Vibrational Spectroscopic Study of Subtle Phase Transitions in Alkali

Article pubs.acs.org/JPCC

Vibrational Spectroscopic Study of Subtle Phase Transitions in Alkali Borohydrides: Comparison with First-Principles Calculations Nina Verdal,*,†,‡ Terrence J. Udovic,† Wei Zhou,†,‡ John J. Rush,†,‡ Daniel J. De Vries,§ and Michael R. Hartman∥ †

NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6102, United States ‡ Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742-2115, United States § Radiation and Isotopes for Health, Delft University of Technology, Mekelweg 15, 2629 JB Delft, Netherlands ∥ Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan 48109-2104, United States S Supporting Information *

ABSTRACT: Neutron vibrational spectra have been measured for the alkali borohydrides KBH4, RbBH4, and CsBH4. The BH4− torsional band for each compound changes noticeably across the corresponding low-temperature phase transition previously identified using thermodynamic (NaBH4, KBH4, RbBH4, and CsBH4) and crystallographic (NaBH4 and KBH4) techniques. Previous neutron diffraction measurements show that the transitions for both NaBH4 and KBH4 are order−disorder transitions involving the relative orientations of the BH4− anions. However, diffraction measurements for both RbBH4 and CsBH4 fail to unequivocally identify long-range-ordered phases below the transitions. The present measurements of BH4− torsional as well as translational optic bands across the transitions, corroborated by first-principles phonon calculations, suggest that the subtle RbBH4 and CsBH4 transitions are indeed analogous to those observed for NaBH4 and KBH4 but of shorter range.

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structures5,12 with tetragonal symmetry in space group P42/ nmc (Figure 1). Above the transitions, they possess a cubic structure in the space group Fm3m ̅ , which is distinguished by the disordering of the BH4− hydrogen atoms onto the eight corners of a cube, each with half occupancy.13,14 This corresponds to two-fold rotational disorder of each tetrahedral BH4− anion in the lattice via 90° reorientations. For the heaviest alkali borohydrides, RbBH4 and CsBH4, evidence of transitions comes mainly from thermodynamics measurements. Stephenson et al.6 reported phase transitions in RbBH4 and CsBH4 near 44 and 27 K, respectively. Later, Gorbunov et al.15,16 using specific heat measurements determined that these transitions occur at 48 and 26.3 K, respectively. For both RbBH4 and CsBH4, diffraction measurements5 at all temperatures indicate a disordered Fm3m ̅ cubic structure as found for NaBH4 and KBH4 above their phase transitions, although additional, very weak neutron powder diffraction (NPD) peaks5 below the RbBH4 and CsBH4 transition temperatures may hint at some partial BH4− orientational ordering. Stephenson et al.6 suggested that because of the identical room-temperature disordered struc-

INTRODUCTION There have been many investigations of alkali borohydrides, previously for their utility as rocket fuel, and recently for their potential to store hydrogen as fuel for mobile applications. The borohydrides are appealing storage materials due to their large gravimetric hydrogen density. The lightest of these materials, LiBH4, has 18 mass % H. Rubidium and cesium borohydrides (RbBH4 and CsBH4) are heavier than LiBH4, NaBH4, and KBH4, resulting in lower hydrogen mass fractions, and are therefore of lesser interest for such applications. The alkali borohydrides as a whole, however, exhibit interesting dynamical and structural properties as a function of cation radius and electronegativity.1−4 For example, as the lattice parameter increases with increasing cation radius,5 the temperature of the characteristic structural transition in these materials decreases.6 For the lightest alkali borohydride, LiBH4, an atypical solid− solid phase transition occurs near 381 K, from a lowtemperature-ordered orthorhombic phase to a high-temperature hexagonal phase entropically stabilized by the c-axis rotational disordering of the BH4− anions.7 For the somewhat heavier NaBH4 and KBH4 compounds, specific heat,6,8,9 neutron spectroscopy,10 total neutron cross section,11 and diffraction measurements5,12 indicate structurally similar order−disorder phase transitions near 190 and 77 K, respectively. Below the transitions, they both possess ordered © 2013 American Chemical Society

Received: November 2, 2012 Revised: December 17, 2012 Published: January 3, 2013 876

dx.doi.org/10.1021/jp310853u | J. Phys. Chem. C 2013, 117, 876−883

The Journal of Physical Chemistry C

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Figure 1. (a) The low-temperature-ordered structure (tetragonal) unit cell of KBH4 is shown within the gray dashes, and the high-temperature disordered structure (cubic) unit cell is within the blue dashes, as viewed along the c axis. K and B atoms are identified by large purple and smaller light blue spheres, respectively. H atoms are shown in pink and white, according to BH4− orientation. (b) The two orientations of BH4−, superimposed to form a cube of hydrogen atoms, as determined by diffraction for the high-temperature-disorder structure, in which each of the hydrogen atoms has half occupancy.

