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PHYSICAL REVIEW B 88, 220102(R) (2013)

First-principles free energy calculations of the structural phase transition in LiBH4

with I, Cl, Na, and K substitution

N. Bernstein and M. D. JohannesCenter for Computational Materials Science, Naval Research Laboratory, Washington, DC 20375, USA

Khang HoangCenter for Computationally Assisted Science and Technology, North Dakota State University, Fargo, North Dakota 58108, USA

(Received 12 September 2013; published 20 December 2013)

LiBH4 is a fast ionic conductor in its high-temperature phase, which is stabilized at room temperature byvarious chemical substitutions, making it a potential solid electrolyte material for Li-ion batteries. Using first-principles variable-cell-shape molecular dynamics simulations, we reproduce the experimentally observed low-and high-temperature structures. Using the height of a structure-factor-like peak as a collective coordinate, wecalculate the free energy differences between the two structures as a function of temperature and substitutionalion concentration. We get good agreement with experiment for I, Cl, and Na, and predict that K is even moreeffective for lowering the critical temperature. Decomposition of the free energy into enthalpy and entropy revealsthat the mechanism driving this lowering varies among substitutional elements. Calculating the full free energy,rather than simply the enthalpy, is therefore crucial to understanding how chemical substitution stabilizes thehigh conductivity phase.

DOI: 10.1103/PhysRevB.88.220102 PACS number(s): 64.70.K−, 82.45.Gj, 71.15.Mb, 71.15.Pd

Lithium borohydride (LiBH4) was originally proposed asa hydrogen storage material,1 but has recently shown promiseas a Li-ion battery solid electrolyte due to its high Li-ionconductivity of over 10−3 S/cm and chemical stability againstLi-metal anodes.2 The complex hydride undergoes a structuralphase transition from the low-temperature (LT) orthorhombicphase to the high-temperature (HT) hexagonal phase around383 ± 2 K.3 The high ionic conductivity is only observed in theHT phase, which in pure LiBH4 is only stable at temperaturesabove the useful operating range.4 However, mixing LiBH4

with LiI, leading to a material with I substituting for BH4 units,stabilizes the HT phase at lower temperatures.5,6 SubstitutingCl or Br for BH4 has also been shown to increase ionicconductivity, presumably by stabilizing the HT phase,5 andthere is preliminary evidence that substituting Na for Li hasthe same effect.7 The fact that I and Na, with larger ionic radii,as well as Cl and Br, with smaller radii, all stabilize the HTphase means that the composition effect cannot be explainedby a simple size effect.8 Computer simulations using first-principles density functional theory (DFT) have been used tostudy LiBH4 in its pure and substituted forms, taking advantageof DFT’s ability to accurately describe the energetics of thesechemically complex materials. Static simulations have shownthat the experimental HT structure has a higher enthalpy atT = 0 K than the LT structure.6,9,10 Dynamic simulationshave been used to study finite temperature properties of theHT structure, such as split occupancy of the Li and B sitesand frequent rotation of the BH4 units,11,12 and Li diffusionpathways and rates.13,14

Here we present a variable-cell-shape DFT moleculardynamics (MD) simulation and free energy calculation ofthe finite temperature structural phase transition itself, andits dependence on I, Cl, Na, and K concentrations. We showthat the computationally obtained structures and transitiontemperatures as a function of I very closely mirror what isexperimentally observed. We also find, again in agreement

with experiment, that Cl and Na stabilize the HT phase.We investigate the efficiency of each substitutional elementin lowering the transition temperature and show that thereis a variation in the underlying mechanism by which eachstabilizes the high-temperature phase. We predict that K substi-tution for Li gives the largest change in free energy differencefor a given substitutional concentration for stabilizing the highconductivity phase of LiBH4.

We use MD simulations at constant temperature andvariable unit-cell shape and size,15 with forces and stressesfrom DFT, to simulate the time evolution of a LiBH4 supercell.We use the MD software from the LIBATOMS/QUIP softwarepackage,16 with DFT forces and stresses from the ViennaAb Initio Simulation Package (VASP) version 5.3.3.17,18 Timeevolution is computed using the velocity Verlet algorithmwith a 0.5 fs time step, to capture the fast dynamics of theB-H bonds. DFT calculations use the Perdew-Burke-Ernzerhof(PBE) functional,19,20 a cutoff energy of 400 eV, and �-pointBrillouin zone sampling. MD simulations are carried outin the NσT ensemble using a Langevin thermostat with adamping time of 100 fs, a barostat allowing unit-cell sizeand shape deformations with Langevin thermalization and adamping time of 1000 fs, and an applied stress of 0 GPa.For our initial conditions we use a 2 × 2 × 2 supercell ofthe experimental LT structure,21 with 32 formula units (f.u.)and a total of 192 atoms. For I-substituted structures wereplace randomly selected BH4 units, i.e., Li(BH4)1−xI IxI , withI atoms. Cl substitution similarly occurs on the BH4 site,whereas Na and K substitutions occur on the Li site. Boltzmannweighted averaging over multiple realizations of the randomchoice of sites would be more accurate, but is not feasiblecomputationally.

