Tan-Lien Pham

Mohammed Guerboub

Assil Bouzid

Mauro Boero

Carlo Massobrio

Young-Han Shin

Guido Ori

Unveiling the structure and ion dynamics of amorphous Na 3-x OH x Cl antiperovskite

Introduction

Over the past decade, low-melting-point Li-and Na-rich antiperovskite solid electrolytes (SEs) have been targeted worldwide as promising materials for solid-state batteries because of their structural flexibility, high ionic conductivity, wide electrochemical window, and stability. [1][2][3][4][5] In particular, sodium oxyhalide and hydroxyhalide antiperovskites hold the potential to be used in the SEs domain due to their low melting point (T m < 300 • C), affordable cost and low environmental impact of the constituent elements (Na, H, O, and Cl), and rapid synthesis. 6 By exploiting their versatile structure, expressed as X 3 OA (X= Li + or Na + ; A= halides (Cl -, Br -, I -) or other anions (BH - 4 , NO - 2 )), the properties, performance and ion dynamics mechanisms of antiperovskite SEs can be tailored through chemical doping, defect tuning, and structural manipulation, thereby providing an exploitable multifaceted chemistry. 1,[7][8][9][10][11][12][13][14][15] By virtue of aliovalent doping or via mixed hydroxides, Li-rich antiperovskites have been shown to yield "superionic" ion conductivities, i.e. ∼0.1 mS/cm. 6,16 Beyond the stoichiometric X 3 OA crystalline systems, attention has been drawn to the use of cation-and anion-doped X 3 OA as well as glassy (or amorphous) antiperovskite electrolytes. 7,17,18 Glassy Li 3 OCl-and Na 3 OCl-based antiperovskites have been originally exploited by the teams of Braga and Goodenough, showing a glass transition temperature in the range 390-450 K and a conductivity of the order of 10 -2 S/cm at room temperature. [19][20][21] However, attempts at reproducing the original laboratory synthesis were not con-clusive and, to date, structure and ion conduction mechanism of glassy antiperovskite electrolytes remain unsolved issues. 1,17,22 The original work of Braga et al. 19 was later questioned by Hanghofer et al. 22 , who ascribed the high conductivity of Li + ions to the presence of impurity and stability of the sample, and the formation of LiCl•H 2 O to the presence of H 2 O. The results of Ref. 19 have been substantiated by ab initio and classical molecular dynamics (MD) simulations agreeing on the existence of a subnanoscale phase separation of Li 3 OCl into Li 2 O and LiCl phases together with an intermixed phase. The formation of such subnanoscale separation was considered to be the promoter of the high conductivity of Li ions as well as a non-negligible Cl ions mobility. 23 Choi et al. further characterized by ab initio MD the structural and electrochemical stability of amorphous Li 3 OCl. 24 Successful use of an electrolyte composite was recently reported by Tian et al.. [25][26][27] In this application, an amorphous Li 3 OCl was used as a matrix embedding Li 6.75 La 3 Zr 1.75 Ta 0.25 O 12 garnet-type oxide particles exhibiting a high room temperature conductivity of 2.27•10 -4 S•cm -1 and an extremely wide electrochemical stability window up to 10 V. In this particular case, the added value promoted by the amorphous X 3 OA phase was the ability to act as a binder and filling agent ensuring the formation of an integrated composite SE with a continuous widespread ionic conductive network. The key role of the amorphous phase was also assigned to the excellent affinity to the lithium metal, greatly decreasing the interfacial resistance between the anode and the electrolyte. [25][26][27] A detailed understanding of the atomic structure is a key step to further capture the details of the transport mechanisms behind its conductivity performance. Recent reviews on the use of anti-perovskites for solid-state batteries underlined the limited nature of the available structural characterization and pointed out the severe need of a quantitative structural assessment to avoid any misinterpretation of the correlations between structure and performances. 1,28,29 Overall, structure and ion dynamics mechanisms are well known for crystalline X 3 OA compounds. This is not the case for amorphous antiperovskites, especially in both hydrogen-free and hydroxylated phases (e.g., oxyhalide X 3 OA vs. hydroxyhalide X 3-x OH x A). In view of these considerations, this work relies on predictive atomic-scale modelling, as first-principles molecular dynamics (FPMD), to elucidate the structure and transport properties of amorphous H-free and hydroxylated Na 3-x OH x Cl systems. [START_REF]In the present manuscript we use the formula Na 3--x OH x Cl to denote the hydroxylated phases[END_REF] In terms of structural analysis, we provide a detailed study of both X-rays and neutrons structure factors. If the X-ray probe is of high importance, neutron studies are particularly needed in the case of antiperovskite electrolyte investigation, being capable of detecting low-Z elements (or light elements, such as H and Li) that are out of reach of X-ray (synchrotron source) techniques. 1,2,17,[START_REF] Gao | [END_REF] Our paper is organized as follows. Sec. 2 describes the methodologies and models systems employed within first-principles molecular dynamics. We give details on the production of the amorphous systems allowing for the calculation of static and dynamical properties. A specific subsection is devoted to a set of definitions for structural (disorder degree) and dynamical (ionic diffusion and conductivities) quantities targeted in this work for their potential applications in the solid-state electrolyte. Results are presented in Sec. 3 and separated into two subsections, the first one (3.1) devoted to structural properties and the second one (3.2) to dynamical properties. The analysis of the structural properties features, firstly, calculated properties in reciprocal and real space (3.1.1 and 3.1.1), namely the structure factors and the pair correlation functions. The following parts are devoted to the analysis of the coordination numbers (3.1.3) and the structural units (3.1.4), with special attention paid to the chemical nature of the various motifs and their variations with the composition. A final part is devoted to the assessment of the structural disorder degree of the studied models (3.1.5). Subsection 3.2 is organized into three parts, describing respectively the different trends taken by the mean square displacements (3.2.1), a dynamical structural rearrangement involving the hydroxyl group (the "paddlewheel" effects, 3.2.2), and the behavior of the diffusion coefficients (3.2.3) leading to the assessment of the ionic conductivity (3.2.4). Concluding remarks are collected in Sec. 4.

