Ion Hopping and Constrained Li Diffusion Pathways in the Superionic State of Antiﬂuorite Li2o

entropy

Article Ion Hopping and Constrained Li Diffusion Pathways in the Superionic State of Antiﬂuorite Li2O

Ajay Annamareddy and Jacob Eapen *

Department of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-919-515-5952

Academic Editors: Giovanni Ciccotti, Mauro Ferrario and Christof Schuette Received: 21 March 2017; Accepted: 15 May 2017; Published: 18 May 2017

Abstract: Li2O belongs to the family of antifluorites that show superionic behavior at high temperatures. While some of the superionic characteristics of Li2O are well-known, the mechanistic details of ionic conduction processes are somewhat nebulous. In this work, we first establish an onset of superionic conduction that is emblematic of a gradual disordering process among the Li ions at a characteristic temperature Tα (~1000 K) using reported neutron diffraction data and atomistic simulations. In the superionic state, the Li ions are observed to portray dynamic disorder by hopping between the tetrahedral lattice sites. We then show that string-like ionic diffusion pathways are established among the Li ions in the superionic state. The diffusivity of these dynamical string-like structures, which have a finite lifetime, shows a remarkable correlation to the bulk diffusivity of the system.

Keywords: Li2O; atomistic simulations; antifluorites; superionics; string-like; ionic conduction; diffusion

1. Introduction

Lithium oxide (Li2O) has long been of interest because of its potential application as a tritium-breeding material in fusion reactors [1]. Li2O is also a superionic (or fast-ion) conductor, attaining liquid-like ionic conductivities within the solid state [2]. It is one of the simplest Li-based superionic conductors having just two species [3–7]. The structural and dynamic characteristics of Li2O thus have implications for important technologies ranging from future fusion reactors to solid state batteries [8]. The focus of this paper is to study the collective dynamics of Li ions in the highly conducting superionic state of Li2O using atomistic simulations and statistical mechanics. Li2O has an antifluorite structure (space group: Fm3m) with oxygen ions positioned on a face-centered cubic (FCC) lattice and the lithium ions occupying all the eight tetrahedral sites of the FCC lattice [9,10]. The crystal structure can be viewed alternatively as cations (Li) arranged on a simple cubic lattice, with anions occupying alternate cube centers. Both depictions are illustrated in Figure1, with cations and anions represented by smaller and larger spheres, respectively. The empty cube centers are the octahedral sites of the FCC lattice and are possible locations for interstitials. The lithium ions diffuse at temperatures lower than the melting point of the material; this leads to the superionic character of Li2O with the lithium ions contributing to the bulk of the ionic conductivity, while the less mobile oxygen ions play the important role of maintaining the crystalline structure. A transposition of cations to the FCC lattice and anions to the tetragonal sites gives rise to a popular variant—the fluorite structure. In Figure1, the anion and cation positions in antifluorites can simply be interchanged to obtain this structure. Fluorite materials such as PbF2, SrCl2, and CaF2 are also superionic conductors with anions making a dominant contribution to the ionic conductivity [9,11,12].

Entropy 2017, 19, 227; doi:10.3390/e19050227 www.mdpi.com/journal/entropy Entropy 2017, 19, 227 2 of 11 Entropy 2017, 19, 227 2 of 11

FigureFigure 1. (1.Left (Left) Lattice) Lattice structure structure of of an an antifluorite antifluorite crystal.crystal. The larger spheres spheres (shown (shown in in red) red) depict depict anionsanions that that form form an an FCC FCC structure structure while while the the smaller smaller spheresspheres (in green) represent represent cations cations occupying occupying allall the the available available tetrahedral tetrahedral sites sites of of the the FCC FCClattice; lattice. ((RightRight)) InIn an alternate illustration, illustration, the the cations cations formform a simple a simple cubic cubic lattice lattice while while the the anions anions occupy occupy alternativealternative cube centers. centers. A A fluorite fluorite structure structure is is realized when the cations and anions are interchanged. Neutron scattering/diffraction experiments realized when the cations and anions are interchanged. Neutron scattering/diffraction experiments and atomistic simulations indicate that the empty cube centers or the octahedral sites of the FCC and atomistic simulations indicate that the empty cube centers or the octahedral sites of the FCC lattice, lattice, which are the interstitial locations, are increasingly avoided in the superionic state for fluorites which are the interstitial locations, are increasingly avoided in the superionic state for fluorites and and antifluorites [9,11,13,14]. antifluorites [9,11,13,14]. Superionic conductors can be categorized into two types depending on the nature of their transitionSuperionic to the conductors conducting can state: be Type categorized I materials into typically two typesshow an depending abrupt transition, on the natureat a particular of their transitiontemperature, to the to conducting the conducting state: state Type through I materials a solid typically-solid phase show transition, an abrupt while transition, Type II at superionics a particular temperature,portray a more to the gradual conducting transition state over through a wide a range solid-solid of temperatures phase transition, [9]. Thewhile superionic Type state II superionics of Type portrayII materials a more is also gradual characterized transition by overa quasi-second-order a wide range ofthermodynamic temperatures transition [9]. The at superionica characteristic state of Typetemperature II materials Tλ, below is alsothe melting characterized point, where by a the quasi-second-order specific heat (cp) shows thermodynamic an anomalous transition peak [15]. at a characteristicSeveral fluorites temperature that fall inT λthe, below Type theII category melting portray point, wherethis anomalous the specific behavior heat ( c[16].p) shows Other an anomalousthermodynamic peak [ 15and]. mechanical Several fluorites properties that also fall exhibit in the a Typechange II in category behavior portray near Tλ: thisfor example, anomalous a behaviorsignificant [16]. decrease Other thermodynamic in the value of andelastic mechanical constant C properties11 is observed also near exhibit Tλ afor change fluorites in behavior[17,18]. Although Li2O is known to be a Type II conductor with a gradual increase in the ionic conductivity near Tλ: for example, a significant decrease in the value of elastic constant C11 is observed [2], the divergent-like behavior of cp has not yet been observed experimentally. Instead, Tλ for Li2O near Tλ for fluorites [17,18]. Although Li2O is known to be a Type II conductor with a gradual has been estimated approximately from neutron diffraction [10], diffusivity measurements [19], and increase in the ionic conductivity [2], the divergent-like behavior of cp has not yet been observed variations in properties such as lattice parameter [10] or elastic constants [20]. While these experimentally. Instead, T for Li O has been estimated approximately from neutron diffraction [10], measurements do not agreeλ on 2the numerical value, it is commonly accepted that λ transition takes diffusivity measurements [19], and variations in properties such as lattice parameter [10] or elastic place in the vicinity of 1200 K [10,19]. constants [20]. While these measurements do not agree on the numerical value, it is commonly accepted Recently, we have shown that many fluorites exhibit an onset of disorder or an onset of that λ transition takes place in the vicinity of 1200 K [10,19]. superionicity at a characteristic temperature (denoted as Tα) that can be obtained from neutron Recently, we have shown that many fluorites exhibit an onset of disorder or an onset diffraction, ionic conductivity, and specific heat data [16], which is distinct from Tλ. Atomistic ofsimulations superionicity have at further a characteristic helped in elucidating temperature the structural (denoted and as dynamicalTα) that changes can be observed obtained across from neutronTα [13,21–23], diffraction, as well ionic as across conductivity, the λ transition. and specific We thus heatregard data a Type [16 ],II ionic which conductor is distinct to be from in theT λ. Atomisticsuperionic simulations state at temperatures have further above helped T inα. In elucidating this work, the we structural first apply and our dynamical theoretical changes and observedexperimental across dataTα [analysis13,21–23 methodologies], as well as acrossto the antifluorite the λ transition. Li2O to show We thusthat the regard onset a of Type superionic II ionic conductorconduction to beoccurs in the at superionicTα ≈ 1000 K. state We then at temperatures show that Li aboveions formTα. string-like In this work, dynamical we first structures apply our theoreticalwith a finite and experimentallifetime, and datathe string analysis diffusivity, methodologies which is to based the antifluorite on peak participation Li2O to show of that Li theions onset in of superionicstrings, depicts conduction a remarkable occurs correlation at Tα ≈ 1000 to the K. Webulk then diffusivity show that of the Li system. ions form string-like dynamical structures with a finite lifetime, and the string diffusivity, which is based on peak participation of Li ions2. inSimulation strings, depicts Methods a remarkable correlation to the bulk diffusivity of the system. Atomistic simulations are widely used today to predict the properties of materials. They are also 2. Simulationimmensely Methodsuseful to uncover atomistic mechanisms, which are difficult to establish solely with experimentalAtomistic simulationstechniques. areIn this widely work, used we todayuse classical to predict atomistic the properties simulations of materials.to investigate They the are alsospatiotemporal immensely useful dynamics to uncover of Li ions atomistic in the superionic mechanisms, state which of Li2 areO. We difficult select toa rigid-ion, establish two-body solely with

