Hydrogen as a probe for defects in materials: Isotherms and related microstructures of palladium-hydrogen thin films

: Metal-hydrogen systems offer grand opportunities for studies on fundamental aspects of alloy thermodynamics. Palladium-hydrogen (Pd-H) thin films of nano crystalline, multi-oriented and epitaxial microstructures, electrolytically charged with hydrogen, serve as model systems. In these films thermodynamics of hydrogen absorption is modified by interface effects related to mechanical stress and to microstructural defects. Since in this respect hydrogen can be utilized to reveal the microstructural constituents of the films, we aim to investigate the distribution of sites (DOS) hydrogen occupies in the films’ solid solution regime. A σ DOS model is proposed, taking the measured substrate-induced stress contribution to the chemical potential into account. This enables the determination of the different sites’ volume fractions and of pure site energy distributions by fitting measured isotherms. Interstitial sites, grain/domain boundary sites and deep traps are distinguished. Dislocations and vacancies are shown to have a minor impact on the films’ trapping of hydrogen atoms, while deep traps are related to the films’ surface. Enhanced binding energies in nano crystalline films can be ascribed to the tensile strain effect of grain boundaries acting on the grains. Measured surface trapping energies fit to the respective bulk values, while the trapping of hydrogen in grain/domain boundaries of the films is significantly increased. This can be interpreted with different grain/domain boundary structures. Different from octahedral interstitial site occupation, tetrahedral site occupation is suggested for grain/domain boundaries of the films.

• exp needs to be fitted to measured chemical potential data of the films. This leaves , , , , , , and as fitting parameters to represent the DOS.
Here we first show that considering two Gaussian site energy distributions next to the interstitial sites is mandatory to fit the measured thin film isotherms. Second, we show that the fits are sensitive to small changes of the fitting parameters, yielding considerably small confidence intervals. Figure S1 shows measured chemical potential data (points) at given hydrogen concentrations of an 80 nm nano crystalline Pd-H thin film, that was corrected by the substrate induced stress impact. The figure reveals modifications of the fitted isotherms resulting from different types of sites taken into account. DOS fits for interstitial sites, grain boundary sites and deep traps, for neglected grain boundary sites as well as neglected deep trap sites are shown, revealing that all three types of sites are necessary to fit the measured chemical potential data. For this film, 0 by preparation. Figure S1. Variation of the kinds of sites considered in the DOS fit of the chemical potential of an 80 nm nano crystalline Pd-H film. All kinds of interstitial sites, grain boundary sites and deep traps are necessary to fit the measured data.
In Figure S2 the interstitial site energy is varied between 0 kJ/mol and 3 kJ/mol. Apparently, small changes in the site energy strongly influence the fitting in the close vicinity of the two-phase region. The best fit is achieved for 1.5 3 kJ/mol. Figure S2. Variation of the interstitial site energy fitting the σDOS model to the measured data. Figure S3 shows fits for modified broad Gaussian distributions around energies , that we relate to the grain/domain boundaries. For the blues line, was changed simultaneously with the site energy . The resulting isotherm is affected at medium H-concentration of the -phase field. The best fit is achieved for 0.5 1 kJ/mol.

Fitted isotherms of Pd-H thin films of different thickness and microstructure
Additional to the 80 nm nano crystalline Pd-H film in Figures 1a and 4 of the main manuscript, Figure S5 shows fitted isotherms according to the quasi-thermodynamic approach and to the σDOS model for Pd-H thin films of different microstructures and different film thicknesses. The figures reveal reasonable σDOS model fits of the chemical potential data in the solid solution regime, with the fitting parameters given in Table 1 of the main text.

Influence of the H-H interaction on the fitting parameters of the σDOS model
To investigate the impact of neglecting the H-H interaction contribution on the fitted model parameters of Eq 14 in more detail, Figure S6 shows the chemical potential of the 80 nm nano crystalline Pd-H film with different corrections: The orange data points represent the original measured data without correction. For the black data points the measured stress impact ∆ was subtracted from the data, like in Figure 3 and Table 1 of the main text. The black line shows the corresponding fit of the data points. The purple dashed line, on the other hand, represents a fit of the data with subtracted ∆ and ∆ , with the parameter 20.5 kJ/mol as determined from the fit of the quasi-thermodynamic model approach to the data, see Figure 1a of the main manuscript. To plot the purple line together with the black data points, ∆ was added back to the fitted isotherm after the fit. All resulting model parameters are summarized in Table S1.  Apparently, the different corrections change the energy scale of the chemical potential to different amounts. Regarding the model parameters, the largest relative effects of preliminary data correction appear in the fitted interstitial site energies and in the grain boundary fractions with peak energy . The relative change of the other parameters is small. The shift of the site energy by 1.7 kJ/mol is largest for the uncorrected data compared to the data corrected by ∆ , shifting towards that of the 80 nm films with other microstructures, see Table 1 of the main text. takes a medium value with a shift of 1.0 kJ/mol for the correction of ∆ ∆ . While fitting the chemical potential without any corrections yields too large site energies with respect to the assumptions of the σDOS model, the correction of ∆ is meaningful from a physical point of view, since the H-H interaction is the initiator of the phase transition in the Pd-H thin films. This is also supported by the consideration of the slopes of the fitted curves in Figures S6  and S7 with respect to the data points close to the solid solution limit : There, the fit is best for the curve corrected for the H-H interaction, because the H-H interaction gradually decreases the slope of the measured chemical potential in that region, indicating the onset of phase transition. Figure S7. Impact of the choice of the site blocking factor on the fitted interstitial site energy of bulk palladium.
However, since the value of was taken from the quasi-thermodynamic model approach but cannot be determined with Eq 14, it is usually neglected in the data evaluation in the present paper. This yields slightly too small site energies and slightly too large grain boundary fractions , compare the second and the third lines of Table S1.
Thereby, the effect of ∆ on the model parameters needs to be compared to the effect of an assumed site blocking factor of 1 / . As shown below, 1 / results in slightly too large site energies , compensating for the effect of ∆ .

