Sorption kinetics in metal hydrides by leaky coating

(cid:1) Partially sealed membrane mimics hydrogen sorption in 3D metal grain. (cid:1) Combination of absorption and permeation yields time-dependent H gradient in the membrane. (cid:1) Formation and growth of incoherent hydride interfaces conﬁrmed.


Introduction
The standard approach for the search of new hydrogen storage materials is to synthesize bulk samples and use gravimetric [1e3] or volumetric [2,4,5] techniques to follow the kinetics of the hydrogenation reaction and to record pressureconcentration isotherms (pcT).Thereafter, the kinetics are modeled along more or less defined assumptions of the time dependent hydrogen distribution inside the metal grains [6,7] e.g., by applying the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model [8e11].
in which the fraction of transformed material f evolves with time t.In addition to the kinetic constant of the process K, the model gives a fit parameter n which is related to the physical phenomenon (diffusion or phase transformation) and the dimensionality of the interface.This procedure is highly debated due to too many degrees of freedom [12], and because additional potentially rate-limiting steps are not included in the JMAK model, such as dissociation [6,13,14], and the finite particle size distribution of technical materials [15].
In addition to the hydrogen sorption kinetics, the formation/decomposition process of the hydride influences the effective equilibrium properties, in particular the absorption/ desorption hysteresis [7].In this paper, we shed light on hydrogen sorption in Pd.Despite being the first system in which hydride formation has been discovered, there is still no consensus on the origin of the hysteresis in this system [16,17].The knowledge gap arises from the same uncertainties described above and owing to the fact that current experimental methods provide information only on either the time-resolved but spatially averaged concentration in a metal (hydride) during the sorption process, or vice versa.Available methods usually rely on sophisticated analytical setups [3,18] such as electron microscopy [19,20], elastic recoil distribution analysis [21], N15-nuclear resonance analysis [22e24] etc., which are difficult to convert into an operando method.
Here we use concurrent hydrogen uptake and hydrogen permeation measurements in/through a palladium membrane to corroborate the build-up of a concentration gradient in the solubility phase and incoherent growth of palladium hydride.A leaky coating on the surface enables simultaneous determination of the experimental hydrogen concentrations with simple methods.

Material & methods
The method introduced in this paper was inspired by chemical engineering approaches to optimize chemical reactors, where the distribution of compounds in a reactor is measured by small probes at various locations in the catalyst bed analyzing the local concentrations.However, introducing a nano-probe into a micrometer sized sample is challenging.Thus, we transform the 3D geometry into a 2D geometry by unfolding the 3D structure with the inner part mimicked by a hydrogen impermeable surface (blue line in Fig. 1 (a) and (b)).The uptake kinetics x H ðtÞ of such a sealed membrane serves as a simple model system for the complex structure of bulk materials.Its measurement is identical to the one used for bulk materials, concretely, the Sieverts method: with x H ðtÞ the total hydrogen content of the membrane at time t, p initial À p f (t) the pressure change of the feed volume V 0 at standard pressure p 0 ¼ 101.325 kPa, V m the standard molar volume, and n Pd the amount of Pd (i.e., the accessible number of Pd atoms in the membrane).The equilibrium hydrogen content x H is calculated from the equilibrium pressure p eq ¼ p f (t / ∞).To probe the hydrogen concentration in the membrane at the sealed side corresponding to the center of a 3D grain, we use a 'leaky' coating, i.e., a coating, which lets hydrogen permeate at fluxes negligible for the overall hydrogen uptake kinetics, but still detectable in ultra-high vacuum (UHV) using a mass spectrometer (MS) (SRS RGA 100).In our case this coating is a 10 nm thick Cu layer deposited by RF sputtering onto the 45 mm thick Pd membrane (Goodfellow, 99.95%) (Fig. 1).The membrane is fixed in a special sample holder that is inserted into a UHV chamber with a mass spectrometer [25].Hydrogen feed pressure is measured with a capacitance pressure gauge (Vacuubrand DVR 5), while the hydrogen pressure in the vacuum chamber is recorded by the mass spectrometer.
The hydrogen desorption from such a surface is surface limited [26e29], and thus the local hydrogen concentration in the membrane c H near to the sealed surface is assumed to be c H ðtÞfp MS ðtÞ: (3) In the following, we first provide evidence for the validity of the approach, i.e., the applicability of eqs.( 2) and ( 3) by probing the hydrogen uptake and hydrogen desorption of the Cu-sealed Pd membrane.Secondly, we discuss the relevance of the results for hydrogen uptake in Pd.