We begin by simulating the evolution of pure LiBH4,initially in the LT experimental structure. At 300 K, ourcalculations reproduce the experimental lattice parametersextremely accurately, with less than 1% error in volume

1098-0121/2013/88(22)/220102(5) 220102-1 ©2013 American Physical Society

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TABLE I. LiBH4 experimental (Ref. 21) and mean MD latticeconstants ai (in A) and unit-cell volume V (in A3) for the simulatedsupercell. MD results for LT structure are averaged over a T = 300 Ktrajectory from 5 to 32 ps, and results for HT structure are averagedover a T = 400 K trajectory from 25 to 40 ps.

a1 a2 a3 V

Expt. room T 14.4 8.87 13.6 1733MD 300 K 14.4 8.97 13.4 1727Expt. 408 K 13.9 8.55 14.8 1761MD 400 K 14.0 8.71 14.5 1761

(Table I). At 400 K, we find that the structure remains close tothe initial one for about 20 ps, after which the lattice constants,plotted in Fig. 1, change to new values (the volume and stressare plotted in the Supplemental Material22). The new latticeconstants are in excellent agreement with the experimentalHT lattice constants at similar temperatures, listed in Table I.Overall the new structure, visualized in Fig. 2, is very similarto the experimental HT structure21 with some additionalfluctuations. In particular, the shifts along the (0001) directionwhen viewed along the (1100) or (1120) directions aredisordered in MD, so that the experimental positions appearto agree with the approximate mean MD position.

Importantly, we reproduce the experimentally observed lowT to high T shift in the Li-B configuration. As can be seen inthe top row of Fig. 2, each “double row” has alternating Li/Bions in the first 10 ps (LT), whereas after the transition to the HTphase, this alternation is replaced by random occupations ofsites just above and below the center of each row. This changein pattern motivates our choice to use the drop in the intensityof a structure-factor-like peak Aq, which gauges the alternationof the Li/B atoms, as a collective coordinate for the structuraltransition. The parameter is defined by

Aq =∣∣∣∣∣

1

NLi

∑

n is Li

exp(i 2π q · rn)

∣∣∣∣∣ , (1)

0 20 40t (ps)

10

12.5

15

|a|(A

)

0

0.5

1

A4,2

,0

a1a2a3

A

FIG. 1. (Color online) LiBH4 lattice constants a1, a2, a3, andstructure-factor-like peak height A4,2,0 for LiBH4 as a function of timeduring a T = 400 K, zero stress variable-cell-shape MD simulation.Structure begins at the experimental LT structure, but at t ≈ 20 psit transforms to a HT structure characterized by different latticeconstants and A4,2,0 value.

MD t=30 ps exper. 400 K

(0001)

(1100)

(1120)

(0001)

(1100)

(1120)

(0001)

(1100)

(1120)

MD t=10 ps

FIG. 2. (Color online) Structures from T = 400 K MD simula-tion and experiment (Ref. 21). MD configurations at t = 30 ps (aftertransition to HT structure, left), experimental HT structure (center),and MD configuration at t = 10 ps (before transition to HT structure,right).

where q is a vector, rn is a the position of Li atom n in latticecoordinates, and the sum is carried out over all NLi Li atoms inthe simulated supercell. The q = (4,2,0) component is chosento pick out the Li/B alternation period in the supercell we use.22

A similar collective coordinate was also recently developedindependently in a different context by Pedersen et al.23

As Fig. 1 shows, the small change in lattice constants isaccompanied by a large and abrupt change in A4,2,0 from∼0.75 to ∼0.1. The difference in values that A4,2,0 takes in thetwo different structures is much larger than the scatter in valueswithin each structure, making this quantity a good collectivecoordinate for distinguishing the two structures and for freeenergy calculations.

