2.1. The Mg–H system
2.1.1. The thermodynamics of Mg –H bulk
Figure 2.1:
Phase dia-gram of the magnesium-hydrogen system ac-cording to Okamoto [70]. The thin films in this work were loaded at 20◦C as indicated by a red line.
temperature/◦ C
weight percent hydrogen
atomic percent hydrogen 20
Mg H
Figure 2.1 shows the bulk phase diagram of the Mg–H system at25 MPaaccording to Okamoto [70]. The two most important phases for this work are the α-phase, in which a small amount of hydrogen is stored in solid solution before the β-phase is formed. In figure 2.1 the α-phase is labeled as Mg and the β-phase is labeled as MgH2. While the α- and β-phase are thermodynamically stable phases, addi-tional phases have been discussed in the literature [24, 21, 71, 72]. A metastable γ-phase is known to form at high pressure conditions. It will be discussed shortly in regard to high stresses that are known to arise in thin films during hydrogen loading [73] (compare section 2.1.2). In addition the literature discusses a δ-phase with a distorted CaF2 structure [74] and a ε-phase with an AlAu2 structure [75].
Both phases are not expected to form during the moderate conditions used in this work and are excluded from the discussion. All in all, the (di-)hydride phase in this work is synonymous with theβ-phase, except where it is specified otherwise. A very informative collection of the different hyride phases and their thermodynamic parameters are given in the thesis of H. Uchida [76]. As the focus of this work is more on the kinetics of the Mg–H system the thermodynamic behavior is discussed in a shorter form.
Metallic magnesium has a hexagonal crystal structure. The lattice parameters are
Phase Structure a /Å b /Å c /Å Ref.
α-Mg hexagonal (P63/mnm) 3.21 =a 5.21 [24]
β-MgH2 tetragonal (P42mnm) 4.50to 4.52 3.01to 3.02 = a [24]
(TiO2 structure) 4.51to 4.52 3.01to 3.02 = a [72]
γ-MgH2 orthorhombic (P bcn) 4.53 5.44to 5.45 4.93 to4.94 [24]
(α-PbO2 structure) 4.51to 4.52 5.43to 5.44 4.92 to4.94 [72]
Table 2.1.: Structural information of theα-,β- and γ-phase of the magnesium-hydrogen system. Only the α- and β-phase are thermodynamically stable. The lattice parameters of all phases were collected by San-Martin and Manchester in 1987 [24] and by Moser et al. in 2011 [72].
given in table 2.1. Theα-phase stores some hydrogen as a solid solution. For ambient conditions the hydrogen forms clusters in hydrogen-vacancy complexes [24]. At low temperatures (T <110 K) hydrogen is solved in the tetrahedral lattice sites and not yet trapped by vacancies. Overall, the solubility of hydrogen is very low for the pure Mg phase. Stampfer et al. gave the relations between the maximum solubilitycα→βH of the α-phase for a given temperature T [77]:
cα→βH = 107·exp
−6225 T
.
This gives a maximal solution of cα→βH =8.4·10−8H/Mg at T =297 K. When more hydrogen is added, the Mg–H system transforms into one of the hydride phases.
At ambient conditions, the transformation will be into the tetragonal β-phase. The crystal structure of the β-phase is also given in table 2.1. The β-phase is a stoichio-metric phase, meaning that the concentration of hydrogen is fixed to cβH = 2 H/Mg and no additional hydrogen can be solved in the bulk [77]. For real systems this may not be completely true; for example grain boundaries may take up different amounts of hydrogen. Borgschulte et al. also showed that oxides can destabilize MgH2 to an under-stoichiometric MgH2−δ-phase [78]. The under-stoichiometric phase was documented before by Schimmel et al. [79].
The hydride formation from a hydrogen gas atmosphere can be plotted by a pressure-composite isotherm (called p-c-T diagram, see figure 2.2) [27]. For a fixed temper-ature T the hydrogen pressure pH and the hydrogen concentration cH taken up by the metal can be measured (for example by the calculations in chapter 3.2.2). The resulting diagram shows a pressure plateau in the two-phase region of the α- and β-phase. By repeating the measurement for different temperatures, a Van’t Hoff plot is created. It shows the logarithm of the hydrogen pressure plotted as function of the reciprocal temperature. The principle is sketched in figure 2.2. The Van’t Hoff plot allows the evaluation of the enthalpy change ∆H and entropy change ∆S
hydrogen concentrationcH
Figure 2.2.: Example of a T diagram and the resulting Van’t Hoff plot. The p-c-T diagram plots the hydrogen concentration cH in the metal as function of the loading hydrogen pressurepH. The plateau of a single isotherm gives the two-phase region width at a given temperature T. Measuring isotherms at different temperatures T1 < T2 <
T3 < T4 < ... allows creating a Van’t Hoff plot of the logarithm of the loading pressure as function of the reciprocal temperature. From this one can evaluate the enthalpy - and entropy change of the phase transformation fromα- to β-phase.
during the phase transformation [80, 13]:
ln
where p0 is the standard pressure and R the gas constant. These entropy and en-thalpy changes may be different for the absorption and desorption of hydrogen, resulting in different plateau pressures. The entropy change∆S comes mostly from the dissociation of hydrogen from the H2 molecule in the gas atmosphere to the atom-ically dissolved hydrogen atom in the metal hydride. After Fukai, the standard en-tropy of hydrogen is approximatelyS0 = 130 JK−1mol−1[27]. Therefore, the entropy change can be estimated as∆S ≈ −130 JK−1(molH2)−1 for all metal-hydrogen sys-tems. For the magnesium-hydrogen system this fits well to the measured changes in entropy. Vigeholm et al. measured an entropy change of∆S =−126 JK−1(molH2)−1 [81], while Klose and Stuke measured ∆S = −146.1 JK−1(molH2)−1 [82]. Other groups measured values between these two, around ∆S = −130 JK−1(molH2)−1 (e.g. references [24, 77, 30, 83]). The enthalpy change for the absorption was found to be about ∆Habs ≈ −70 kJ(molH2)−1 in the bulk system [78, 81]. For the des-orption enthalpy change most groups measured values of approximately ∆Hdes = (−75±5) kJ(molH2)−1 in bulk systems (see references [24, 77, 78, 30, 83]).
With the literature data and equation 2.1 the absorption pressure expected in bulk
systems can be calculated for a temperature of T = 297 K: Furthermore, equation 2.1 allows estimating the necessary temperature to release hydrogen from the hydride at ambient pressure using the values for ∆Hdes and ∆S given above (⇒pH/p0 ≈1):
This shows the thermodynamic stability of the β-phase, as it needs low pressures to form at T = 20◦C or high temperatures of about 300◦C to dehydride at 1 bar hydrogen pressure.