Results & discussion
The verification of eq. ( 2) boils down to confirming that the hydrogen loss through the sealed side does not influence the thermodynamic properties of the material.Therefore, we compare the equilibrium pressure composition isotherms of the sealed Pd membrane with those published in literature [30] (Fig. 2).There is very good agreement in the solubility phase.The deviation in the two-phase region results from non-equilibrium conditions, as we avoid full hydrogenation of the membrane to prevent its embrittlement.Since main influence from hydrogen loss can be expected at small hydrogen concentrations, the influence of the leaky membrane side is negligible.
For equation (3), we compare the experimental x H ðtÞ and c H (t) with simulations based on the 1D diffusion [31].The total uptake x H ðtÞ into a membrane with thickness 1 2 l and diffusion constant D is while the concentration c H (t) is The two curves converge for large t (see Fig. 3(a)) indicating attainment of equilibrium, i.e., Main difference is the time lag at initial time scale, which is the consequence of the spatial gradient inside the membrane.The qualitative agreement of simulation and measurement (Fig. 3) is very good, given the uncertainty that the feed pressure and thus the hydrogen concentration at the feed side of the membrane is slightly time dependent (see also absolute feed pressure evolution in Fig. 2).Best quantitative agreement was reached using D ¼ 5 , 10 À11 m 2 s À1 , which is approximately factor two off (literature value for hydrogen diffusion in Pd: D ¼ 12 , 10 À11 m 2 s À1 , Ref. [32]).

The result that
x H ¼ c H for t ¼ ∞ can be independently confirmed by measuring x H and c H for various equilibrium pressures.
Fig. 4(a) and (c) show the changes of p f from which x H is derived, and p MS during absorption and desorption.The final values x H and c H are compared in Fig. 4(b).In the solid solution, p MS and x H follow a linear behavior, which is experimental evidence that p MS is proportional to the local hydrogen concentration near the sealed membrane side (eq.( 3)).Furthermore, all uptake curves follow the same function as representatively shown for one case in Fig. 3, i.e., in the  x H (from feed pressure in (a)) correlated with H 2 MS signal p MS plotted on a linear scale.The inset shows the correlation on a double log plot.In this case the slope is m ¼ 1.08 corroborating eq. ( 3).(d) Sketch of hydrogen absorption and desorption in Pd (letters "A" to "G" correspond to the points indicated in (c)).A diffusion gradient forms in the solid solution, followed by the nucleation and growth of the hydride phase ("A" to "C").solid solution, the uptake is rate-limited by hydrogen diffusion in the membrane.
Once the two-phase regime is reached, the uptake x H ðtÞ behaves different from that of p MS (t).Assuming that eq. ( 3) is still valid, this is indicative of a complex absorption process.Most striking differences are observed during desorption.During desorption, the feed side is kept under vacuum (i.e., p f ¼ 0), and thus p f cannot be used to determine the total loss of hydrogen.However, p MS still provides information on c H .In solid solution, c H during desorption displays the mirrored behavior of the one during absorption, which is evidence for diffusion rate-limited desorption.In the twophase regime, a plateau phase is observed (see "D 00 to "E 00 in Fig. 4(c)).The plateau is also indicated during the absorption process ("A 00 to "B 00 ).After the plateau, p MS continues to increasedrespectively decreases down to the plateau during desorption (from "C 00 to "D 00 ).These observations match the expected path through the HePd phase diagram from solid solution to two-phase regime and hydride phase and vice versa.However, under the conditions used, the membrane does not completely reach the hydride phase ( x H x0:1[H]/[Pd]).The fact that c H increases, indicates that some Pd in the solid solution is supersaturated, and hydride formation (and decomposition) takes place time-delayed at various locations inside the membrane (for illustration see sketches "A 00 to "C 00 and "C 00 to "E" in Fig. 4(d), respectively).The hydride is formed at various nucleation sites near the feed side, and then grows.After coalescence, a hydride interface moves through the crystal as defined by Ho et al. [33].These observations are in perfect agreement with the explanation of the hysteresis in Pd by the thermodynamic constraints of a growing hydride interface [17].The corresponding hydrogen kinetics via interface motion v is estimated by the temporal existence of the plateau phase t measured by p MS (see also Fig. 4(a, c)): which gives v x 2 , 10 À6 ms À1 at 80 C and v x 2 , 10 À5 m s À1 at 105 C, respectively.If the interface velocity v was determined by diffusion, the corresponding velocity of a diffusion front with coefficient D int would be with k 0 x 6 and t ¼ 1000 s [31].D int is in the same range as the diffusion of H in the solid solution (D ¼ 1.2 , 10 À10 m 2 s À1 , Ref. [32]) as well as in the hydride phase of Pd (D ¼ 8 , 10 À10 m 2 s À1 , Ref. [34]).However, diffusion as the ratelimiting process of the interface growth is unlikely, because in that case the interface velocity should depend on time t.
As the plateau length scales with t and x H , the interface velocity v is constant.Recent measurements find interface velocities in single Pd nanoparticles of the order of 10 À9 m s À1 , i.e. three orders of magnitude lower [35].The authors attribute the low velocity to strain formation impeding the propagation of the interface.Defects such as dislocations further impede the hydrogenation as observed in defective nanoparticles [20].However, our value of v is in good agreement with measurements of the hydride interface velocity in polycrystalline Pd foils using electron microscopy measurements (v x 10 À6 m s À1 ) [33].Hydride formation in polycrystalline materials can also proceed via incoherent precipitates, in contrast to the coherent phase transformation in small nanoparticles as above.Once formed, the propagation of incoherent precipitates is much faster than the coherent ones because it is not hindered by elastic strains anymore [33].This matches the observation in Fig. 4: a fast hydride growth with constant interface velocity after a nucleation period (both during absorption and desorption).The similarity of D and D int is no coincidence: since after nucleation, the growth of the hydride is not rate-limiting anymore, mass transport becomes rate-limiting, i.e., hydrogen diffusion.