The powdered precipitate was then rinsed with two separate 50 mL portions of isopropyl alcohol to dissolve any remaining Na11BH4 and any undissolved NaOH. The precipitate was collected and washed with diethyl ether to help remove excess H2O. The product was then dried under vacuum at 523 K for 24 h. The final Rb11BH4 product was verified by FTIR and XRD. Cs11BH4 was prepared by mixing a cold (253 K), saturated solution of CsOH-H2O in methanol with slightly greater than one equivalent of Na11BH4. The resulting precipitate was vacuum filtered using two 50 mL washes of cold (253 K) methanol and evacuated overnight at 393 K, modeled after a previous synthesis.18 The dry material was handled in an inertgas glovebox. The Cs11BH4 product was confirmed by XRD. K11BH4 was prepared by a similar metathesis reaction as described previously.10 Vibrational spectra of K11BH4, Rb11BH4, and Cs11BH4 (which throughout this article will be referred to simply as KBH4, RbBH4, and CsBH4) were measured at the NIST Center for Neutron Research (NCNR) on the filter analyzer neutron spectrometer (FANS)19 over energy transfers of 25 to 178 meV (1 meV = 8.066 cm−1). Horizontal collimations of 20 min before and after either a pyrolytic graphite PG(002) monochromator (for measurements between 15 and 32 meV) or a Cu(220) monochromator (for measurements between 25 and 178 meV) resulted in resolutions (full-width at half maximum, fwhm) ranging from 1.2 to 1.6 meV for energy transfers of 15 to 32 meV using PG(002) and 1.1 to 5.5 meV for energy transfers of 25 to 168 meV using Cu(220). For these measurements, each sample was contained in an aluminum foil packet arranged annularly in a He-filled cylindrical aluminum can. Sample temperatures were controlled using a top-loading, closed-cycle He refrigerator with lowpressure He gas acting as a heat-exchange medium between the

tures for NaBH4, KBH4, RbBH4, and CsBH4, similar order− disorder transitions involving BH4− anions are to be expected in all of these materials and at progressively lower temperatures, as the lattice constant increases and the interactions between BH4− anions become weaker. If changes in the relative orientations of the BH4− anions are involved in these transitions, even only among the nearestneighbor anions, then such transitions should perturb the BH4− rotational potentials. Vibrational spectroscopic methods, unlike diffraction, are local probes most sensitive to changes in the immediate environment surrounding each anion. As such they are appropriate for characterizing possible short-range-ordered arrangements of BH4− anions. Hence, changes in the BH4− torsional vibrations across the transition should be observable by neutron vibrational spectroscopy (NVS), as reported previously for NaBH4 and KBH4.10,11 We present NVS data for KBH4 and the heavier borohydrides across the transition in conjunction with first-principles phonon calculations for KBH4. Such calculations were found to be in good agreement with experiment and also proved to be useful for better understanding the nature of the analogous BH4− orientational arrangements in RbBH4 and CsBH4.

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EXPERIMENTAL SECTION Rb BH4 was prepared through a metathesis reaction between 11 B-enriched sodium borohydride (Na11BH4) and rubidium hydroxide (RbOH). RbOH was dissolved in deionized H2O to give a 9 mol/L solution. Na11BH4, synthesized from 99.83 atom % 11B-enriched boric acid as previously reported,17 was added in a 2:1 mol ratio (NaBH4/RbOH) to ensure that all of the rubidium hydroxide was converted by the following reaction: RbOH + Na11BH4 → Rb11BH4 + NaOH. The solution was shaken vigorously for ∼1 min. The resulting solution and precipitate were filtered through a fine glass frit Büchner funnel. 11