Impact of the site blocking factor on model parameters
In the DOS fits in the main text the site blocking factor , describing the blocking of sites for the occupation by hydrogen atoms for electronic and microstructural reasons, was set to 1. In Figure S7 we evaluate the effect of setting 1 / , allowing for the occupation of all octahedral sites in the Pd metal, compared to the correct value of 0.6 / . Chemical potential data of 100 µm bulk Pd-H are shown, together with a fit of the quasi-thermodynamic approach of Eq 4 in the main manuscript.
One can see that the choice of affects the magnitude of the fitted site energy , yielding smaller site energies for smaller blocking factors. For the fitting of the quasi-thermodynamic approach the blocking factor was set to 0.62 / as well, yielding a similar site energy like the DOS model fitted with the same . In total, we find 5.7 kJ/mol for 1 / , but 4.6 kJ/mol for 0.6 / . This effect balances the -increase due to the negligence of ∆ in the σDOS model, see above.

Impact of dislocations on the measured chemical potential
In the fit of the density of site energies to the measured chemical potentials of Pd-H thin films we neglected the influence of dislocations. This is justified by the observation that the chemical potential of thin films increases much slower with the hydrogen concentration in the film than predicted by the dislocation impact. Hence, dislocations seem not to play a major role as trapping sites affecting the chemical potential in Pd-H thin films. This is shown in Figure S8, where the increase of the chemical potential with hydrogen concentration is compared for strongly deformed bulk Pd to that of an 80 nm nano crystalline Pd thin film. The bulk data were adapted from [1]. Figure S8. Comparison of the chemical potential increase with hydrogen concentration of heavily deformed bulk Pd [1] and of an 80 nm nano crystalline Pd thin film.
The bulk sample possesses a fitted areal dislocation density of 7.4 10 cm [1], close to the maximum geometrically possible dislocation density of metals. It is shown in [1] that the deviation of the chemical potential increase from the ideal slope, which is represented by in Figure S8, is caused by the trapping of hydrogen atoms in the dislocations in the bulk sample. This is also shown in Figure S9.
The chemical potential of the nano crystalline Pd thin film, however, increases much slower with hydrogen concentration than that of the bulk sample; the chemical potential increase is shifted by two orders of magnitude in concentration. This hints on thin film trap sites different from dislocations. Figure S9. Comparison of the chemical potential of bulk Pd of [1] with the approximation of Eq 19 in the main manuscript. Different from the Pd thin film data in Fig. 8 of the main text, Eq 19 fits the bulk data reasonably well.

Offset of the chemical potential
The chemical potential of hydrogen in metals contains half of the standard potential of gaseous hydrogen [2,3], /2 2 /2 with 1.013 bar at 7.55 K , / , constant 8.57 10 bar/K / and the dissociation energy 4.476 eV/ 431.9 kJ/mol of molecules. Hence, the chemical potential of hydrogen in a metal is proportional to an equivalent outside hydrogen gas pressure. Targeting site energies of hydrogen in metals, the energy scale often is shifted by /2, practically neglecting the offset term. Thereby it is /2 231.8 kJ/mol at T 297 K.

Oxide potential of a Pd thin film
All Pd thin films investigated in the present manuscript were electrolytically charged with hydrogen in an electrolyte consisting of 2/3 distilled water and 1/3 80% phosphoric acid. The electromotive force (EMF) of the samples was measured in reference to a saturated Ag/AgCl electrode. The of the electrolyte was pH 4. To investigate the oxide potential, Pd thin films were oxidized by anodically discharging. This is shown in Figure S10 for the example of a 100 nm nano crystalline Pd thin film. There, the EMF as well as the corresponding chemical potential are plotted as a function of the negative electric charge applied to the sample. The oxide potential starts at 45 kJ/mol and possesses a plateau at 70 kJ/mol.