To compute free energy differences we use the umbrella in-tegration (UI) method,24 where the gradients of the free energy∂F/∂A with respect to a collective coordinate A are sampledfrom a biased simulation, and then numerically integrated withrespect to A. We use A4,2,0 as the collective coordinate. Moredetails are provided in the Supplemental Material.22 For eachtemperature and I, Cl, Na, or K concentration we evaluate∂F/∂A at values of A from 0.0 to 1.0 in intervals of 0.125,as well as at 0.0625 and 0.6875. We sample the gradientsfor at least 25 ps per trajectory, skipping the initial 2.5 psof each trajectory to allow for equilibration, and compute thefree energy by numerically integrating the gradients with thetrapezoidal rule. Note that since the bias potential only affectsa single peak in Aq, it is possible, but rarely observed, forthe system to minimize the bias potential (i.e., approximatelysatisfy the restraint) without actually being in the HT structure.In our simulations this only happens for part of the trajectory

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0 0.5 1A4,2,0

-0.2

0.0

0.2

0.4

F(e

V)

350K375K400K425K450K

-2.0

0.0

2.0∂F

/∂A

(eV

)

FIG. 3. (Color online) Gradient of the free energy (top) andcorresponding integrated free energy (bottom) for pure LiBH4 asa function of collective coordinate A4,2,0, for a range of temperatures.Note that energy differences along each curve are meaningful, butdifferences between curves are not.

for one set of conditions (pure material, T = 350 K), and weexclude that portion of the trajectory.

The results of the UI calculations of the derivative ofthe free energy and the corresponding integrated free energyprofiles for pure LiBH4 are plotted in Fig. 3. It is important tonote that the variation of F as a function of A at a giventemperature is meaningful, but differences between curvesat different temperatures are not. At the lowest temperaturesthere are two minima: a deeper one at a high value of A4,2,0,corresponding to the LT structure, and a shallower one at alow value of A4,2,0, corresponding to the HT structure. Asthe temperature increases, the free energy at low A4,2,0 (HTstructure) decreases relative to high A4,2,0 (LT structure), untilthe two minima are equally deep, and then switch order. Atsufficiently high temperatures the minimum at high A4,2,0 (LTstructure) disappears, and only the HT structure is stable.

We define �F as the difference between the free energiesat A4,2,0 ≈ 0.125 and A4,2,0 ≈ 0.75, which are at or very nearthe positions of the two minima in every case (temperatureand substitutional concentration) where both minima arepresent. The value of �F (T ,x), where x is the substitutionalconcentration, is plotted in Fig. 4. The temperature where�F is equal to zero is the temperature at which the freeenergy ordering of the two phases switches, and is thereforeby definition the structural transition temperature Ts . At each Iconcentration �F decreases with increasing temperature andas the I concentration increases, Ts decreases. At T = 350 K,12.5% Na substitution lowers the free energy difference by thesame amount as 12.5% I, while 12.5% Cl substitution has abouthalf the effect, and 12.5% K substitution has about double theeffect. A plot of Ts extracted from the x axis intercepts of thevarious curves in Fig. 4, as a function of I concentration xI isshown in the inset. The transition temperature goes down fromabout 415 K for xI = 0 to 320 K at xI = 25%. The simulationresults are consistently about 35 K above the experimentallymeasured transition temperature. It is likely that the free energycalculation suffers from some systematic shift because some

xI (%)

Ts

(K)

300 400T (K)

−10

−5

0

5

ΔF

(meV

)

pure12.5% I25% I12.5% Na12.5% Cl12.5% K

0 25

350

400

450simul.exper.

FIG. 4. (Color online) LiBH4 free energy differences per formulaunit �F between HT and LT structures as a function of systemtemperature and substitutional concentration. The temperature atwhich a given curve intersects �F = 0 eV defines the transitiontemperature at that concentration. Inset: Li(BH4)1−xI IxI transitiontemperature, extracted from zero intercepts in the main panel, asa function of I concentration. Experimental data is from Table I inRef. 6.

unbiased variable-cell-shape trajectories, for example, the pureLiBH4 at T = 400 K shown in Fig. 1, spontaneously transformto the HT structure despite being slightly under the calculatedtransition temperature. The sample error bars suggest that theduration of the trajectories is sufficient, but other sources oferror such as the finite number of points (choices of restraintcenters) for the numerical integration of the gradients, and theuse of a single configuration to represent random disorder,remain.

We can estimate the effect of Cl, Na, and K substitutionson transition temperature if we assume that the same lineardependence of �F vs T seen in Fig. 4 also holds. Under thisassumption, Na substitution would lead to the same transitiontemperature shift as an equal concentration of I, while thesame concentration of Cl would lead to shifts that are halfas large, and K substitution to shifts that are twice as large.Experimental results5–7 are consistent with these estimates forCl and Na. Our results for K are a prediction which could betested in future experiments.