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CsBH4 is observed near 137.8 meV, and the scissor mode is found near 151.1 meV. Less intense, three-phonon processes are likely responsible for scattering between roughly 85 and 130 meV for both samples. Figure 3 displays the torsional bands of all four alkali borohydride compounds at 4 K. It is clear that the lattice

sample can and refrigerator. All data were reduced using the DAVE software package.20 First-principles calculations were performed for various structural models for KBH4 within the plane-wave implementation of the generalized gradient approximation to DFT using a Vanderbilt-type ultrasoft potential with Perdew−Burke− Ernzerhof exchange correlation (Quantum-ESPRESSO package: www.pwscf.org).21 Structural optimizations were performed with respect to atomic positions with the unit cell volume fixed at the observed 10 K value. For each model, the one+two-phonon density of states (PDOS) was calculated from the DFT-optimized structure using the supercell method with finite displacements22,23 and appropriately weighted to take into account the total neutron scattering cross sections of the elements involved. For direct comparison with the neutron vibrational spectrum, the calculated PDOS was appropriately averaged over momentum (Q)-space and convoluted with the instrumental resolution.

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RESULTS AND DISCUSSION 1. Spectroscopic Results. The RbBH4 and CsBH4 neutron vibrational spectra over the range of 15 to 168 meV at 4 K are shown in Figure 2. The spectra are very similar to those of

Figure 3. Torsional bands at 4 K for CsBH4, RbBH4, and KBH4 (from this study) and NaBH4 (from ref 10). Instrumental resolution is ∼1.2 meV fwhm. Vertical error bars denote ±1 σ.

spacing impacts the width and energy of the torsional band; i.e., the larger the lattice parameter, the more the neighboring ions are isolated from one another. This, in addition to the decreasing electronegativity with increasing cation size, causes the torsional band to shift to lower energy. The broader bands for lighter alkali borohydrides can be partially attributed to greater dispersion of the torsional modes due to stronger interactions of the closer, aligned BH4− anions. Hence, the alkali borohydride with the largest lattice parameter, CsBH4, has a torsional band centered near 36.8 meV with an fwhm of ∼1.3 meV compared with that with the smallest lattice parameter, NaBH4, which has a torsional band peaked near 44.1 meV with an fwhm of ∼7.0 meV. It is interesting to note that the increasing width of the torsional band correlates somewhat with the width of the BH4− scissor mode as a function of cation size as observed by Raman spectroscopy,18 suggesting that these may indeed be anharmonically coupled. Additional RbBH4 vibrational spectra were measured at temperatures between 4 and 65 K at 1 K intervals near the structural transition at 48 K suggested by thermodynamic measurements. A sudden change in the shape of the torsional band (shown in Figure 4) is observed between 48 and 49 K. Such a change is not observed for the BH4− bending-mode vibrations, as exemplified in Figure S1 in the Supporting Information (SI) for the fundamental mode at 139 meV. The abrupt increase in width and shift to lower energies in the torsional band upon only a 1 K increase in temperature is indicative of a change in the potential energy surface along the torsional coordinate rather than a simple increase in thermal population. This is emphasized in the inset of Figure 4b, which shows the intensity of the torsional band at 37.5 meV as a function of temperature. Between the temperatures of 48 and 49 K, the peak intensity changes dramatically and the band profile shifts to lower energy. Similarly, spectra were collected for CsBH4 over the temperature range of the phase transition between 4 and 29 K. Six of these spectra are plotted in the region of the torsional

Figure 2. Neutron vibrational spectra of RbBH4 (top) and CsBH4 (bottom) at 4 K. Vertical tick marks in these spectra represent uncertainties of ±1 σ.

NaBH4 and KBH4 previously published.10,24 RbBH4 exhibits an intense BH4− torsional band peaked near 39.1 meV and combination and overtone modes associated with this band near 60.9, 72.6, and 76.1 meV, demonstrating a slight anharmonicity in the torsion mode. A BH4− umbrella mode is observed near 138.8 meV, and a BH4− scissor mode is observed near 153.9 meV. The CsBH4 torsional band appears near 36.8 meV, with related combination and overtone modes found near 56.6, 67.9, and 71.4 meV. The umbrella mode of 878