Calculation methodology and models

Producing Na 3-x OH x Cl amorphous models at 300K

The Car-Parrinello (CP) method 32 was used to produce dynamical trajectories of the targeted Na 3-x OH x Cl systems. The exchange-correlation functional selected is the generalized gradient approximation (GGA) proposed by Perdew, Burke and Ernzerhof (PBE). 33 The valence-core interaction was described by numerical norm-conserving Troullier-Martins (TM) 34 pseudopotentials for all elements (Na, O, H, and Cl). In the case of Na, semicore states were included to ensure a good description of the energetics and electronic features dependent on its cationic nature. This amounts to electronic configurations He 2s 2 , 2p 6 , 3s 1 for Na; He 2s 2 , 2p 4 for O; 1s 1 for H and Ne 3s 2 , 3p 5 for Cl. Valence electrons are treated explicitly and represented on a plane-wave basis set with the sampling of the Brillouin restricted to the Γ point. A fictitious electron mass of 600 a.u. and a time step of 0.12 fs ensured optimal conservation of the constants of motion. FPMD simulations were performed in the canonical NVT ensemble with the ionic temperature controlled with a Nosé-Hoover [35][36][37] thermostat chain. 38 For the simulations performed at high temperature (>1000 K) we used a Blöchl-Parrinello electronic thermostat 39 with a target kinetic energy of 0.1 a.u. to control the fictitious motion of the electronic degrees of freedom. Our FPMD approach 32 has been extensively used and benchmarked in the last decades and contributed to the improvement of our understanding of both the physical and chemical properties of chalcogenide and chalcohalide glasses, [40][41][42] hybrid 43,44 and porous 45 systems, and nanomaterials. [46][47][48][49] The Na 3 OCl glass model was generated by quenching from the melt. The initial configuration consisted of a Na 3 OCl crystal cubic unit cell replicated 3 × 3 × 3 times to obtain a 135 atoms model (81 Na, 27 O, and 27 Cl) in a cubic simulation cell of side 13.6148 Å. 50 The initial box was expanded by 15% with respect to crystal density in order to ease melting at a high temperature. A similar approach was followed for the case of glassy Li 3 OCl. 23 Periodic boundary conditions were applied throughout. The model under-went a thermal cycle via canonical NVT simulations, according to the following protocol: 1.3ps ps at T = 300 K, 1.2 ps at T = 1200 K, 3.1 ps at T = 1500 K, 46.5 ps at T = 1800 K, 38.6 ps at T = 1000 K, 33.7 ps at T = 750 K and 35 ps at T = 300 K (see Fig. 1). At the end of the cycle of T = 300 K, the density of the system was set to 2.04 g/cm 3 (corresponding to a final volume expansion of +5% with respect to the crystal phase) in order to reduce the stress tensor to values very close to 0 GPa. Along the thermal cycle the energy cutoff for the plane wave expansion was R Ec = 80 Ry. This value was increased at T = 300 K (R Ec = 160 Ry) to achieve convergence of the stress tensor. By using the final configuration of amorphous Na 3 OCl at T = 300K, we randomly replaced the proper amount of Na atoms with H atoms to obtain Na 2.5 OH 0.5 Cl and Na 2 OHCl (corresponding to ∼17% and ∼33% of Na replaced, respectively). These systems undertook a thermal cycle via canonical NVT simulations with the following time schedule: 20 ps at T = 1800 K, 22 ps at T = 1000 K, 19 ps at T = 750 K, and 30 ps at T = 300 K. To obtain vanishing pressure at T = 300 K, the density of the system was set to 1.93 g/cm 3 (Na 2.5 OH 0.5 Cl), and 1.75 g/cm 3 (Na 2 OHCl) (Fig. 1