Entropy 2017, 19, 227 3 of 11 experimental techniques. In this work, we use classical atomistic simulations to investigate the spatiotemporal dynamics of Li ions in the superionic state of Li2O. We select a rigid-ion, two-body Buckingham-type potential developed by Asahi et al. [24] from ﬁrst principles electronic structure simulations. The potential takes the following form: 1 zαzβ Cαβ V r = + A e(−rαβ/ραβ) − (1) αβ αβ r αβ 6 4πε0 αβ rαβ

The indices α and β label the ionic species, r denotes the distance between the ions, and z is the effective charge. The ﬁrst term in Equation (1) represents the Coulombic interaction between the charged ions, while the second and third terms represent the repulsive interaction due to the overlap of the electron clouds and the attractive van der Waals or dispersion interaction, respectively. In the parametrization used by Asahi et al. [24], the dispersion forces are excluded for all ion-pairs, while the short-range repulsive interaction is attributed to Li-O pair only. It is interesting to note that the training set for developing the potential is mainly driven by the melting point for Li2O (Tm = 1711 K). While other potentials have been developed in the past [20,25–29] for studying the superionic behavior of Li2O, the potential developed more recently by Asahi et al. [24] reproduces the experimentally measured density of Li2O in the entire solid phase and more importantly forecasts the λ transition, based on thermal expansivity, to occur at ~1200 K [24], which is close to what is deduced from neutron diffraction experiments. Remarkably, the Asahi et al. potential also predicts the onset of disorder at a lower temperature (Tα in our terminology), which is nearly identical to the temperature at which a change of slope is detected in the ionic conductivity [2]. Thus with a minimal training set and parameters (see Table A1 in AppendixA), the potential developed by Asahi et al. [ 24] predicts the transitions that are most critical to explaining the atomistic mechanisms that underpin the superionic state of Li2O. A comparison of different transition temperatures in Li2O is shown in Table1.

Table 1. Comparison of transition temperatures in Li2O. Tλ and Tm refer to the λ transition temperature and the melting point, respectively, and Tα represents the disorder-onset temperature.

Experiments Atomistic (MD) Simulations † Tα (K) 1000 [2,10] 1000 Tλ (K) 1200 [10] 1200 [24] Tm (K) 1711 [24] 1715 [24] † Estimated from the long-range order parameter (LRO)—see Figure 4.

An 8 × 8 × 8 antifluorite system containing 6144 ions is used for performing the atomistic simulations. The Newton equations of motion are integrated using Velocity-Verlet algorithm under periodic boundary conditions. A time step of 1 fs is observed to maintain long-time energy conservation. The system is first equilibrated in a NPT ensemble (constant number of ions, pressure and temperature) under zero pressure. The dynamical quantities of interest are then evaluated in the energy conserving micro-canonical ensemble. The Coulomb interactions are evaluated using the Wolf-summation method [30], which we have benchmarked to the standard Ewald method (see AppendixB). In the Wolf method, both the Coulomb energies and forces are evaluated by applying a spherical truncation at a finite cut-off radius with charge compensation on the surface of the truncation sphere [30]. It is computationally more efficient than the widely used Ewald or fast-multipole methods, with comparable accuracy for bulk and nanostructures. A cut-off radius of 10 Å is applied in the evaluation of both the short-range and electrostatic interactions. All the simulations are performed using the atomistic code—qBox that has been developed in-house. Entropy 2017, 19, 227 4 of 11

3. Results

3.1. Onset of Disorder from Reported Experimental Data

We begin by estimating the crossover temperature (Tα) in Li2O that corresponds to the onset of disorderEntropy 2017 from, 19, 227 experimental data. For this purpose, we utilize the reported data on neutron4 of 11 diffraction [10] and ionic conductivity of Li ions using the nuclear magnetic resonance (NMR) [10] and ionic conductivity of Li ions using the nuclear magnetic resonance (NMR) method [2]. The method [2]. The neutron diffraction measurements give quantitative information on disorder neutron diffraction measurements give quantitative information on disorder by measuring the by measuring the fraction of Li ions that are displaced from their native lattice sites at different fraction of Li ions that are displaced from their native lattice sites at different temperatures, as shown temperatures, as shown in Figure2a. Using the best ﬁt to the diffraction data (shown by the dotted in Figure 2a. Using the best fit to the diffraction data (shown by the dotted line), the onset of disorder line), the onset of disorder can be estimated to occur at Tα ≈ 1000 K. In our earlier study, we had can be estimated to occur at Tα ≈ 1000 K. In our earlier study, we had established that fluorite established that ﬂuorite conductors such as PbF , SrCl , and CaF show a rapid rise in the ionic conductors such as PbF2, SrCl2, and CaF2 show a rapid2 rise2 in the ionic2 conductivity at Tα though with conductivity at T though with a higher activation energy [16,21]. As shown in Figure2b, the ionic a higher activationα energy [16,21]. As shown in Figure 2b, the ionic conductivity of Li2O broadly falls conductivityinto two Arrhenius of Li2O broadly segments falls with into a twonoticeable Arrhenius change segments in slope with at Tα a≈ noticeable1000 K. Both change neutron in slopescattering at T α ≈ 1000and K.ionic Both conductivity neutron scattering measurements, and ionic therefore, conductivity establish measurements, that the onset of therefore, Li disorder establish takes place that theat onsetthe of characteristic Li disorder takestemperature place at T theα ≈ 1000 characteristic K. temperature Tα ≈ 1000 K. ) f 2

n (a) Neutron Diffraction Data (b) Ionic Conductvity 0.100 1 )