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ments of NaBH4 and KBH4,4,10,25,26 further indicating a change in the rotational potentials across the phase transition. Again, in contrast, the BH4− bending modes are insensitive to the phase change (Figure S1 in the SI). As previously mentioned, KBH4 and NaBH4 are ordered in the tetragonal phase at low temperatures and upon heating transform to an orientationally disordered cubic phase (Figure 1). For these compounds, the disordered cubic phase results in shorter distances on average between H atoms of neighboring BH4− ions.27 Going up the alkali group, progressively higher transition temperatures are required for the entropic term to compensate for the unfavorable overlap energy as the lattice parameter decreases. Thus, as suggested in thermodynamic measurements, RbBH4 and CsBH4 undergo transitions at temperatures lower than those for NaBH4 and KBH4, although these “transitions” are not accompanied by a clearly observed change to a long-range-ordered structure. The behavior with temperature of the NaBH4 torsional band (shown previously11) is similar to that of KBH4 but less pronounced, presumably due to thermal population and changes in the more rapid rotational dynamics at and above the higher transition temperature near 190 K. In conjunction with the torsional bands, the lower-energy translational optic bands involving the relative motions of the alkali metal cations and BH4− anions also proved to be sensitive to the nature of the BH4− reorientational order. A comparison of these lower-energy bands below and above the phase transition as well as at 4 K is displayed in Figure 5 for KBH4, RbBH4, and CsBH4. Noticeable changes in the translational bands across the transition, particularly for the lower-energy band, are evident for KBH4 and to a lesser extent for RbBH4. For CsBH4, the relatively weaker anion−anion interactions resulted in a significantly narrower lower-energy band. Hence the 1.2 meV fwhm instrumental resolution was not good enough to satisfactorily resolve analogous changes in this band. 2. Comparison with First-Principles Calculations. A. KBH4 Phonon Calculations. In an attempt to shed further light on the nature of the dynamical transitions observed in our spectroscopic study, representative first-principles phonon calculations for KBH4 were performed to predict the behaviors of the vibrational bands observed across the phase boundary for the borohydrides in Figures 4 and 5 and Figure S1 in the Supporting Information. Starting with the low-temperatureordered structure (with P42/nmc symmetry) from Renaudin et al.5 (see Figure 1), the atomic positions were optimized for a = 4.684 Å and c = 6.571 Å, assuming a tetragonal unit cell. (N.B., a√2 = 6.624 Å, is only 0.8% larger than c, indicating that the unit cell is very close to cubic, like the higher-temperaturedisordered structure, with respect to the packing of the BH4− anions and K+ cations.) The perfectly ordered structure consists of ab planar sublattices of identically oriented BH4− anions, with alternating orientations for successively stacked sublattices in the c direction. The calculated, Q-space-averaged PDOS from this optimized ordered KBH4 structure over a broad energy range is shown in Figure S2 in the SI compared with the 4 K neutron vibrational spectrum, indicating good agreement between theory and experiment. The torsional band is shown in more detail in Figure 6 (blue line). Although first-principles calculations are straightforward when dealing with a perfectly ordered structure, using such a periodicity-based method to determine the PDOS of a statistically disordered structure is computationally much more difficult. In an attempt to estimate the effect of BH4−

Figure 4. Neutron vibrational spectra of the torsional band above (red) and below (blue) the phase transition temperature for (a) KBH4 between 70 and 81 K at 1 K intervals (inset: the intensity at 41.6 meV as a function of temperature), (b) RbBH4 between 46 and 55 K (inset: the intensity at 39.5 meV as a function of temperature), and (c) CsBH4 between 25 and 28 K (inset: the intensity at 36.7 meV as a function of temperature). The horizontal line represents the instrumental resolution fwhm. Vertical error bars denote ±1 σ.

band in Figure 4. Despite the more compromising convolution of a narrower PDOS with the 1.2 meV fwhm instrumental resolution, the change in shape and intensity of this mode across the phase transition is still very pronounced, similar in character to what is observed for RbBH4. The intensity at 36.7 meV as a function of temperature is presented in the inset of Figure 4c. An abrupt intensity change occurs very near 26.3 K, the temperature first reported16 for the phase transition in CsBH4. As in the case of RbBH4, the bending modes are essentially unchanged (Figure S1 in the SI) at temperatures above and below the transition. Interestingly, KBH4 behaves in an analogous way to CsBH4 and RbBH4. Aided by the relatively good instrumental resolution compared with the broadness of the KBH4 torsional band, the shape of the band changes even more dramatically across the transition, as is shown by the temperature dependence in Figure 4. An increase in the dynamic reorientational disorder of the BH4− groups has been observed in NMR and quasielastic neutron scattering (QENS) measure879



