XRD: in situ hydrogen loading

Samples have been studied at two different synchrotron radiation sources. At the Deutsches Elektronen-Synchrotron (DESY), beamline PETRA P08, Hamburg a wavelength of λ =0.9998Å was used. The samples were loaded with a loading setup similar to the one presented above (section 3.2.1) with a Kapton window to allow for synchrotron radiation to pass into and out of the sample chamber. In con-trary to the setup presented above, the volume of the reservoir and sample chamber is not known for this setup, but the volume is much larger. Therefore, the condi-tions for diffusion from a limited volume do not apply (compare section 3.2.2). The pressure does not change after beginning the loading procedure, indicating that the reservoir is large enough to assume a constant surface concentration. The solution of Fick’s law for these condition is given by equation 3.7. A four-point resistance measurement was carried out during hydrogen loading.

At the European Synchrotron Radiation Facility (ESRF), beamline BM20, Grenoble a similar measurement was done using a wavelength of λ = 1.078Å. The setup employed at the ESRF was already presented by H. Uchida who also used it forin situ hydrogen loading experiments of Mg thin films [76]. The main difference to the other presented loading setups is that this one works with a constant hydrogen flow, artificially holding a constant pressure at the sample surface. This is an ideal situation for the solution given by equation 3.7. However, it takes up to2 hto reach the highest measured pressure of8·10⁵Pa, because the hydrogen is limited by a flow controller. During this time the concentration can only be assumed to be constant in a short enough time frame. The setup does also include a four-point resistance measurement and offers the possibility to heat the sample.

The measurements at both beamlines allow to measure the development of one Bragg peak during hydride formation over time. This allows measuring the XRD peak areaA^peak over time. The XRD peak area is proportional to the volume of the material in the sample [215]. Assuming the model of Uchida et al. [44] the MgH2

grows as a (after some time) closed layer starting from the top of the Mg thin film.

Therefore, the volume of Mg and MgH2 in the thin films can be simplified to only the layer thickness of the magnesium d_{M g} and magnesiumdihydride d_{M gH2} layer:

d_{M gH2} =d₀−d_{M g}

⇔d_{M gH2} =d₀−d₀A^peak_{M g} A^peak₀

⇔ d_{M gH2}

d₀ = 1− A^peak_{M g} A^peak₀ .

A^peak₀ is the Mg peak area before loading where d₀ = d_{M g}. Assuming that the hydride formation is diffusion limited and the diffusion process is one-dimensional, the hydride thickness over time can be described asd_{M gH2} =p

2D_{M gH2}·√

t. D_{M gH2} is the diffusion coefficient of hydrogen diffusing through MgH2. This finally leads to a description of the peak area over time:

A^peak_{M g}

A^peak₀ = 1−

p2DM gH2

d₀ ·√

t. (3.20)

Plotting the normalized peak areaA^peak_{M g} /A^peak₀ over the square root of time results in the diffusion constant of hydrogen in magnesiumdihydride as the slope of a straight line with a y-intersect of one. If the data points do not form a straight line in this plot it can be seen as an indication that the hydride formation is not limited by a one-dimensional diffusion process.

3.6. Finite-element simulations

In the following section finite-element-simulations (FEM simulations) performed for this work are introduced. These are employed to gain additional insight into how the grain boundary diffusion and grain diffusion contribute to the overall diffusion in the sample. The experimental techniques introduced above only allow to measure the overall diffusion coefficient of hydrogen in the sample. A comparison with FEM simulations may help to separate the influence of the single components. Finite-element simulations divide a given geometry by means of a mesh into subdivisions.

For each of the mesh nodes the relevant differential equations are solved (e.g.the diffusion equations as given by equation 2.10 and equation 2.11). By combining the equations of each subdivision into a larger system the behavior of the overall geometry can be calculated. In this work, the FEM simulations were performed using the commercial software COMSOL Multiphysics^R Modeling Software, version 5.2. The first part introduces the geometries modeled as well as the meshing used. In

addition the stability of the simulations is tested in dependence on the mesh. In the second section the data evaluation of the simulation is discussed. The experimental procedure and the results are published in "FEM simulation supported evaluation of a hydrogen grain boundary diffusion coefficient in MgH2" [221].

3.6.1. Simulation setup

constant flux: 1/m³

heigh

t:100nm lateralgrainsize:

grain boundary size δ=: 1 nm 2nm,4nm, 10nm,20nm, 50nm

(a) Schematic drawing of the FEM simulation

5 nmx y

z

(b) Meshing of the FEM simulation Figure 3.12.: Geometry (a) and mesh(b) used with the COMSOL Multiphysics ^R Mod-eling Software, version 5.2. The geometry employs periodic boundary conditions on the sides, resulting in an infinite 100 nm thin film. The mesh was set on the surface, refined and swept in the diffusion direction (z-direction). The figure is reproduced from [221].

The 3-D Finite-Element simulations were performed using the “Transport of diluted Species” physics module. This models the transport of a single species of low con-centrations. This seems to be well applicable as hydrogen has low solubility in magnesium and magnesiumhydride (MgH2 being a stoichiometric phase). For this work, convection, as an additional transport mechanism, was deactivated. Figure 3.12 shows the modeled geometry. The geometry is build up out of 100 nm high grains that are separated by 1 nm thick slabs of grain boundaries. Periodic bound-ary conditions are applied on the outside of the grains so that the overall geometry models an infinite 100 nm thin film with columnar, box-shaped grains. Five differ-ent grain sizes d of 2 nm, 4 nm, 10 nm, 20 nm and 50 nm were studied. On top of this, a constant fluxJ of hydrogen ofJ =1 m⁻³ is applied. The diffusion coefficient of the grain boundary was set to D_GB =10⁻¹⁷m²/s, as a rough mean value of the different literature values at room temperature (compare figure 2.6). The diffusion

coefficient of the grain was varied, starting at D_V =10⁻¹⁸m²s⁻¹ and decreasing in orders of magnitude up to D_V =10⁻²²m²s⁻¹. This contains the assumption done by Fisher and others after him, that the grain boundary diffusion is faster than the grain diffusion (compare section 2.2.3). The initial concentration c_H in the whole sample was c_H = 0/m³. The simulation gives the concentration in arbitrary units of1/m³ at chosen time levels oft=10⁻²s,10⁻¹s,1 s,10 s and100 s. The simulation times and simulated lengths were adjusted to the modeled diffusion coefficients, but the results of the simulation can easily be transferred to other diffusion coefficients.

The same is true for different time and length scales. D ≈ l²/t allows recalculat-ing the diffusion coefficient D, time t and distance l in relation to each other. An example of the meshing is shown in figure 3.12. For all grain sizes a free triangular mesh was applied on the top plane (lying in x-y-direction) of the geometry which was swept in the direction of diffusion (z-direction). To control the influence of the meshing on the results the concentration in the grains, in the grain boundaries and in some random cut planes, containing both, were studied. The mesh was refined until the relative change in the average concentration was below 0.2 % between two refinement steps for all regions and for all times. The resulting meshing was used in the final simulations. In z-direction this was always the case for a division into 75 or more elements. The number of elements and their size of the free triangular mesh in x- and y-direction was different for the different grain sizes. The total number of elements increased from 231 600 elements for a grain size of 2 nm to 839 550 for a grain size of 50 nm.