Dynamics of porous and amorphous magnesium borohydride to understand solid state Mg-ion-conductors

Rechargeable solid-state magnesium batteries are considered for high energy density storage and usage in mobile applications as well as to store energy from intermittent energy sources, triggering intense research for suitable electrode and electrolyte materials. Recently, magnesium borohydride, Mg(BH4)2, was found to be an effective precursor for solid-state Mg-ion conductors. During the mechanochemical synthesis of these Mg-ion conductors, amorphous Mg(BH4)2 is typically formed and it was postulated that this amorphous phase promotes the conductivity. Here, electrochemical impedance spectroscopy of as-received γ-Mg(BH4)2 and ball milled, amorphous Mg(BH4)2 confirmed that the conductivity of the latter is ~2 orders of magnitude higher than in as-received γ-Mg(BH4)2 at 353 K. Pair distribution function (PDF) analysis of the local structure shows striking similarities up to a length scale of 5.1 Å, suggesting similar conduction pathways in both the crystalline and amorphous sample. Up to 12.27 Å the PDF indicates that a 3D net of interpenetrating channels might still be present in the amorphous phase although less ordered compared to the as-received γ-phase. However, quasi elastic neutron scattering experiments (QENS) were used to study the rotational mobility of the [BH4] units, revealing a much larger fraction of activated [BH4] rotations in amorphous Mg(BH4)2. These findings suggest that the conduction process in amorphous Mg(BH4)2 is supported by stronger rotational mobility, which is proposed to be the so-called “paddle-wheel” mechanism.


Neutron data analysis
Due to the high incoherent scattering cross section of hydrogen, the signal registered at the detector is predominantly from hydrogen. Generally, in the incoherent approximation, the scattering function S(Q,E) for particles performing simultaneous but independent motions can be described as: ( , ∆ ) = ( , ∆ ) ⊗ ( , ∆ ) ⊗ ( , ∆ ) i.e. it is given by the convolution of the scattering functions for translational diffusion, rotational and vibrational motions, respectively. In borohydrides, in the temperature range studied here, relevant for the analysis are jump rotational diffusion processes of the BH4 tetrahedra and fast vibrational processes, but no translational diffusion processes have been observed below 500 K, similarly to αand β-Mg(BH4)2 1,2 and LiBH4 3 . For the BH4 tetrahedra, rotations around the C2 or C3 symmetry axis are most likely, and the scattering function for this rotation can be described by: ( , ∆ ) = 0 ( ) (Δ ) + 1 1 (Γ 1 ) i.e. the sum of an elastic contribution and a Lorentzian L1(1) that is characterized by its half-width at half maximum (HWHM), 1, that is constant for all Q-values. 1 is related to the mean residence time between subsequent jumps, 1 ~ 1/, the proportionality depends on whether it is C2 or C3 rotation. Because of the normalization condition of the scattering function, it is 0 ( ) = 1 − 10 . A0(Q) is the Q-dependent elastic incoherent structure factor (EISF) of the rotation which describes the spatial extension and the symmetry of the respective local motion. A0(Q) for rotations around the C2 or C3 symmetry axis is given for both cases by: dB-H is the boronhydrogen distance and j0(x) is the zero order Bessel function. Expressions for further rotational motions can be found in Ref. 4 . Vibrational motions are generally described by: where D(Q) denotes the Debye-Waller factor. In the harmonic and isotropic approximation, it is given by ( ) = exp ( , where <u 2 > is the mean square displacement of the hydrogen atom. In the one-phonon harmonic approximation, the inelastic scattering function Sinel(Q,E) can be described by a damped harmonic oscillator. From the measurements, it appears that vibrational and rotational motions are on similar energy scales. Therefore, for the analysis, the convolution of Srot and Svib had to be considered in full. The measured S(Q,E) has thus been described using: Res is the instrument resolution function measured at 3.5 K. The inelastic contribution has been modelled using a series of (over-) damped harmonic oscillators (DHO).
To focus on the low energy excitation spectrum, the data have also been converted to the imaginary part of the dynamical susceptibility, -′′ which is calculated (for system energy gain) from 5-7 : nB is the Bose occupation factor, = [exp ( Δ ) − 1] −1 , and k is the Boltzmann constant. Data were summed over all Q-values.
For the data analysis of the measured QENS spectra, the dynamic susceptibility -′′ has been analysed to obtain the characteristic frequencies of the inelastic contributions.-′′ has been fitted by a weighted sum of four damped harmonic oscillators (DHO) (plus background) whose imaginary part of the dynamical susceptibility have the form: ED is the characteristic energy of the DHO and D the damping factor. To account for the finite energy resolution of the instrument, the spectra was convoluted with a Gaussian resolution function of the appropriate width at the characteristic energy ED obtained analytically from the instrument settings ( 8 . The obtained values for ED were taken as fixed input to reduce the number of free fit parameters in the analysis of S(Q,E).