0.075 -1 0

/Sm -1 0.050 σ -2 log(

0.025 -3 -4

Fractiondisplaced of Li ions ( 0.000 -5 800 1000 1200 1400 1600 1.0 1.5 2.0 Temperature (K) 1000/T (K-1)

Figure 2. (a) Fraction of lithium ions (nf) that are displaced from regular sites, obtained from neutron Figure 2. (a) Fraction of lithium ions (nf) that are displaced from regular sites, obtained from neutron diffraction measurements in Li2O [10]. The onset of disorder estimated from the best-fit to the diffraction measurements in Li2O[10]. The onset of disorder estimated from the best-fit to the diffraction diffraction data occurs at Tα ≈ 1000 K. (b) Variation of ionic conductivity with temperature in Li2O data occurs at Tα ≈ 1000 K; (b) Variation of ionic conductivity with temperature in Li2O measured measured using nuclear magnetic resonance (NMR) [2]. As evident, the ionic conductivity also shows using nuclear magnetic resonance (NMR) [2]. As evident, the ionic conductivity also shows a noticeable a noticeable change in slope at Tα ≈ 1000 K. A quantitative correlation between the onset of disorder change in slope at Tα ≈ 1000 K. A quantitative correlation between the onset of disorder and the change and the change in the slope of ionic conductivity is observed in several fluorite conductors such as in the slope of ionic conductivity is observed in several fluorite conductors such as SrCl2, PbF2 and SrCl2, PbF2 and CaF2 [16]. CaF2 [16]. We will now use atomistic simulations to determine the structural disorder among the Li cations inWe Li2 willO. Similar now useto our atomistic earlier simulationsinvestigation to on determine UO2, we thecalculate structural the long-range disorder amongorder (LRO) the Li cationsparameter in Li2O. [13,31] Similar for tothe our Li sub-lattice. earlier investigation The LRO, me onasured UO2, through we calculate the Fourier the long-range transform orderof the (LRO)local N iq⋅r parameterdensity [given13,31 ]by for ρ the()q = Li sub-lattice.e j where The q is LRO, the wave measured vector,through is a simple the metric Fourier to assess transform quasi-static of the  j=1 · local density given by ρ(q) = ∑N eiq rj where q is the wave vector, is a simple metric to assess disorder at different scales. For j=crystalline1 solids, the magnitude of LRO remains steady with time quasi-static disorder at different scales. For crystalline solids, the magnitude of LRO remains steady and tends to approach unity with decreasing temperature while for liquids it quickly decays to zero withindicating time and the approaches absence of unity any long-range with decreasing order. As temperature shown in Figure while for3, the liquids magnitude it quickly of LRO decays for Li to zeroions indicating remains the constant absence in oftime any for long-range all temperatures order.As tillshown melting in indicating Figure3, thean magnitudeunderlying ofcrystalline LRO for Li ionsstructure. remains However, constant the in timedecrease for all in temperatures magnitude with till meltingtemperature indicating suggests an underlyingperceptible crystallinedisorder structure.among However, Li ions while the decreasepreserving in magnitudea crystalline with sub- temperaturelattice, which suggests is somewhat perceptible counter-intuitive. disorder among One Li ionspossible while way preserving to accommodate a crystalline disorder sub-lattice, among whichLi ions iswithin somewhat the Li counter-intuitive.sub-lattice is through One dynamic possible wayhopping to accommodate of Li ions disorder between amonglattice sites, Li ions a mechanism within the Lithat sub-lattice has been is observed through in dynamic prior atomistic hopping of Lisimulations ions between of Li lattice2O [26,27], sites, and a mechanismubiquitous among that has the been anions observed of fluorite in superionic prior atomistic conductors simulations in the of Lisuperionic2O[26,27], state and [13,21,23]. ubiquitous We among will demonstrate the anions oflate fluoriter that such superionic cation hopping conductors between in the native superionic sites stateindeed [13,21 ,takes23]. We place will and demonstrate Li ions diffuse later thatthrough such string-like cation hopping dynamical between structures. native sites In contrast, indeed takesthe placeoxygen and Li anions ions diffuse show throughlimited variation string-like in dynamicalthe LRO magnitude structures. with In contrast, temperature the oxygen indicating anions a rigid show limitedanion variation sub-lattice. in the LRO magnitude with temperature indicating a rigid anion sub-lattice. Figure 4a shows the time-averaged LRO magnitude against temperature (scaled to Tλ = 1200 K). The growth of disorder on the Li sub-lattice accelerates at ~1000 K, which is the characteristic temperature at which the neutron diffraction data shows a discernible increase in the disorder. Interestingly, at temperatures below Tα (1000 K), and above Tλ (1200 K), the LRO shows a near-linear variation for the range of temperatures considered in this study. As expected, the LRO magnitude hovers near unity and does not change significantly for the oxygen anions.

Entropy 2017, 19, 227 5 of 11

Figure4a shows the time-averaged LRO magnitude against temperature (scaled to Tλ = 1200 K). The growth of disorder on the Li sub-lattice accelerates at ~1000 K, which is the characteristic temperature at which the neutron diffraction data shows a discernible increase in the disorder. Interestingly, at temperatures below Tα (1000 K), and above Tλ (1200 K), the LRO shows a near-linear variation for the range of temperatures considered in this study. As expected, the LRO magnitude hoversEntropy near unity2017, 19 and, 227 does not change signiﬁcantly for the oxygen anions. 5 of 11 Entropy 2017, 19, 227 5 of 11 1.0 1.0 (a) Lithium 1.0 1.0 0.9 (a) Lithium 0.9 0.9 0.9 0.8 0.8 0.8 0.8 (b) Oxygen 0.7 0.7 (b) Oxygen 0.7 0.7 0.6 0.6 800 K 900 K 1000 K 1100 K 1200 K 1300 K 0.6 0.6 1400 800 KK 900 K 1000 K 800 K 900 K 1000 K 1100 K 1200 K 1300 K

Long-range order (LRO) order Long-range 0.5

Long-range order (LRO) order Long-range 0.5 1100 K 1200 K 1300 K 1400 K 1400 800 KK 900 K 1000 K