inset).

In what follows, the results on the atomic structure are presented as time-average values over the last 20 ps of the thermal cycles at T= 300 K. For all the simulations performed here, we used the developer version of the CPMD package. 51 With respect to the quantification of the disorder degree of the different models, herein we report the main feature of the smooth overlap of atomic positions (SOAP) descriptor employed. In the SOAP formalism, the density of the neighborhood surrounding the i-th atom is smoothed using descriptor functions that encode regions of atomic geometries with respect to atomic neighbor environments within a finite cutoff radius using a local expansion of a Gaussian smeared atomic density ρ i,a (r) with orthonormal functions based onspherical harmonics Y m l ( r) and radial basis functions R n (r). 52,53 In this study, we computed the average summation p(r) vector for the Cl atoms, 54 being the only element present in equal amounts in all three models. The formula for computing the SOAP p(r) vector is given by:

p i,aa ′ nn ′ l (r) = 1 √ 2l + 1 ∑ m (c i,a nlm ) * c i,a ′ n ′ lm (1)
where c i,a nlm is defined by:

c i,a nlm (r) = R n (r) * Y m l ( r) * ρ i,a (r)dr and : ρ i,a (r) = ∑ j δ aa j exp[ -|r i -r j | 2 2σ 2 a ] f cut (r i j ) (2)
where the sum is over neighbours j of element a that is within the cutoff and f cut (r) is a cutoff function that smoothly goes to zero at r cut . The hyperparameter σ a has units of length and determines the regularity (smoothness) of the representation. The absolute value of the SOAP p(r) vector depends on the number of species, number of descriptors calculated, and cutoff radius, making it a valuable tool for analyzing specific systems. For this study, the a=Cl and a ′ is one of those elements Cl, Na, O, or H with a cutoff radius of 5.5 Å that was chosen to encompass the next- nearest neighbor environments. σ a = 1 and with n max = 10 and l max = 8. The final number of descriptors computed correspond to 4185 for Na 3 OCl and 7380 for the two hydroxylated models.

Dynamical simulations at finite temperatures

FPMD simulations allow to assess the transport properties of mobile ions in amorphous materials. [START_REF] Massobrio | Theory and Simulation in Physics for Materials Applications: Cutting-edge Techniques in Theoretical and Computational Materials Science[END_REF]56 In this work, the ions dynamics and transport properties were studied at different temperatures, specifically T = 300 K, 450 K, 600 K, 800 K, 1000 K, and 1200 K (Na 3 OCl only). The three systems were equilibrated in the NVT canonical ensemble for a total simulation time of ∼50 ps. Trajectories were collected every 20 fs. The mean square displacements (MSD) of all ions were computed according to

r 2 (t) = N ∑ i=1 |r i (t) -r i (0)| 2 = ∑ i (∆R i (t)) 2 ( 3
)
where N is the number of the atoms of interest and