Figure A9
Characteristic timescales τi for the Lorentzians Γi (i = 1, 2). a) the temperature range was from 310 K but below the ε-phase transition, T < Tε, b) temperature range was from 310 K to above the ε-phase transition, T > Tε.
The fraction of hindered rotations p is plotted in Fig. A10. Heating of amorphous Mg(BH4)2 yields to a decrease of activated rotations up to the crystallization temperature. During the 2 nd heat, the crystallized sample and -Mg(BH4)2 exhibit a continuous increase of activated BH4 rotations until all C2/C3 modes are active (i.e. p = 0) at a temperature that coincides with the -phase transition temperature.

Figure A10
Fraction of the hindered rotations p derived from EISF and QISF. a) the temperature range was from 310 K but below the ε-phase transition, T < Tε b) temperature range was from 310 K to above the ε-phase transition, T > Tε.
The mean square displacement (<u 2 >) of the hydrogen atoms has been calculated from where F is the scaling factor. The results for <u 2 > are shown in Fig. A11. Figure  In the beginning of the experiment, during the 1 st heating, the difference between the two samples is obvious. The γ-Mg(BH4)2 shows an continuous increase in <u 2 > while the <u 2 > of the ball milled phase is larger, and remains almost constant upon heating to 380 K. In the amorphous material, fast vibrations appear to be enhanced compared to crystalline -Mg(BH4)2 while both samples appear to be quite similar in the temperature range 380 -455 K. In the initially amorphous material, a jump like increase of <u 2 > is observed at the phase transition to the '. No such jump has been observed for γ-Mg(BH4)2, however, the latter showed signs of hydrogen loss during the QENS measurements at temperatures above 470 K and these data points will therefore not be discussed any further. The last data point of ball milled sample in the β'-modification at 485 K is also showing a decrease in <u2>. However, the QENS data did not show a decrease in intensity, which would be indicative for hydrogen loss. SR-PXD data after heating, shown in Fig. A4, confirm the β'-Mg(BH4)2 phase at room temperature while TG data in Fig. A5 show a 2-3 wt.% hydrogen loss, which can explain the loss in intensity here 9 . The internal bending and stretching modes of the [BH4] tetrahedra have been reported to be very similar for most of the polymorphs in Mg(BH4)2 and this trend is also shown in Fig. A6 10 . This was stated to result from the related phase transition of the sample. In our measurements, between RT and 450 K only linear dependencies were found for the three phase transitions (from amorphous into gamma, epsilon and beta'). A stepwise function was only found for the irreversible phase transition towards β'phase. In agreement with literature, this behaviour was reported as well in α-to β-phase transition as well as in orthorhombic-LiBH4 to hexagonal phase transition 1 . The latter is conductive 11 . Furthermore, a stepwise function was found in β-modification between 200 and 300 K 2 . Nevertheless, with those results in mind, a correlation might be imaginable between the high temperature β and β'-phase and the reported phase transitions in Mg(BH4)2-diglyme0.5.