Long-range order (LRO) order Long-range 0.5

Long-range order (LRO) order Long-range 0.5 0.4 1100 K 1200 K 1300 K 0.4 0 20 1400 K 40 60 80 100 0 20406080100 0.4 0.4 0 20Time 40 (ps) 60 80 100 0 20406080100Time (ps) Time (ps) Time (ps) Figure 3. Magnitude of long-range order (LRO) parameter for (a) lithium and (b) oxygen ions across Figure 3. Magnitude of long-range order (LRO) parameter for (a) lithium and (b) oxygen ions across severalFigure 3.temperatures Magnitude ofin long-rangeLi2O. The orderreciprocal (LRO) lattice parameter vector for q (isa )chosen lithium along and ( b[1) oxygen0 0] and ions [1 across−1 1] several temperatures in Li2O. The reciprocal lattice vector q is chosen along [1 0 0] and [1 −1 1] directionsseveral temperatures for the lithium in Liand2O. oxygen The reciprocal sub-lattices, lattice respectively, vector q isand chosen the magnitude along [1 0of 0] q correspondsand [1 −1 1] directionstodirections the for longest the for lithium wavelengththe lithium and oxygen inand the oxygen system. sub-lattices, sub-la ttices, respectively, respectively, and and the the magnitude magnitude of ofq qcorresponds corresponds to the longestto the wavelength longest wavelength in the system. in the system. 5 (a) (b) 5 Neutron diffraction 0.9 (a) 4 (b) LRO Neutron diffraction 0.9 4 LRO 0.8 3 0.8 3 Lithium 2 0.7 Oxygen Lithium 2 1 0.7 Oxygen Scaled disorder 1 0.6 Scaled disorder

Long range order (LRO) 0 ≈ 0.6 Tλ=1200 K Tα 1000 K

Long range order (LRO) 0 ≈ 1000 1100 1200 1300 1400 0.9 1.0Tλ=1200 1.1 K 1.2Tα 1.3 1000 K 1.4 1.5 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1000 1100Temperature 1200 1300(K) 1400 Tλ/T Temperature (K) Tλ/T Figure 4. (a) Variation of the time-averaged long-range order (LRO) for lithium and oxygen ions from 800Figure to 14004. (a) K.Variation (b) Comparison of the time-averaged of the (scaled) long-range neutron order diffr (LRO)action for data lithium with and the oxygen (shifted/scaled) ions from Figure 4. (a) Variation of the time-averaged long-range order (LRO) for lithium and oxygen ions from disorder800 to 1400 metric K. evaluated(b) Comparison from the of LRO the of(scaled) Li ions neutron. The statistical diffraction error data for LRO with is theless (shifted/scaled)than 1%. 800 to 1400 K; (b) Comparison of the (scaled) neutron diffraction data with the (shifted/scaled) disorder disorder metric evaluated from the LRO of Li ions. The statistical error for LRO is less than 1%. metric evaluatedWe have fromcalculated the LRO a metric of Li ions. for TheLi-ion statistical disorder error as forthe LRO difference is less than(modulus) 1%. between the magnitudeWe have of LRO calculated and its ideala metric value for (unity) Li-ion and disorder shifted itas to thegive difference a null value (modulus) at Tα. In Figure between 4b, wethe Wehavemagnitude have compared calculated of LRO this a anddisorder metric its ideal for metric Li-ion value from (unity) disorder simulations and as shifted the to difference the it tofraction give (modulus)a of null Li ionsvalue leaving between at Tα. In the Figure the native magnitude 4b, sites we evaluated from neutron scattering data, both scaled to their respective magnitudes at Tλ. of LROhave and compared its ideal valuethis disorder (unity) metric and from shifted simulations it to give to athe null fraction value of Li at ionsTα. leaving In Figure the4 nativeb, we sites have comparedInterestingly,evaluated this disorder from the disorderneutron metric metric fromscattering simulationsfrom data, atomistic both to si thescaledmulation fraction to istheir able of Litorespective ionscapture leaving themagnitudes variation the native ofat theT sitesλ. neutronInterestingly, diffraction the disorder data at temperatures metric from rangingatomistic from simulation 1000 to ~1300is able K to(shown capture by the shadedvariation region); of the evaluated from neutron scattering data, both scaled to their respective magnitudes at Tλ. Interestingly, atneutron higher diffractiontemperatures, data LRO at temperatures predicts a larger ranging disorder from eventually1000 to ~1300 leading K (shown to melting by the at shaded 1715 K, region); which the disorder metric from atomistic simulation is able to capture the variation of the neutron diffraction isat close higher to temperatures,the experimental LRO melting predicts point a larger (1711 disorder K). eventually leading to melting at 1715 K, which data atis temperatures close to the experimental ranging frommelting 1000 point to (1711 ~1300 K). K (shown by the shaded region); at higher temperatures,3.2. Hopping LRO and predicts Cooperativity a larger among disorder LI Ions eventually leading to melting at 1715 K, which is close to3.2. the Hopping experimental and Cooperativity melting pointamong (1711LI Ions K). A number of atomistic simulations on fluorites and antifluorites have shown that the mobile anions hop from one native lattice site to another at temperatures close to Tλ [13,26,27,32,33]. We have 3.2. HoppingA and number Cooperativity of atomistic among simulations LI Ions on fluorites and antifluorites have shown that the mobile observedanions hop similar from onehopping native of lattice Li ions, site asto anothershown in at Figuretemperatures 5. The closeLi ions to Tmostlyλ [13,26,27,32,33]. vibrate about We theirhave Alattice numberobserved sites, ofsimilar punctuated atomistic hopping simulationsby of sporadic Li ions, hoppingonas shown ﬂuorites to in a Figure andneighboring antiﬂuorites 5. The site.Li ions Occasionally, have mostly shown vibrate concerted that about the ionic mobiletheir hopping can also be noticed, in addition to a few large amplitude displacements. The characteristic anions hoplattice from sites, one punctuated native lattice by sporadic site to another hopping at to temperatures a neighboring close site. toOccasionally,Tλ [13,26,27 concerted,32,33]. We ionic have motionhopping of can all Lialso ions, be noticed,on average, in addition can be extracted to a few largeusing amplitude Gs(r,t), the displacements.self-part of the Thevan characteristicHove space- timemotion correlation of all Li function.ions, on average,Gs(r,t) gives can thebe extractedconditional using probability Gs(r,t), theof finding self-part an of ion the in van an infinitesimal Hove space- volumetime correlation dr around function. r at time G ts,( givenr,t) gives that the it has conditional occupied probability the origin atof time finding t = 0. an Figure ion in 5b an shows infinitesimal Gs(r,t) forvolume the Li d ionsr around for a rcorrelation at time t, given time thatof 30 it ps has at occupied 1200 K. The the self-correlationorigin at time t =decays 0. Figure rapidly 5b shows with Gr, sbut(r,t) for the Li ions for a correlation time of 30 ps at 1200 K. The self-correlation decays rapidly with r, but

