66

67 rati f quantities (defrmed/anneald) c f /c D/D ρ Η / ρ Η cncentratin atppm H/Pd Fig. 6: Hydrgen resistivity increment, hydrgen diffusivity and the cncentratin f free hydrgen in heavily defrmed palladium divided by the crrespnding values in well annealed palladium. These ratis are pltted versus H-cncentratin and in agreement with Eq. IV.9 they are abut the same ver the whle range f cncentratins with in the α-phase f Pd. 67

68 diffusin cefficeint (cm /s) cncentratin H/Pd Fig. 7 Diffusin cefficient f hydrgen in heavily defrmed Pd (clsed circles) and well annealed Pd (pen circles). The line fllwing the values f defrmed Pd was calculated frm measured chemical ptentials withut a fitting parameter (cf. text). 68

69 ln(q d Σ /d Ωnet 10 m ) Q [Å-] Fig. 8: A mdified Guinier Plt f the macrscpic scattering crss sectin dσ/dω (cf. Eq. IV.10) measured by small angle neutrn scattering fr defrmed Pd with 1 at.-% H. The slpe f the linear part yields the radius f hydrgen enriched cylinders frmed by segregatin at the dislcatin line. 69

70 radius [Å] c [10 H/Pd] Fig. 9: Radii btained frm small angle neutrn scattering fr a defrmed Pd-sample at varius H- cncentratins. The line is calculated by assuming the frmatin f a hydride f cylindrical shape in the strain field f dislcatins. The dislcatin density (ρ= cm - ) was used as a fitting parameter t btain agreement with experimental data. 70

71 -1 net-d Σ /d Ω [cm ] D/Pd H/Pd Fig. 10: Macrscpic crss sectin fr a defrmed Pd sample cntaining 0.8 at.-% hydrgen (clsed sircles) r deuterium (pen circles). The lines are calculated by using Eq. IV.10. The difference f dσ/dω between the tw istpes is less than expected frm their scattering length which is evidence fr a vlume expansin during the hydride frmatin at the dislcatin lines (cf. text). 71

72 H -equilibrium pressure (Pa) single cryst. Pd nancryst. Pd H-cncentratin (H/Pd) 10-1 Fig. 11: H -equilibrium pressure versus H-cncentratin fr single crystalline ( ) and nancrystalline Pd () [81] at 95 K. The dtted line has a slpe f within the α-phase accrding t Sieverts Law fr c<0.01 and it becmes a plateau within the α+β tw phase regin fr c> The slid and dashed lines fr the nancrystalline sample are calculated assuming a distributin f site energies (cf. Fig. IV.9) and including r excluding H-H interactin as explained in the text. 7

73 E 0 E seg σ n(e) Fig. 1: Site energy distributin fr H in nancrystalline Pd used in rder t calculate the line in Fig. IV.8. The bimdal distributin cvers sites within the grains with the same energy E=0 by definitin and sites with a Gaussian distributin within the grain bundaries. The Gaussian has the average site energy E seg and the width σ. 73

74 effective diffusin cefficient (cm /s) nancrystalline Pd single crystalline Pd H-cncentratin (H/Pd) H-H interactin withut with Fig. 13: Effective diffusin cefficient f hydrgen at 95 K in nancrystalline Pd () and single crystalline Pd () as a functin f the ttal H-cncentratin. The hrizntal line thrugh the single crystalline data crrespnds t H-diffusin f nn-interacting H-atms. The curves thrugh the nancrystalline data are calculated using Eqs. II.13 and III.17 and assuming that the effective diffusin cefficient crrespnds t the grain bundary diffusin cefficient (cf. text). H-H interactin is included by adding a term Wc gb t the chemical ptential, where W equals -30 kj/ml as in plycrystalline Pd [75] and c gb is the lcal cncentratin in the grain bundaries. Nte that grain bundary diffusin f interstitials at lw cncentratins is slwer than in single crystals. 74

75 y d d x c(x,y) c(x,y) δ A1 δ A c(x,y) c(x,y) B C Fig. 14: Schematic presentatin f three limiting cases f grain bundary diffusin with bundaries perpendicular t the surface, where d is the distance f grain bundaries r the grain size, respectively. δ is the thickness f grain bundaries and c(x,y) is the cncentratin prfile. With diffusin cefficients D g and D gb fr grains and grain bundaries and t as the diffusin time the cnditins fr the three limiting cases are [85]: A1 and A: D g t>>d, B: 100δ< D g t<d/0 and C: D g t>δ/0 75

76 Fig. 15: Mdel f the (111) Ag/MgO interface at lw (a) and high (b) partial pressure f xygen. In a) structural vacancies are frmed in the terminating xygen layer, in rder t maintain stichimetry f MgO (cf. text). At high xygen pressures the vacancies are filled with excess xygen which gets its electrns frm Ag frming a kind f silver xide at the interface. In the presence f excess xygen strng trapping f hydrgen ccurs at the metal/xide interface. 76

Materials Engineering 272-C Fall 2001, Lecture 7 & 8 Fundamentals of Diffusion

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