Interplay of diffusion and dissociation mechanisms during hydrogen absorption in metals

Kinetic measurements of gas-solid reactions, in particular hydrogen-sorption kinetics, are usually interpreted according to single rate-limiting step models. However, recent studies gave clear evidence for the interdependence of fundamentally different steps involved in hydrogen sorption. This interdependence is explored in a one-dimensional continuum model in order to estimate the time, temperature, and pressure dependences of the sorption kinetics and the relevance of several materials parameters involved in the process. Quantitative descriptions of the extraction of physical parameters for various scenarios are given. The model is successfully tested for several experimental cases, ranging from model systems (thin films) to practical systems (${\text{LaNi}}_5{\text{H}}_x$ and ${\text{MgH}}_2$ powder samples).

Figure 1

(Color online) Top and bottom panels: The various layers considered in the model. The surface layer consists of hydrogen adsorbed on surface atoms (amount of H is described by $\theta$) attached to subsurface atoms, which belong to the diffusion layer. The hydrogen concentration in the diffusion layer is labeled $c$; the concentration in the metal hydride is $c_{\text{MH}}$. Two typical situations are sketched: The top panel shows a metal hydride with fast diffusion inside the metal/metal hydride, where the metal hydride nucleates homogeneously. The main diffusion barrier is the surface. The bottom panel sketches the formation of a hydride with a very slow diffusion of hydrogen in the metal hydride. The main diffusion barrier is the metal hydride. Middle panel: Schematic representation of the concentration dependence of the hydrogen chemical potential in (left) the diffusion layer and (right) in the metal hydride for a given temperature.

Figure 2

(Color online) Scheme of the two-step model used to define the variables used in this work. Top, middle, and bottom panels: Potential energy, chemical potential, and hydrogen concentration as a function of the location in the diffusion layer and in the metal hydride, considering also the inherent hysteresis of the $p\text{−}c$ isotherms of any metal hydride. In the metal (hydride) the diffusion is $\infty$. Therefore, the concentration of hydrogen is determined by the current $j_{\text{diff}}\left(x=L,t\right)$. The diffusion current $j_{\text{diff}}\left(x=0,t\right)$ is equal to the current $j_{\text{barrier}}$ through the surface barrier with height $E_1$.

Figure 3

(Color online) Pressure dependence at 600 K of the surface coverage (calculations from Ref. 19) and hydrogen bulk concentration (experimental values from Ref. 25) of Ni. The lines are extrapolations used to plot the bulk concentration as a function of the surface coverage (inset). A good fit (line) to these values is $c\left(\theta \right)\propto \theta /\left(1+a\theta \right)$, with $a\simeq -1.25$, in fair agreement with the used approximation [Eq. (5)].

Figure 4

(Color online) Time dependence of the hydrogen uptake, limited by dissociation (case A) and limited by the solubility of the surface layers (case B). The dotted line refers to the condition expressed in Eq. (26).

Figure 5

(Color online) Graphical solution of Eq. (28) for various pressures and diffusion parameters $D_4/L_4>D_3/L_3>D_2/L_2>D_1/L_1$, highlighting the interdependence of the uptake rate and the concentration at the surface.

Figure 6

(Color online) (a) Pressure dependence of H absorption of ${\text{Mg}}_{1-y}{\text{Ni}}_y$ thin films for various compositions $y$ measured at room temperature [$y=0.74$ (black) open diamonds; 0.79 (green) stars; 0.84 (red) empty circles; 0.95 (blue) filled circles]. The solid lines are linear fits, from which the plateau pressures are derived. The regression coefficient decreases from nearly 1 (0.9997, black dots) to 0.9011 (blue dots). The inset is an enlargement of the Mg-rich sample, which clearly shows the nonlinear pressure dependence. The extrapolation to zero rate defines the equilibrium pressure $p_{\text{pl}}$. Data were from Ref. 38. (b) The plateau pressures from (a) are used to plot the rate as a function of $p/p_{\text{pl}}$. The solid curves are linear fits to the experimental data, revealing the same slope for all curves.

Figure 7

(Color online) H-uptake rate of ${\text{Mg}}_2\text{Ni}$ at various pressures and temperatures measured optically. Points marked (1) and (2) indicate pressure regimes, where a negative and a positive apparent activation energy, respectively, are found (see text for details).

Figure 8

van’t Hoff plots of ${\text{Mg}}_2\text{Ni}$. Equilibrium pressures were from Fig. 7 (thin films) and Ref. 44 (bulk).

Figure 9

(Color online) Surface activation energy $E_1$ of Pd-capped ${\text{Mg}}_2\text{Ni}$ thin films as obtained from an Arrhenius plot of the parameter $A^{\prime }=R/\left(p-p_{\text{pl}}\right)$ [Eq. (32)].

Figure 10

Bottom panel: Hydrogen-uptake rates of $d$ metal–coated Ta wires (hollow dots) and of surface-oxidized yttrium films (full dots). Middle panel: Corresponding apparent activation energies and the dissociation barrier heights and activation energies for permeation through the metal coating. Top panel: rate limiting parameter $r$ as derived from Eq. (51).

Figure 11

(Color online) Pressure dependence of the H absorption of activated and unactivated ${\text{LaNi}}_5{\text{H}}_x$ at various temperatures. Data were from Refs. 53, 54. The data are fitted to Eq. (31) (dashed curves) and to Eq. (58) (full curves).

Figure 12

(Color online) Absorption and desorption kinetics of ball-milled ${\text{MgH}}_2$. For desorption into vacuum and absorption at 5 bar ${\text{H}}_2$, the temperature is in both cases $300°\text{C}$. The absorption behavior is best described by a three-dimensional controlled contracting volume $\left(L\to L\ast \right)$ model with decreasing interface velocity (i.e., $dL/dt\ne \text{const}$).

Figure 13

(Color online) Impact $R/R_0$ of additives on the absorption properties of metal hydrides. (a) Dependence of the impact on the diffusion path length $L$ as simulated by the two-step model with model parameters for diffusion and surface parameters $A$ and $B$. (b) The effect of various additives on the absorption kinetics relative to that of as-milled ${\text{MgH}}_2$ is plotted as a function of the filling factor. Data were taken from Ref. 65.