Entropy 2017, 19, 227 6 of 11

Entropy 2017, 19, 227 6 of 11 observed similar hopping of Li ions, as shown in Figure5. The Li ions mostly vibrate about their lattice sites,the punctuatedmultiple peaks by show sporadic the proclivity hopping of to the a neighboring Li ions to settle site. at Occasionally, these positions concerted simultaneously. ionic hopping To canidentify also be the noticed, peak locations, in addition the radial to a distribution few large amplitude function (RDF) displacements. of the Li ions at The 1200 characteristic K is also shown motion in Figure 5b. Interestingly, the peak positions of Gs(r,t) show excellent agreement with the nearest of all Li ions, on average, can be extracted using Gs(r,t), the self-part of the van Hove space-time neighbor distances obtained from RDF, indicating that Li ions engage in discrete hops from one correlation function. Gs(r,t) gives the conditional probability of ﬁnding an ion in an inﬁnitesimal native lattice site to another. volume dr around r at time t, given that it has occupied the origin at time t = 0. Figure5b shows G (r,t) The ionic jumps are thermally driven and do not depend on having permanent vacancies/defects s forin the the Li system. ions for In our a correlation recent work time on fluorites, of 30 ps we at have 1200 established K. The self-correlation that such concerted decays ionic rapidly hops can with r, butpersist the multiple for hundreds peaks of show picoseconds the proclivity and are ofinstru themental Li ions in to forming settle at quasi- theseone-dimensional positions simultaneously. string- Tolike identify structures the peak [23]. Not locations, surprisingly, the radial a similar distribution mechanism function is seen to (RDF) be operational of the Li among ions at the 1200 Li ions K is also shownabove in the Figure characteristic5b. Interestingly, order-disorder the peak transition positions temperature of Gs(r,t) showTα. In excellent a departure agreement from withthe the nearestconventional neighbor interpretation distances obtained of defects, from we RDF, will indicatingnow show that that Li disorder ions engage among in discretethe Li ions hops is from onemanifested native lattice through site todynamic another. string-like structures, which have a finite lifetime.

3 2.0 (a) Lithium ions (b) Lithium ions

2 -2 10 1.5 1 /2 a )

t -3

, 10

r 1.0 (t), 0 ( s x Δ G

-1 -4 10 0.5

-2 T = 1200 K

T = 1200 K Function Distribution Radial 10-5 0.0 0 20406080100 12345678 Time (ps) r(Å) Figure 5. (a) Normalized x-displacement of four randomly chosen lithium ions at 1200 K. Recall that Figure 5. (a) Normalized x-displacement of four randomly chosen lithium ions at 1200 K. Recall that the closest cation sites are separated by a/2, along the [1 0 0] direction. (b) The van Hove self- the closest cation sites are separated by a/2, along the [1 0 0] direction; (b) The van Hove self-correlation correlation function Gs(r,t) of lithium ions for a correlation time of 30 ps at a temperature of 1200 K. function Gs(r,t) of lithium ions for a correlation time of 30 ps at a temperature of 1200 K. The Gs(r,t) The Gs(r,t) peaks are nearly coincident with the nearest neighbor peaks of the partial Li radial peaksdistribution are nearly function coincident (RDF). with the nearest neighbor peaks of the partial Li radial distribution function (RDF). 3.3. String-Like Dynamic Structures TheThe ionic hopping jumps of areLi ions thermally mostly drivenoccurs cooperativ and do notely depend between on two having or more permanent neighbors. vacancies/defects We observe in thefrom system. our simulations In our recent that the work mobile on fluorites, Li ions tend we haveto follow established each other that in suchquasi-one-dimensional concerted ionic hops or can persiststring-like for hundreds trajectories. of picoseconds We identify a and string are as instrumental an ordered set in of forming ions in quasi-one-dimensionalwhich an ion replaces the string-like next structuresone over [a23 certain]. Not time surprisingly, interval; a string a similar thus mechanismcontains at least is seen two ions. to be This operational criterion is among similar theto the Li ions aboveone theused characteristic in the study of order-disorder supercooled liquids transition to anal temperatureyze string-likeTα .cooperative In a departure motion from in the arrested conventional interpretationstate [34,35]. If of P defects,(L) is the weprobability will now of showdetecting that a string disorder of length among L over the a Li time ions interval is manifested δt, the mean through dynamicstring length string-like (Δ) is structures,computed as which Σ()/Σ() have a finite. We lifetime.have varied the time interval of observation (δt) from 0.1 ps to several hundred ps; these time intervals are generally sufficient to establish the kinetics 3.3.of String-Like string formation Dynamic and Structures annihilation. For computational efficiency, we have limited our analysis to the top 5% mobile Li ions. As observed in our previous work on fluorites [23], the string characteristics areThe dynamically hopping ofsimilar Li ions among mostly different occurs mobile cooperatively groups. between two or more neighbors. We observe from ourFigure simulations 6a shows thatthe variation the mobile of the Li mean ions tend string to length follow (Δ each) with other time, infor quasi-one-dimensional temperatures ranging or string-likefrom 1000 trajectories. to 1400 K. WeThe identifystrings have a string a lifetime as an orderedwith the settime of needed ions in for which establishing an ion replaces the longest the next onestring, over adenoted certain timeas τ* interval;(T), decreasing a string with thus increasing contains at temp leasterature. two ions. For This any criterion temperature, is similar with to the oneincreasing used in the time, study the strings of supercooled latch on to liquids new ions to to analyze grow in string-like length until cooperative the peak time motion τ*. Clearly, in the this arrested statedynamic [34,35 ].process If P(L )is is favorable the probability at lower of temperatures detecting a rather string than of length at higherL over temperatures. a time interval Thus,δ stringst, the mean stringof different length (lengths∆) is computed can be formed as ΣLP at( allL) /temperatures,ΣP(L). We have but longer varied ones the timeare more interval probable of observation at lower (δt) fromtemperatures. 0.1 ps to several It is interesting hundred to ps; note these that time the peak intervals length are of the generally strings sufficientapproaches to two establish at the highest the kinetics temperatures, which indicates that the string motion at these temperatures mostly involves a set of of string formation and annihilation. For computational efficiency, we have limited our analysis to the two Li ions where one replaces the other. Since the formation of strings is a random process, coherent top 5% mobile Li ions. As observed in our previous work on fluorites [23], the string characteristics are dynamically similar among different mobile groups. Entropy 2017, 19, 227 7 of 11

Figure6a shows the variation of the mean string length ( ∆) with time, for temperatures ranging from 1000 to 1400 K. The strings have a lifetime with the time needed for establishing the longest string, denoted as τ*(T), decreasing with increasing temperature. For any temperature, with increasing time, the strings latch on to new ions to grow in length until the peak time τ*. Clearly, this dynamic process is favorable at lower temperatures rather than at higher temperatures. Thus, strings of different lengths can be formed at all temperatures, but longer ones are more probable at lower temperatures. It is interesting to note that the peak length of the strings approaches two at the highest temperatures, whichEntropy indicates 2017, 19 that, 227 the string motion at these temperatures mostly involves a set of two Li ions7 of where11 one replaces the other. Since the formation of strings is a random process, coherent and well-defined stringand formation well-defined follows string an formation exponential follows distribution—a an exponential distribution—a feature that is alsofeature observed that is also in observed supercooled in supercooled liquids and jammed systems [34,36]. As noted in Figure 6b, the probability of string liquids and jammed systems [34,36]. As noted in Figure6b, the probability of string formation formation follows an exponential distribution above the order-disorder transition temperature Tα follows an exponential distribution above the order-disorder transition temperature T (~1000 K). (~1000 K). At lower temperatures, the distribution is uneven, which suggests a lack of spatio-temporalα At lowercoherence temperatures, among the the Li ions distribution in forming is a uneven, string-like which structure. suggests A similar a lack behavior of spatio-temporal is also observed coherence in amongfluorites, the Li wherein ions in well-characterized forming a string-like strings structure. are identified A similar only above behavior the characteristic is also observed temperature in fluorites, whereinTα [23]. well-characterized strings are identified only above the characteristic temperature Tα [23].