Conclusion
We have demonstrated a simple method to study the hydrogen gradient in hydrogen absorbing metals by probing the total hydrogen uptake and simultaneously the desorption from a partially sealed membrane.The strength of the method is the simplicity of the experimental setup, consisting of a UHVmembrane setup with established technology [27,36,37] (mass spectrometer, Sieverts method).Such a setup can be combined with additional characterization methods [25].Crux of the method is that the model membrane must reflect the physical properties of the corresponding real world material.For the materials system at hand, Pd, this does not pose a problem.We used a commercial Pd foil, with 10 nm sputter coated Cu layer.Similar materials such as d-metal alloys for hydrogen selective membranes [38] and for hydrogen storage (e.g., TiFe [39]) as well as hydrogen resistant steel alloys [40] are suitable samples as well.Particularly in the latter cases, the influence of the mechanical treatment is an interesting subject of investigation, e.g., microstructure changes due to rolling and its influence on hydrogen permeation and thus embrittlement.Further aspect to be studied in the future is the dimension of the system, i.e., the thickness of the membrane.
The applicability of the Sieverts method to measure total hydrogen uptake was experimentally confirmed and is reasonable given a negligible hydrogen loss through the Cu coated surface.We proved empirically that the mass spectrometer signal is a good measure of the hydrogen concentration near the coated membrane.Here, the theoretical background is less clear, but this method may be a future tool to shed light on the controversially discussed topic of hydrogen desorption rates from metal hydrides.By measuring the two properties x H and c H , the construction of the time-dependent hydrogen gradient in a membrane is possible.For Pd at low hydrogen concentrations, i.e., in the solid solution phase, a diffusion gradient builds up and disappears when equilibrium is reached.At higher concentrations, the hydride is formed.The kinetics are governed by incoherent precipitates formed at various sites, coalesce, and subsequently, the growing hydride interface moves i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 3 3 4 0 3 e3 3 4 0 9 through the membrane.It is worth noting that the uptake kinetics alone would not have allowed to draw these conclusions.

Fig. 1 eFig. 2 e
Fig.1e (a) While absorbing hydrogen from the gas phase (red arrows), an uneven hydrogen distribution is formed inside a metal (hydride) grain.(b) The situation is mimicked using a membrane sealed on one side with a leaky coating.The membrane is embedded into a UHV-system equipped with surface characterization (not shown), mass spectrometer, and a Sieverts type hydrogen supply[25].(c) Shows a typical measurement of the feed pressure p f (t) and H 2 MS signal p MS (t).(For interpretation of the references to color/colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3 eFig. 4 e
Fig. 3 e (a) Simulation of averaged hydrogen concentration x ̄HðtÞ and c H (t) at the sealed membrane side in red and blue, respectively (equations (4) and (5)) with D ¼ 5 , 10 ¡11 m 2 s ¡1 and l ¼ 90 mm.(b) Measurement of x H ðtÞ and p MS (t) at 40 mbar and 82 C corresponding to hydrogen uptake in the solid solution phase of Pd. (For interpretation of the references to color/colour in this figure legend, the reader is referred to the Web version of this article.)