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larger (containing 32 BH4− anions each) than the cubic unit cell (in Figure 1) of the high-temperature-disordered phase containing four BH4− anions. Unfortunately, our computations were limited to supercells of this size, which already contained a total of 192 atoms. Nonetheless, even within 2 × 2 × 2 supercells, one can simulate sufficient BH4− orientational disorder to estimate its effect on the torsional band upon performing a periodic DFT phonon calculation. This supercell approach proved useful in previous studies modeling the PDOS of (disordered) nonstoichiometric palladium hydrides.28,29 Model I started with the structural parameters and BH4− orientations of the low-temperature-ordered phase. Next, only one of the 32 BH4− anion orientations was flipped (at the center of the supercell), leading to P42̅ m symmetry. The structure was then optimized. Although periodic, this model attempted to represent, albeit somewhat simplistically, the effect of introducing a diluted orientational defect into the otherwise low-temperature-ordered BH4− sublattice. It is apparent from Figure 6 that the resulting torsional band (black line) softened slightly compared with the calculated band of the low-temperature-ordered structure (blue line), transferring some intensity from higher to lower energies. The calculated static internal energy of the optimized model I structure was only 25.0 meV/2 × 2 × 2 supercell greater than that of the low-temperature-ordered supercell. In other words, it required ∼25 meV to create a single orientational defect in the ordered supercell. Model II was similar to model I except that an additional BH4− anion (besides the center anion) was flipped at the corner of the supercell, resulting in a total of two anions flipped out of 32. The symmetry remained the same (P4̅2m), and the structure was optimized. The static internal energy of the optimized structure for model II was 48.3 meV greater than the ordered structure. The doubling of the defects resulted in a “doublet” torsional band (Figure 6, cyan line) with an additional shift of some of the higher energy intensity present for model I to lower energies. Because of limited computational resources, no attempt was made to modify this model by placing the additional defect in other positions of the 2 × 2 × 2 supercell. It was assumed that the results would be qualitatively similar. Model III also started with the structural parameters and BH4− orientations of the low-temperature-ordered phase. Next, four BH4− anions out of eight in the middle (z = 0.5) ab-planar anion layer (for a total of 4 out of the 32 anions in the supercell) were flipped at the boron fractional positions of x = ±0.25 and y = ±0.25 with respect to the center of the supercell, again yielding a P4̅2m-symmetric structure. The calculated torsional band (Figure 6, orange line) after optimization was even more softened and attenuated at higher energies than model II. The static internal energy was 107.3 meV higher than that of the low-temperature-ordered structure. Finally, model IV attempted to emulate a statistically disordered, high-temperature cubic structure as was suggested by KBH4 diffraction studies. Starting with a fully cubic supercell with the same cell volume as model I and all BH4− anions identically aligned, 16 of the 32 BH4− anions were chosen using a randomizing routine and orientationally flipped. This led to P1 symmetry, meaning that each of the 192 atoms in the unit cell was allowed to relax during the optimization. A total of 10 different random arrangements were optimized, yielding an average static internal energy 180.8 meV greater than that for the low-temperature-ordered structure. The torsional bands

Figure 5. Neutron vibrational spectra for (a) KBH4 at 4 (gray), 75 (blue, below the phase transition temperature, Tpt), and 77 K (red, above Tpt); (b) RbBH4 at 4 (gray), 40 (blue, below Tpt), and 50 K (red, above Tpt); and (c) CsBH4 at 4 (gray), 24.3 (blue, below Tpt), and 26.8 K (red, above Tpt). Vertical error bars denote ±1 σ.