1 14 1000 K (a) 900 K (b) ) 12 1050 K 1100 K Δ 1150 K 1200 K 1000 K 10 1250 K 1300 K 0.1 1200 K 8 1350 K 1400 K

6 ) L

( 0.01 P 4 0.001 Mean string length ( length Mean string

2 1E-4 0.1 1 10 100 0 204060 Time-interval, δt (ps) String length, L

FigureFigure 6. ( a6.) ( Thea) The evolution evolution of of string-like string-like dynamicaldynamical structures structures in in time time among among the the most most mobile mobile (top (top 5%) Li ions. The mean string length (Δ) is measured by the number of ions constituting the string. 5%) Li ions. The mean string length (∆) is measured by the number of ions constituting the string; (b) The probability of finding a string of length L at different temperatures during the string lifetime (b) The probability of finding a string of length L at different temperatures during the string lifetime τ*(T), which is defined as the time needed for establishing the longest string. τ*(T), which is defined as the time needed for establishing the longest string. The population of lithium ions that are involved in strings throws more light on the string kinetics.The population Figure 7a of shows lithium the ions mean that population are involved of ions in strings that are throws participating more light in the on string the string process. kinetics. FigureStrikingly,7a shows near the Tα mean (1000 K), population close to 80% of of ions the that Li ions are (among participating the top 5%) in the participate string process. in the formation Strikingly, of strings. At temperatures above Tλ (1200 K), as the peak string length decreases, the peak near Tα (1000 K), close to 80% of the Li ions (among the top 5%) participate in the formation of strings. participation also drops significantly. Interestingly, the peak participation lifetime (τP), which we At temperatures above Tλ (1200 K), as the peak string length decreases, the peak participation also dropsdefine significantly. as the time Interestingly,when most ions the participate, peak participation is somewhat lifetimelarger than (τP the), which peak string we define length aslifetime the time τ* as shown in Figure 7b for temperatures below Tλ. While in typical supercooled liquids, these when most ions participate, is somewhat larger than the peak string length lifetime τ* as shown in lifetimes are nearly identical [23,34], our earlier work on fluorites (CaF2 and UO2) also shows τP to be Figure7b for temperatures below Tλ. While in typical supercooled liquids, these lifetimes are nearly larger than τ* at temperatures below Tλ [23]. However, at a temperature of Tλ and above, both identical [23,34], our earlier work on fluorites (CaF and UO ) also shows τP to be larger than τ* at timescales are almost coincident, a behavior that is identical2 in2 CaF2 and UO2. It is instructive to note temperaturesthat the lifetime below Ttimescalesλ [23]. However, are O(1) at ps a temperatureat Tλ, which ofis Taλ signatureand above, for both liquid-like timescales diffusivity. are almost coincident,Summarizing, a behavior two distinct that is timescales identical can in be CaF recognized2 and UO in2. antifluorites It is instructive (and fluorites to note [23]): that the time lifetime timescalescorresponding are O(1) to ps the at peakTλ, which string islength a signature (τ*), and for the liquid-like time when diffusivity.most ions participate Summarizing, in strings two ( distinctτP). timescalesWhereas can τ* is be associated recognized with in individual antifluorites strings, (and τP fluoritesdictates the [23 collective]): the time rela correspondingxation of the strings. to the We peak stringtherefore length anticipate (τ*), and that the τ timeP (and when not τ* most) will control ions participate the diffusive in stringsrelaxation (τ Pof). the Whereas system:τ in* is fluorites, associated withwe individual have shown strings, that thisτP dictatesis indeed the the collective case [23]. relaxation of the strings. We therefore anticipate that τP (and not τ*) will control the diffusive relaxation of the system: in fluorites, we have shown that this is indeed the case [23].

Entropy 2017, 19, 227 8 of 11 Entropy 2017, 19, 227 8 of 11

0.8 (a) τ* (b) Entropy 2017, 19, 227 8 of 11 τP 0.6 10

0.8 (a) (ps) τ* (b) P 0.4 τ τP 0.6 10 and *

0.2 (ps) τ P 0.4 τ 1 Fractionions of

0.0 1000 K 1050 K 1100 K and 0.2 * 1150 K 1200 K 1250 K τ 1300 K 1350 K 1400 K 1 Fractionions of 0.1 1 10 100 0.0 1000 K 1050 K 1100 K 0.8 0.9 1.0 1.1 1.2 1.3 Time 1150 interval, K 1200 δ tK (ps) 1250 K T /T 1300 K 1350 K 1400 K λ 0.1 1 10 100 0.8 0.9 1.0 1.1 1.2 1.3 Figure 7. (a) The fraction of ions participating in string-like dynamical structures among the most Figure 7. (a) The fractionTime interval, of ions δt participating(ps) in string-like dynamical structures among the most mobile (top 5%) Li ions. (b) Lifetimes corresponding to peak string length (τT*λ)/ Tand maximal Li ion mobile (top 5%) Li ions; (b) Lifetimes corresponding to peak string length (τ*) and maximal Li ion P participationFigure 7. (a P()τ The). While fraction these of timescalesions participating are nearly in string-likeidentical above dynamical Tλ (1200 structures K), they amongdiverge the below most T λ. participation (τ ). While these timescales are nearly identical above Tλ (1200 K), they diverge below Tλ. mobile (top 5%) Li ions. (b) Lifetimes corresponding to peak string length (τ*) and maximal Li ion