Our UI simulations give the free energy as a functionof collective coordinate A4,2,0 for each temperature andsubstitutional concentration. The free energy differences �F

can, by definition, be decomposed into enthalpy differences�H and entropy differences �S, related through �F =�H − T �S. While it is straightforward to directly computethe enthalpy difference from the simulated DFT potentialenergies, it is unfortunately not feasible to directly compute theentropy differences for this material system. The conventionalapproach, using the vibrational density of states to computethe entropy25 which is accurate if the atomic motion isapproximately harmonic, will not work for this material,

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12.5%K

12.5%Na

pure 12.5%I

25%I

12.5%Cl

composition

0

ΔF

(meV

/f.u

.)

25

50

ΔH

orT

ΔS

(meV

/f.u

.)ΔF ΔH TΔS

FIG. 5. (Color online) Differences at T = 350 K between HTand LT structures of free energy �F (left axis), enthalpy �H

(right axis), and derived entropy contribution to free energy T �S ≡�H − �F (right axis), in meV/f.u. as a function of chemicalsubstitution.

since, as seen in previous simulations,11,12 the motion ofthe atoms is very anharmonic. The Li and B positionsshow a bimodal position distribution, far from the Gaussiandistribution expected for a quasiharmonic system. The B-Hbond vibrations are approximately harmonic, but especially inthe HT structure the four H atoms rotate about the B center ina completely anharmonic manner. It is nevertheless possibleto calculate T �S ≡ �F − �H (note that the entropy andenthalpy differences are therefore correlated), and we plotthese three quantities for T = 350 K in Fig. 5. The enthalpydifference of 50 meV/f.u. for the pure material is in excellentagreement with experimental results3 of 53 ± 9 meV. Our useof MD simulations is important for achieving this accuracy:Static DFT calculations on the experimental structures, whichcannot be fully relaxed because the HT structure is not stable atT = 0 K, give higher values of 77 meV (Ref. 6) to 128 meV.9

As can be seen by the slopes of the lines in Fig. 5,the effect of adding I is to strongly reduce the enthalpydifference between the HT and LT structures, while less rapidlydecreasing the entropy difference. The net and somewhatsurprising result is that the entropy term, which drives thephase transition, changes upon substitution so as to counteractthe tendency of I to lower the free energy and therefore thetransition temperature. K substitution affects the free energyand enthalpy similarly to I, but more strongly at a givenconcentration. Na and Cl substitutions, on the other hand,

behave quite differently. For Na and Cl the enthalpy differencereduction is much lower than that resulting from an equalconcentration of I, but the entropy change is also much smaller,so it does not counteract the enthalpy change as much. Thenet effect is that Cl is about half as effective and Na equallyeffective as I at lowering the free energy difference, a resultthat is essentially uncorrelated with the substitution inducedenthalpy difference. It has previously been noted that a simplesize effect of the substituting element is not consistent withthe ability of both larger (Na, I, K) and smaller (Cl, Br) ionicradius species to stabilize the HT phase.8 Our results show thateven the separate enthalpy and entropy contributions are notsimply correlated with the substituting species’ ionic radius.An explicit calculation of the free energy such as we presenthere is required to explore this system and its dependence oncomposition.

In conclusion, our variable-cell-shape first-principles DFTMD simulations and UI free energy calculations reproduce theexperimentally observed structural phase transition in LiBH4.The structures, transition temperature, and its dependence onchemical substitution are in very good agreement with theexperimental results. Our simulation show that the LT-HTenthalpy and free energy differences are affected differently byeach substitutional element, thereby showing the importanceof free energy calculations, rather than the easier to computeenthalpy, in explaining the structural transition in LiBH4.Because we use DFT and no physically important adjustableparameters, our method is transferable to other chemical sub-stitutions in this material as well as to other materials, and canbe used to efficiently test, without experimental input, possiblesubstitutions for their effects on transition temperatures. Weuse this capability to predict that substituting K for Li stabilizesthe high conductivity phase of LiBH4 with a higher efficiencythan the previously proposed elements I, Cl, and Na.

N.B. and M.D.J. acknowledge funding for this projectby the Office of Naval Research (ONR) through the NavalResearch Laboratory’s Basic Research Program. K.H. wassupported by the US Department of Energy (Grant No. DE-FG52-08NA28921) and by the Center for ComputationallyAssisted Science and Technology (CCAST) at North DakotaState University. Computation was carried out at the AFRLDoD Major Shared Resource Center.

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