Figure 6. Calculated, Q-space-averaged, torsional bands for models incorporating increasing numbers of BH4− orientational defects: The low-temperature-ordered structure (blue), model I with one of 32 BH4− anions flipped (black), model II with two of 32 anions flipped (cyan), model III with four of 32 anions flipped (orange), model IV with random orientations of all BH4− anions (red), and model V in which all BH4− anions are aligned in a cubic cell (gray).

orientational disorder on the PDOS, various supercell models were constructed (Figure S3 of the Supporting Information), starting with quasi-cubic (models I, II, and III) or cubic (models IV and V) unit cells. These supercells were 2 × 2 × 2 880



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were calculated for the first two random arrangements and found to be almost identical to each other. The average of these two torsional bands (Figure 6, red line) showed a markedly softer, broader distribution of torsional energies compared with that of the low-temperature-ordered structure, as experimentally observed. Figure 7 compares the calculated torsional bands

might be expected to shift a portion of the original band from higher to lower energies, as observed. In addition to the torsional bands, the calculated and measured PDOSs for the low-energy BH4− translational optic modes of the low-temperature-ordered and statistically disordered KBH4 structure models are worthy of comparison. As was the case for the torsional bands, the calculated translational bands for the low-temperature-ordered and model IV structures differ from one another in shape, and both show good qualitative agreement with the measured translational bands both below and above the phase transition in Figure 8.

Figure 7. (a) The observed neutron vibrational spectra of the KBH4 torsional band above (red) and below (blue) the phase transition temperature. (b) The calculated, Q-space-averaged torsional band for the low-temperature-ordered structure (blue line) and the randomly disordered model IV (red). Vertical error bars in panel a denote ±1 σ.

for the low-temperature-ordered structure and model IV with the measured torsional bands of the ordered and disordered KBH4 structures, more clearly demonstrating the good qualitative agreement between theory and experiment. For the alkali-metal borohydrides of this study, each BH4− anion has 12 nearest-neighbor anions surrounding it in a cuboctahedral arrangement (Figure S4 in the Supporting Information). In the low-temperature-ordered structure, only one-third (four) of these anions have the same orientation as the central BH4− anion; the other two-thirds (eight) have the opposite orientation. In the statistically disordered structure, the relative orientations of the 12 nearest-neighbor anions will be, on average, evenly split (six and six). Hence, it seems reasonable to speculate that the characteristic softening of a portion of the torsional band across the order−disorder transition is partially a manifestation of the increase in the number of nearest BH4− anion neighbors with like orientations in the disordered phase. This speculation was explored by calculations for an additional model V, starting with a tetragonal unit cell (with a√2 = b√2 = c) with the same cell volume as the unit cell of the low-temperature-ordered phase but with all BH4− anions identically oriented, thus leading to F4̅3m symmetry. The structure was then optimized, yielding a static internal energy 819.8 meV higher than that of the lowtemperature-ordered structure. The resulting torsional band (see Figure 6, gray line) shows a dramatic softening compared with that of the low-temperature ordered structure. It is clear that this mode softening is indeed largely a consequence of identically aligned BH4− neighbors and that any increase in these types of neighbors at the expense of oppositely aligned neighbors (as is the case to varying degrees for all models I−V)

Figure 8. (a) Observed neutron vibrational spectra for KBH4 at 75 (blue, below the phase transition temperature, Tpt) and 77 K (red, above Tpt) and (b) the calculated, Q-space-averaged, translational optic bands for the defect-free, low-temperature ordered structure (blue); and the randomly disordered model IV (red). Vertical error bars in panel a denote ±1 σ.

B. Implications for RbBH4 and CsBH4. It is apparent that the calculated KBH4 torsional band transformation from the lowtemperature-ordered structure to model IV (the statistically disordered cubic phase) is qualitatively very similar to that observed experimentally for all three borohydrides in Figure 4, suggesting that they are all undergoing similar order−disorder transitions. Moreover, as previously mentioned, the observed low-energy translational optic band (in Figure 5) for RbBH4 displays similar characteristics and transition behavior to that of KBH4, again implying structural similarities. (Unfortunately, although some minor changes across the phase transition were also observed for the analogous CsBH4 band in Figure 5, it was not measured with adequate resolution to enable a more clearcut comparison with KBH4.) Unlike the marked sensitivity of the calculated PDOS associated with the BH4− torsional and translational modes to the details of the BH4− sublattice orientations, the calculated PDOS of the BH4− bending modes (as seen in Figure S2 in the Supporting Information for the low-temperature-ordered KBH4 structure) was found to be largely insensitive to the incorporation of disorder as in models I−IV. This is consistent with what we experimentally observed 881