Weparticipation will now (τ P).attempt While these to timescalesevaluate arean nearly effective identical string above diffusivity.Tλ (1200 K), they As diverge strings below are T λ.transient structures,We will nowdiffusivity attempt cannot to evaluate be computed an effective as the string longtime diffusivity. (t →∞ As) limit strings of the are slope transient of the structures, mean diffusivitysquare Wedisplacement cannot will now be ( computedattemptΔr2) [31]. to Instead, evaluate as the we longtimean will effective assume (t→ string that∞ ) the limitdiffusivity. string of diffusivity the As slope strings is of controlled theare meantransient by square the 2 →∞ displacementpeakstructures, string participation (diffusivity∆r )[31]. cannot Instead,lifetime be τcomputed weP [23]. will Since assumeas thestrings longtime that experience the (t string a) characteristiclimit diffusivity of the slope isdisplacement controlled of the mean of by δP the peakin squaretime string τ Pdisplacement, participation we will regard (Δ lifetimer2 )the [31]. string Instead,τP [ 23diffu]. we Sincesivity will stringsassumeto be proportional experiencethat the string ato characteristicdiffusivity (δP)2/τP, analogous is controlled displacement to theby the slope of δP inof time peaktheτ meanP string, we square will participation regard displacement. the lifetime string InτP diffusivity Figure[23]. Since 8, we strings to delineate be proportional experience the (scaled) a characteristic to (δ bulkP)2/ τdiffusivityP, analogousdisplacement of the to lithiumof the δP slope in time τP, we will regard2 the string diffusivity to be proportional to (δP)2/τP, analogous to the slope ofions the mean evaluated square as displacement.lim()Δrt 6 and In Figure compare8, we it delineateagainst the the (scaled) (scaled) string bulk diffusivity, diffusivity as of a the function lithium of the mean squaret→∞ displacement. In Figure 8, we delineate the (scaled) bulk diffusivity of the lithium ions evaluated as lim ∆r2/6t and compare it against the (scaled) string diffusivity, as a function of ionstemperature. evaluated Thetas→ ∞ limcorrelation()Δrt2 6 between and compare the scaled it against string the and (scaled) bulk stringdiffusivities diffusivity, is quite as a striking. function As →∞ ofshown temperature. in the inset The oft correlation Figure 8, the between average thestring scaled displacement string and δP at bulk τP is diffusivities only slightly ismore quite than striking. the of temperature. The correlation between the scaled string and bulk diffusivities is quite striking. As Asclosest shown cation-cation in the inset separation of Figure8 ,distance, the average and they string differ displacement little across theδP at temperatures.τP is only slightly Thus the more string than shown in the inset of Figure 8, the average string displacement δP at τP is only slightly more than the thediffusivity closest cation-cation is dominated separation by the peak distance, string and participation they differ lifetime little across τP. The the temperatures.close correspondence Thus the closest cation-cation separation distance, and they differ little across the temperatures. Thus the string stringbetween diffusivity the bulk is dominatedand string bydiffusivities the peak stringemphasizes participation the influence lifetime of τlow-dimensionalP. The close correspondence string-like diffusivity is dominated by the peak string participation lifetime τP. The close correspondence betweenstructuresbetween the inthe bulk the bulk andsuperionic and string string diffusion diffusivities diffusivities processes. emphasizesemphasizes the the influence inﬂuence of oflow-dimensional low-dimensional string-like string-like structuresstructures in the in the superionic superionic diffusion diffusion processes. processes. 10 10 Bulk Diffusivity Bulk String Diffusivity Diffusivity 1 String Diffusivity 1 P τ 0.1 3 P 0.1 τ

(Å) at (Å) 3

δ 2 (Å) at (Å)

Scaled Diffusivities P

δ 2

Scaled Diffusivities 1 0.01 Lithium ions Average Average 1 0.01 Lithium ions Average Average 0 0.80 0.9 1.0 1.1 1.2 1.3 1.4 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Tλ/T Tλ/T 0.8 0.9 1.0 1.1 1.2 1.3 0.8 0.9 1.0 1.1 1.2 1.3 T /T T λ/T λ Figure 8. Bulk and string diffusivities, scaled to the corresponding magnitudes at Tλ, for different Figure 8. Bulk and string diffusivities, scaled to the corresponding magnitudes at Tλ, for different Figure 8. Bulk and string diffusivities, scaled to the correspondingP magnitudes at Tλ, for different temperatures. The inset shows the mean string displacement (Pδ ) at peak string participation lifetime temperatures. The inset shows the mean string displacement (δ ) at peakP string participation lifetime temperatures.P The inset shows the mean string displacement (δ ) at peak string participation (τ ().τP ). lifetime (τP).

Entropy 2017, 19, 227 9 of 11

4. Discussion and Conclusions Early neutron diffraction investigations have focused on determining the most probable defect positions for the disordered state in fluorites [11,37] and antifluorites [10]. While several quasi-static defect cluster models have been proposed before [38], none of them has the ability to identify realistic ionic conduction or diffusive pathways. The goal of this paper has been to investigate the dynamics of Li ions in the superionic state using atomistic simulations and propose a mechanism that can potentially pinpoint the origin of enhanced diffusivity of Li ions in Li2O. Our first endeavor has been to define the extent of the superionic state of Li2O. Using the reported neutron diffraction [10] and ionic conductivity [2] data, we have established that the disorder among Li ions is first manifested at a characteristic temperature Tα (~1000 K), which is below the λ transition temperature Tλ (1200 K). The disorder estimated from the atomistic simulations shows excellent agreement with the neutron diffraction data and has provided an additional confirmation for the order-disorder transition or the onset of superionicity at Tα. The hopping of Li ions across the native tetrahedral sites has been reported before, as well as in this investigation. We have examined the kinetics of hopping over longer timescales and have determined that Li ions form string-like dynamical structures with a finite lifetime. We have then shown that the string diffusivity, which is based on peak participation of Li ions in strings, depicts a remarkable correlation to the bulk diffusivity of the system. Thus, a plausible atomistic mechanism is proposed that accounts for both the disorder, which is inherently dynamic, and the enhanced ionic diffusion in the superionic state of Li2O.

Acknowledgments: The authors acknowledge funding from the US Department of Energy (DOE) through the Nuclear Energy University Program (NEUP). Insightful observations from Professors Juan Garrahan and Srikanth Sastry are gratefully acknowledged. Author Contributions: Ajay Annamareddy conceived the project, performed the simulations, and analyzed the data. Ajay Annamareddy and Jacob Eapen jointly wrote the paper. Both authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A The parameters for the inter-ionic potential are shown in Table A1.

Table A1. Inter-ionic potential parameters [24].

z+ (|e|) z− (|e|) A++ (eV) A−− (eV) A+− (eV) ρ+− (Å) 0.790909 −1.581818 0.0 0.0 1425.4833 0.236303

C++ = C−− = C+− = 0.