∑ i (∆R i (t)) 2
is the MSD averaged on all realizations of Eq. (3) for a given value of the time t. It should be noted that in its standard definition the mean square displacement is normalized to the number of particles, say N, so as to appear as an intensive property resulting from an average over the whole system. However, for sake of convenience, we prefer to take advantage of a definition of MSD taking the form of Eq. (3). To assess the socalled tracer diffusive D * behavior of the ions, MSD is analysed in log-log plots via the slopes β (t) as a function of time,

β (t) = d log ∑ i (∆R i (t)) 2 d log t (4)
At very short t, a value of β (t) equal to two is expected, corresponding to a free, ideally ballistic, motion of the ions, while at very long times, β (t) should reach a value equal to one, corresponding to a real diffusive regime, with the MSD growing linearly with time. We would like to stress here that this analysis is crucial to determine the effective achievement of diffusive behavior and identify the portion of the MSD plot that should be used to extract diffusion coefficients. This is particularly important in the case of disordered materials such as glasses, [START_REF] Habasaki | [END_REF] ionic liquids, 58 and molten salts. 59 In the present work, the lower and upper bounds of the real diffusive regime were defined within the interval delimited by (lower boundary) the t at which β (t) reaches a value of one up to (upper boundary) the value of t corresponding to a length of trajectory 15% shorter than the total one. 60 For the systems that reach the diffusive regime at a given temperature, the ions diffusion coefficients (e.g. tracer diffusivity) were calculated by using the Einstein relation given by: 61,62

D * = lim t→∞ 1 N2d ∂ r 2 (t) ∂t ≃ ∑ i ∆R 2 i N2dt ( 5
)
where d is the dimensionality factor (d = 3 for three dimensional systems) and we have dropped the explicit dependence on time in Eq. ( 5) as a result of the infinite t limit. When the diffusive regime is reached, the Arrhenius equation can be used to calculate the activation energy E a barrier for diffusion (conductivity) by fitting the data of logD * (logσ * ) vs. 1/T as:

D * = D 0 exp - ∆E a kT (6)
From the tracer diffusivity D * , the idealized ionic conductivity can be calculated based on the Nernst-Einstein relation:

σ * = D * Nq 2 V kT = q 2 2dV kT ∑ i ∆R 2 i ( 7
)
where V is the total volume of the model system, q is the charge of mobile-ion species, T is temperature, k is the Boltzmann constant.

A more accurate estimation of the relevant conductivity is instead given by the Einstein formulation of the net charge migration as:

σ ion = D σ Nq 2 V kT = q 2 2dV kT ∑ i ∆R i 2 (8)
The charge-diffusion coefficient D σ , which refers to the displacement of the center of mass of all the diffusing ions and is used here to consider possible correlation effects, is defined as:

D σ = (∑ i ∆R i ) 2 2dNt = ∑ i ∆R 2 i 2dNt + ∑ i ∑ ii̸ =i ⟨∆R i ∆R ii ⟩ 2dNt ( 9
)
where d is the dimension of lattice where diffusion takes place and N refers to the number of diffusing ions. [START_REF] Der Ven | [END_REF][64][65][66] The numerator on the left-hand side was split into a self-diffusion part and a distinct-ion part, the latter accounting for correlations between the movements of distinct ions of the same type (i.e. Na ions). The Haven ratio is a measure of the degree by which the trajectories of different atoms are correlated to each other. [START_REF] Der Ven | [END_REF]67 For single ion conductors, it is

H σ R = D * D σ ( 10
)
becoming equal to one in the absence of any correlation between different diffusing atoms. This is asymptotically approached in the dilute limit, where diffusing atoms rarely encounter each other during ther motion through a crystal. If distinct ions move preferentially in the same direction (positive correlation), then H R > 0 and H R < 1, which is usually observed for single-ion conductors. 66,[68][69][70] At non-dilute concentrations, as in the majority of the solid electrolytes, positive correlations between different diffusing ions become particularly relevant, resulting in H R < 1, whereas negative correlations between different diffusing ions (distinct ions move preferentially in the opposite directions) give H R > 1. To consider all types of ions correlations (positive and negative) involved in the system, namely cation-cation correlations (σ dist ++ ), anion-anion correlations (σ dist --) and cation-anion correlations (σ dist +-) contributing to the Haven ratio, the following equation is used:

H tot R = σ * tot σ tot ion = σ sel f + + σ sel f - σ tot ion = σ sel f Na + σ sel f O + σ sel f Cl σ sel f Na,O,Cl + σ dist NaNa,OO,ClCl + 2σ dist NaCl,NaO,ClO (11)
where

σ tot ion = q 2 2dV kT N ∑ i z i ∆R i 2 (12) σ sel f A = q 2 z 2 A 2dV kT ∑ i ∆R 2 i ( 13
)
σ dist AA = q 2 z 2 A 2dV kT ∑ i ∑ ii̸ =i ⟨∆R i ∆R ii ⟩ ( 14
)
σ dist AB = q 2 z A z B 2dV kT ∑ i ∆R i ∑ j ∆R j (15)
N is total number of ions in the electrolyte, and z i is the charge number of ion i. 58 The sum in Eq. ( 10) can be re-expressed as shown in Eqs. ( 11)-( 13). H σ R is computed by considering correlation effects only promoted by Na ions whereas with H tot R the contribution of possible correlation effects due to O and Cl ions are also taken into account. These equations are an alternative to the methods based on quantifying collective correlated effects by integration of the ions velocities autocorrelation functions. [71][72][73] 3 Results and Discussion

Structural properties

Total X-rays and neutrons and structure factors

The total X-rays and neutrons structure factors (S X T (k) and S N T (k), respectively) can be directly calculated on the equilibrium trajectory in the reciprocal space. 74,75 Alternatively, S X T (k) can be obtained by Fourier transform (FFT) of the real space total pair correlation function g tot (r). We underline that the calculation of S T (k) directly in the k-space is, generally, to be preferred over the Fourier transform (FFT) of the real-space pair distribution functions, since the former approach avoids the effects of the finite range of integration. However, the FFT procedure is currently adopted in the literature to smooth out the noise affecting the structure factors computed directly in reciprocal space, especially in the case of glassy configurations at room temperature. In the present work, we employed the FFT method and the calculated S X T (k) and S N T (k) are shown in Fig. 2. In the following, we analyze X-rays and neutrons FFT S T (k) in terms of peaks positions and intensities comparing the three Na 3-x OH x Cl systems. We remark that S X T (k) of Na 3 OCl features a first peak at about k ∼2.5 Å -1 with a shoulder at k ∼2.0 Å -1 and a second peak at k ∼3.5 Å -1 . These first two peaks are followed by much broader maxima and minima and rapidly damped oscillations up to k ∼8.0 Å -1 . Na 2.5 OH 0.5 Cl and Na 2 OHCl S X T (k) have very similar profiles and minor differences with respect to pristine Na 3 OCl. In contrast, while the X-rays structure factor S X T (k) exhibits similarities across all three cases, the neutron structure factor S N T (k) displays significant distinctions, indicating that it may be a better diagnostic tool for discerning models containing varying amounts of hydrogen. Higher intensities are found in S N T (k) for Na 3 OCl. By increasing x, maxima and minima are reduced and different profiles appear in between k ∼2 Å -1 and k ∼4 Å -1 to signify a lower degree of structural organization. The relatively small peak at about k ∼1 Å -1 reveal that the intermediate range order is very limited in these amorphous systems. Overall, Fig. 2 is indicative of modifications taking place within the networks affecting an ordered structural sequence made of distinct shells of neighbors.