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c direction); therefore, Sconf is assumed to be zero. For the randomly disordered structure, we assume that KBH4 behaves orientationally like a regular solution (i.e., ideal Sconf values despite potentially nonzero internal energy values due to orientational disorder). Thus, for a dynamically disordered lattice with xA and xB number fractions of A-type and B-type BH4− orientations, Sconf = −kB (xA ln xA + xB ln xB). For the randomly disordered model, xA = xB = 0.5 and Sconf = kB ln 2 = 1.911 meV/K/2 × 2 × 2 supercell. If we assume that T = 76.2 K, then TSconf = 145.6 meV. From the optimized structure and PDOS calculated for model IV, the first three terms in eq 1 add up to 193.6 meV/2 × 2 × 2 supercell. The resulting ΔF (193.6 − 145.6 = +48.0 meV) is positive. For a spontaneous transformation, the free-energy change must be zero. According to the calculation, this requires a transition temperature near 100 K, around 30% higher than observed. This is another way of saying that the calculations overestimate the nonconfigurational free-energy increase with disorder by ∼30%. This is not unreasonable considering the limitations inherent in using the DFT method. Moreover, we have chosen model IV configurations using a relatively small supercell to calculate the average free-energy change. It is probable that with a much larger supercell there are numerous disordered configurations with somewhat lower static internal energies, leading to transition temperatures closer to the observed value. Such calculations are beyond the scope of the present study. To complicate matters further, the early heat capacity studies of KBH48,30 measured an apparent entropy change across the transition that is slightly more than half of that expected for fully random disorder. Furukawa et al.8 suggested that this was an indication of some persistent local ordering, that is, reorientational pairing of BH4− anions just above the phase transition. This was in contrast with the corresponding entropy change observed for NaBH4,9,27 which was more in agreement with full disorder. A configurational entropy change close to that reported for KBH4 corresponds to randomly flipping four out of 32 BH4− anions in the 2 × 2 × 2 ordered supercell. For this case, we define A-type BH4− orientations as corresponding to the anions of the perfectly ordered lattice and B-type BH4− orientations as corresponding to those anions that are flipped. Therefore, xA = 0.875, xB = 0.125, and Sconf = 1.039 meV/2 × 2 × 2 supercell. Model III (with 4 anions flipped out of 32 and already closely approaching the torsion band observed above the phase transition) is the most appropriate model we have to estimate the nonconfigurational terms of the free energy. Using this model, the first three terms in eq 1 add up to 140.5 meV/2 × 2 × 2 supercell, and the resulting ΔF = 140.5 − 79.2 = +61.3 meV. Now the calculations overestimate the nonconfigurational free-energy increase with disorder by ∼77%. Again, on the basis of the DFT limitations and the probable existence of multiple lower-energy configurations, this may not be unreasonable. In closing, although the fully random model IV is in overall better agreement with respect to the observed torsional band behavior and calculated free-energy changes, one cannot fully discount the previous configurational entropy measurements alluding to the presence of partial orientational order above the phase transition for KBH4, RbBH4, or CsBH4.

across the phase boundary for KBH4, RbBH4, and CsBH4 in Figure S1 in the Supporting Information. Hence, the totality of the first-principles phonon calculations bolsters our contention that the PDOS behaviors for RbBH4 and CsBH4 across the phase boundary reflect analogous order− disorder transformations as for KBH4, at least on a short-range scale largely unobservable by diffraction techniques. The main differences lie in the narrowing of both the torsional (and translational) band widths and the magnitude of the accompanying mode softening for RbBH 4 and CsBH 4 compared with KBH4. This is dictated by the weaker anion− anion interactions for the heavier alkali compounds due to their relatively larger nearest-neighbor anion−anion separations. The lack of clear diffraction evidence of a structural transformation at the low-temperature order−disorder transition for the heavy borohydrides is quite unusual. It is possible that the BH4 realignments required among independent short-range-ordered domains for coalescence into a long-range-ordered structure are kinetically impeded at the low (≤50 K) transition temperatures. Nonetheless, the first-principles phonon calculations as exemplified for KBH4 provide rough guidance to the amount of disorder that can be tolerated in the low-temperature “ordered” phases of RbBH4 and CsBH4. The progression of the calculated torsional band upon increasing the amount of disorder to the low-temperature-ordered structure (i.e., models I−III) as shown in Figure 6 suggests that these “ordered” RbBH4 and CsBH4 phases cannot possess more than ∼3% orientational defects as present for model I (with one of 32 BH4− anions flipped). Anything more disordered than this (such as model II with its 6% defects) leads to poorer qualitative agreement with the measured low-temperature bands, particularly at higher energies. Moreover the abrupt torsion-band changes upon crossing the transition would be largely diminished if the torsional band of the low-temperature phase was already softened by the presence of this or higher levels of orientational defects (e.g., model III with its 12% defects). C. Free-Energy Considerations. It is also of interest to estimate the KBH4 free-energy change upon transitioning between the low-temperature-ordered structure and the disordered model IV. Assuming that the final disordered state is in equilibrium, dynamically sampling all of configurational space, one must consider the additional entropic stabilization due to orientational disorder for counterbalancing the increase in internal energy. As such, the free energy F can be defined as: F = Estatic +