Appendix B. Comparison of Wolf and Ewald Summation Methods for Coulombic Interactions The short range Wolf method [30] is an attractive choice for the evaluation of electrostatic interactions in ionic systems. We have veriﬁed that the accuracy of the Wolf method is comparable to that of the Ewald sum method, which is typically used in ionic molecular simulations. The Ewald sum simulations are performed using LAMMPS [39] while the Wolf method is implemented using our in-house code—qbox. As delineated in Figure A1, both methods predict nearly identical structure (radial distribution function) and dynamics (mean square displacement). The methods are tested on identical systems with 6144 ions with periodic boundary conditions applied along all three orthogonal directions. A timestep of 1 fs is employed for all the simulations. For the Ewald sum calculations, a relative error of 10−7 is assigned (for forces), while for the Wolf method, the damping parameter (α) is chosen as 0.3. Our earlier simulations [13,21–23] on superionic UO2 and CaF2 also showed excellent agreement between these methods. Entropy 2017, 19, 227 10 of 11 Entropy 2017, 19, 227 10 of 11

2.5 (a) Ewald sum 10 (b) Ewald sum Wolf method Wolf method )

2 1200 K 2.0 900 K

1.5 1200 K 1 900 K

Li-Li g(r) 1.0

0.1 0.5 ions (Å MSD of Li

0.0 0246810 0.01 0.1 1 10 r (Å) Time (ps)

Figure A1. Structure and dynamics of lithium ions in Li2O using Ewald sum and Wolf methods. Figure A1. Structure and dynamics of lithium ions in Li2O using Ewald sum and Wolf methods. (a) Radial distribution function, g(r) of lithium ions at temperatures on either side of the superionic- (a) Radial distribution function, g(r) of lithium ions at temperatures on either side of the superionic-onset onset temperature (Tα ~ 1000 K). (b) Mean square displacement of the lithium ions for the same temperature (Tα ~1000 K); (b) Mean square displacement of the lithium ions for the same temperature range. temperature range. References References 1. Stacey, W.M. Fusion: An Introduction to the Physics and Technology of Magnetic Confinement Fusion, 2nd ed.; 1. Stacey,Wiley-VCH: W.M. Fusion: Weinheim, An Introduction Germany, to 2010. the Physics and Technology of Magnetic Conﬁnement Fusion, 2nd ed.; Wiley-VCH:2. Strange, Weinheim,J.H.; Rageb, Germany, S.M.; Chadwick, 2010. A.V.; Flack, K.W.; Harding, J.H. Conductivity and NMR study of 2. Strange,ionic J.H.; mobility Rageb, in S.M.;lithium Chadwick, oxide. Faraday A.V.; Flack,Trans. 1990 K.W.;, 86 Harding,, 1239–1241. J.H. Conductivity and NMR study of ionic 3.mobility Kumar, in lithium B.; Kumar, oxide. J.; FaradayLeese, R.; Trans. Fellner,1990 J.P.;, 86 ,Rodrigue 1239–1241.s, S.J.; [CrossRef Abraham,] K.M. A solid-state, rechargeable, 3. Kumar,long B.; cycle Kumar, life lithium-air J.; Leese, R.; battery. Fellner, J. J.P.;Electrochem. Rodrigues, Soc.S.J.; 2010 Abraham,, 157, A50–A54. K.M. A solid-state, rechargeable, long 4.cycle Xu, life M.; lithium-air Ding, J.; battery.Ma, E. One-dimensionalJ. Electrochem. Soc. stringlike2010, 157 cooperative, A50–A54. migration [CrossRef of] lithium ions in an ultrafast 4. Xu, M.;ionic Ding, conductor. J.; Ma,E. Appl. One-dimensional Phys. Lett. 2012 stringlike, 101, 031901. cooperative migration of lithium ions in an ultrafast ionic 5. Mo, Y.; Ong, S.P.; Ceder, G. First principles study of the Li10GeP2S12 lithium super ionic conductor material. conductor. Appl. Phys. Lett. 2012, 101, 031901. [CrossRef] Chem. Mater. 2012, 24, 15–17. 5. Mo, Y.; Ong, S.P.; Ceder, G. First principles study of the Li10GeP2S12 lithium super ionic conductor material. 6. Salanne, M.; Marrocchelli, D.; Watson, G.W. Cooperative mechanism for the diffusion of Li+ ions in Chem. Mater. 2012, 24, 15–17. [CrossRef] LiMgSO4F. J. Phys. Chem. C 2012, 116, 18618–18625. 6. Salanne, M.; Marrocchelli, D.; Watson, G.W. Cooperative mechanism for the diffusion of Li+ ions in LiMgSO F. 7. Wang, Y.; Richards, W.D.; Ong, S.P.; Miara, L.J.; Kim, J.C.; Mo, Y.; Ceder, G. Design principles for solid-4 J. Phys. Chem. C 2012, 116, 18618–18625. [CrossRef] state lithium superionic conductors. Nat. Mater. 2015, 14, 1026–1031. 7. Wang, Y.; Richards, W.D.; Ong, S.P.; Miara, L.J.; Kim, J.C.; Mo, Y.; Ceder, G. Design principles for solid-state 8. Ranganath, S.B.; Hassan, A.S.; Ramachandran, B.R.; Wick, C.D. Role of metal-lithium oxide interfaces in lithiumthe superionicextra lithium conductors. capacity ofNat. metal Mater. oxide2015 lithium-ion, 14, 1026–1031. battery anode [CrossRef materials.][PubMed J. Electrochem.] Soc. 2016, 163, 8. Ranganath,A2172–A2178. S.B.; Hassan, A.S.; Ramachandran, B.R.; Wick, C.D. Role of metal-lithium oxide interfaces in 9.the extraHull, lithium S. Superionics: capacity Crystal of metal structures oxide lithium-ion and conduction battery processes. anode materials.Rep. Prog. Phys.J. Electrochem. 2004, 67, 1233–1314. Soc. 2016, 163, A2172–A2178.10. Farley, T.W.D.; [CrossRef Hayes,] W.; Hull, S.; Hutchings, M.T.; Vrtis, M. Investigation of thermally induced Li+ ion 9. Hull,disorder S. Superionics: in Li2O using Crystal neutron structures diffraction. and conductionJ. Phys. Condens. processes. Matter 1991Rep., 3 Prog., 4761–4781. Phys. 2004 , 67, 1233–1314. 11.[CrossRef Hutchings,] M.T.; Clausen, K.; Dickens, M.H.; Hayes, W.; Kjems, J.K.; Schnabel, P.G.; Smith, C. Investigation 10. Farley,of T.W.D.;thermally Hayes, induced W.; anion Hull, disorder S.; Hutchings, in fluorites M.T.; using Vrtis, neutron M. Investigation scattering techniques. of thermally J. Phys. induced C Solid Li +Stateion disorderPhys. in 1984 Li2O, 17 using, 3903–3940. neutron diffraction. J. Phys. Condens. Matter 1991, 3, 4761–4781. [CrossRef] 11. 12.Hutchings, Voronin, M.T.; B.M.; Clausen, Volkov, K.; S.V. Dickens, Ionic conductivity M.H.; Hayes, of W.; fluorite Kjems, type J.K.; crystals Schnabel, CaF P.G.;2, SrF Smith,2, BaF2 C., and Investigation SrCl2 at high of thermallytemperatures. induced anionJ. Phys. disorder Chem. Solids in ﬂuorites 2001, 62 using, 1349–1358. neutron scattering techniques. J. Phys. C Solid State Phys. 198413. ,Annamareddy,17, 3903–3940. [A.;CrossRef Eapen,] J. Disordering and dynamic self-organization in stoichiometric UO2 at high

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