∑ 1 ℏωi + kBT ∑ ln(1 − eℏω /k T ) i

i

− TSconf

2

B

i

(1)

where Estatic is the static internal energy, ℏωi are the vibrational normal mode energies, kB is Boltzmann’s constant, T is the temperature, and Sconf is the configurational entropy. The second and third terms are the vibrational energy contributions to the free energy, the second associated with the zero-point energies of the phonons and the third associated with the energy changes upon raising the temperature from 0 K to T. The last term is the contribution from the configurational entropy due to orientational disorder. For the ordered structure, although there are two possible orientations for each BH4− anion (which can be labeled for convenience as Atype and B-type orientations), there is only one configuration of anion orientations (i.e., alternating A and B-type layers in the

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SUMMARY AND CONCLUSIONS Neutron vibrational spectra were measured for KBH4, RbBH4, and CsBH4, at temperatures just above and below the proposed structural transitions near 76.2 (KBH4), 48.5 (RbBH4), and 26.3 K (CsBH4), respectively. In each case, the shapes of the 882



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torsional (and to a lesser extent translational) bands observed with NVS changed abruptly and in a similar manner across the phase transition temperature. This indicates abrupt changes in the torsional (and translational) potential energy for each borohydride and offers spectroscopic confirmation of a transition previously detected with thermodynamic measurements.6,15,16,31 The results for the heavier borohydrides are analogous to the order−disorder transition of KBH4 across its crystallographic phase transition, and such behavior is found to be consistent with first-principles phonon calculations. Because no clear-cut change in BH4− site symmetry marking a longrange order−disorder transition is observed for RbBH4 and CsBH4 by NPD5 (although it is possible that higher-resolution diffraction measurements would reveal a small change in structure below the transition), we suggest that the BH4− anion ordering is of a more short-ranged nature. If the coherence length of the ordering is well below ∼10 nm, then any splitting or addition of Bragg peaks due to local ordering and concomitant tetragonal distortion may be more difficult to discern by a diffraction measurement. Yet it is clear from a comparison of the spectroscopic and computational results that the coherence length of the short-ranged order in RbBH4 and CsBH4 must be at least as large as a 2 × 2 × 2 supercell (with only minor orientational defects) to enable the abrupt spectroscopic changes observed across the phase boundary. In short, the ordering process associated with the transition in RbBH4 and CsBH4 is obviously more subtle than that observed for the lighter alkali borohydrides, and any expected transformation from short-range to long-range order may be kinetically impeded by the relatively low transition temperatures. For such short-range-ordered lattices, application of pair distribution function analysis across the phase transition might provide additional insights concerning the local atomic arrangements and should be pursued. Moreover, to more closely model by first-principles calculations an incoherent arrangement of short-range-ordered domains would require much larger supercells than currently used. Nonetheless, with additional computational resources, this may also be an avenue to explore further. Finally, these observations and our combined methods may apply to a range of molecular solids with low-temperature phase transitions, where long-range phase changes could be impeded by dynamical “freezing”.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The observed and calculated PDOS for the low-temperature ordered KBH4. Also shown are the borohydride bending mode vibrational peaks at temperatures above and below the transition, a schematic of the 2 × 2 × 2 KBH4 supercell, and a schematic of the cuboctahedral arrangement of nearestneighbor BH4− groups. This material is available free of charge via the Internet at http://pubs.acs.org.

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Article

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by the DOE EERE through grant no. DE-EE0002978. 883


Vibrational Spectroscopic Study of Subtle Phase Transitions in Alkali

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