Total and partial pair-correlation functions

The total and partial pair correlation functions for the Na 3-x OH x Cl (x= 0, x=0.5, x=1) systems are shown in Fig. 3 and4, respectively. In terms of total pair correlation functions, the Hfree Na 3 OCl one shows a first narrow peak centered at ∼2.28 Å (with a full width at half maximum (FWHM) of 0.28 Å) and a broader second peak at a lower intensity at ∼2.77 Å (FWHM of 0.47 Å). These two peaks are present also in the hydroxylated systems, although with a lower ratio between the intensities for Na 2.5 OH 0.5 Cl, whereas similar values are observed in Na 2 OHCl. As expected, the presence of H leads to a very sharp peak at about ∼0.98 Å, with FWHM of 0.02 Å. In addition, there is a very small maximum in Na 2 OHCl at ∼1.5 Å. Analyzing the partial pair correlation functions g αβ (r) (Fig. 4) allows us to trace back the role of each individual pair contribution with respect to the total one. In Tab. 1 we report the nearest-neighbor distances r i j identified by the position of the first maximum of the g αβ (r) for the three Na 3-x OH x Cl systems. For comparison, we also report the nearest-neighbor distances found in synthesized pure crystalline Na 3 OCl. 15,76 The first peak of g tot (r) for amorphous Na 3 OCl can be ascribed entirely to the Na-O pair (∼2.28 Å), whereas the second peak is mainly attributed to the Na-Cl pair (∼2.75 Å) and only partially to Na-O and Na-Na (∼3.14 Å) g αβ (r) contributions. The first peak of g NaNa (r) is responsible for the shoulder occurring after the second peak (∼2.8-3.1 Å) found in the total pair correlation function. Analyzing the peak positions of g NaO (r) for amorphous Na 3 OCl, the Na-O bond distance is found slightly larger than the typical ionic bond reported for crystalline Na 3 OCl (∼2.25 Å), whereas the Na-Cl ionic bond distance is shorter than that of crystalline one (∼3.18 Å). The partial pair correlation In Na 2 OHCl, such a peak is minimally visible but a small peak at a shorter distance (∼1.5 Å) is discernible. This latter peak is the one that generally identifies H atoms (or protons) hopping between neighboring hydroxyl groups and giving rise to stable or metastable water molecules. [77][78][79][80][81][82] In terms of anion-anion pair correlation functions, all three g OO (r), g ClCl (r) and g OCl (r) are characterized by a first main peak centered at around ∼3.7-4.0 Å. There are interesting differences to be pointed out when comparing g OO (r) for the three Na 3-x OH x Cl models. In particular, Na 3 OCl exhibits a relatively broad first peak at ∼3.85 Å (FWHM of ∼0.86 Å) whereas the two hydroxylated models feature intense peaks at ∼3.72 Å and ∼3.61 Å (FWHM of ∼0.77 Å and ∼0.72 Å, respectively) followed by a shoulder centered at about ∼4.5 Å. Also, worth of notice is a small peak at ∼2.5 Å for Na 2.5 OH 0.5 Cl and a shoulder at about ∼2.9 Å for Na 3-x OH x Cl while in the case of Na 3 OCl g OO (r) goes to zero for distances <3.2 Å. In terms of g ClCl (r) and g OCl (r), the three models show very similar profiles, although the intensity of the main peak of the two hydroxylated systems is smaller for a higher degree of disorder with respect to Na 3 OCl.

Total and partial coordination numbers

More insights into the Na 3-x OH x Cl networks can be obtained by looking at the total and partial coordination numbers (n i and n i j , respectively), obtained by relying on the pair correlation functions. More precisely, n i j is obtained by integrating the first peak of g i j (r) up to a given cutoff distance corresponding to the position of the first minimum. The total Na, O, Cl, and H coordination numbers are then obtained as follows:

n Na = n NaO + n NaCl + n NaNa + n NaH , n O = n ONa + n OH + n OCl + n OO , n Cl = n ClNa + n ClH + n OCl + n ClCl and n H = n HO + n HCl + n HNa + n HH .
When the first peak of the corresponding g i j (r) is followed by a clear minimum well separated from a second maximum, n i j does not suffer from any ambiguities for its definition, since the running coordination number exhibits a clear plateau (see the behavior in the case of Na-O interactions, inset of Fig. 4). In the opposite case, the determination of n i j is less clearcut (see the Na-Na case in the inset of Fig. 4) and n i j is much more sensitive to the choice of the cutoff. For these reasons, in addition to the individual pair cutoffs taken as the minima after the first peak of g i j (r), we adopted a second cutoff (a "total" one) defined by the minimum after the second peak of g tot (r). The value found (3.3 Å) is determined as the average between the three amorphous models and it allows us to consider within the first coordination shell of Na atoms all the counter-ions (O and Cl) and a number of neighboring Na atoms. With this procedure, we can compare the coordination numbers of amorphous Na 3-x OH x Cl models to those of the crystalline phase of Na 3 OCl. We remind that crystalline Na 3 OCl (cubic phase, stable at T= 300K 15,50,76 ) is characterized by Na atoms coordinated by six counter-ions (two oxygen and four chlorine atoms) within the nearest-neighbor shell, O atoms are coordinated by six Na atoms and Cl atoms are coordinated by twelve Na atoms. Within the cutoff employed above (3.3 Å) each Na atom has also eight neighboring Na atoms, leading to a "total" Na coordination of 14. 50,76 Therefore, the local environment of Na in crystalline Na 3 OC can be thought of as made by a total of 14 neighboring atoms, with neighboring Na interacting through O atoms via Na-O-Na linkages (see Fig. 5a). On the basis of these considerations, we performed the analysis of the coordination numbers n i j of amorphous Na 3-x OH x Cl systems by considering the "partial" coordination number n p ij (definition 1) well adapted to describe the so-called "cation-anion" (i.e. counter-Table 1 Upper part: nearest-neighbour interatomic distances r i j (in Å) identified by the position of the first maximum of the pair correlation functions g i j (r). We also give the first peak full width at half maximum (FWHM). For comparison, we report the values of the crystalline Na 3 OCl phase obtained by experiments. 50,76 Lower part: total and pair coordination numbers as defined by taking different cutoff radii for the definition of a total or a "cation-anion" shell of interactions (see Sec. 3.1.3). Values in parenthesis correspond to the statistical uncertainty on the last reported digits.

Exp. 50,76 FPMD, this work Crystal Na ions) interactions and the "total" coordination number n t ij (definition 2) that is most appropriate to describe the whole set of Na neighbors. The same distinction has been introduced to obtain the distribution of the structural coordination units characterizing the environment of each atom. For a given number of neighbors l and a given atomic species, one can extract from each configuration the chemical nature of the neighbors, providing a detailed description of the network organization. It is important to underline that a coordination number gives an average behavior stemming from all neighbors with no insight into the detailed chemical nature of bonding, while the information provided by the coordination structural units shows how each atomic species organizes itself when connecting to atoms of the same or of a different kind. To help understanding these definitions, Fig. 5 contains snapshots of the local coordinations of the Na, O, and Cl atoms found in crystalline Na OCl as well as typical structural units in amorphous Na 3-x OH x Cl systems issued from our calculations. Also, the insets of Figure 4 show the running integrals of the individual partial correlation function g i j (r) from which we extracted the coordination numbers n t ij and n p ij given in Tab. 1. Focusing on the structural units, Fig. 6 provides their distribution in terms of the number l neighbors for Na, on the basis of definition 2. In terms of total number of neighbours of Na atoms (within 3.3 Å), we observe that amorphous Na 3 OCl exhibits a markedly distinct behavior when compared to its crystalline counterpart.

Specifically, the unit distribution for Na atoms is remarkably wide (from 4 to 11) and centered at l = 8.0, with the number of neighboring counter-ions decreasing from six to four, and the number of Na neighbors halved from eight to four. The partial replacement of Na with H atoms induces a shift of the l units distribution surrounding Na atoms to lower values, centered at ∼7.3 and ∼6.0 for Na 2.5 OH 0.5 Cl and Na 2 OHCl, respectively. By using definition 1, the distributions of structural units for Na, O, and Cl that account for what we termed counter-ions interactions are reported in Tab. 2, together with the breakdown of the chemical composition of each structural unit for a given l.

Chemical identification of the structural units for the three compositions

In the case of Na ). The value of n t O in amorphous Na 3 OCl is about ∼6.3 (Tab. 1), close to the one in crystalline Na 3 OCl. 15,50,76 We find ∼69.1% of O atoms coordinated by six Na atoms and ∼27.6% by seven Na atoms (Tab. 2). For Cl, n t

Cl is equal to ∼5.7, almost half of that found in the crystalline phase (12). Several structural units connected to Cl are noticeable in Tab. 2. In fact, ∼39.8% of Cl atoms are coordinated by six, ∼31.6% by five, ∼17.1% by seven and ∼8.2% by four Na atoms. Having established the main features of the structural units in Na 3 OCl, we can turn to the analogous description for the case of the hydroxylated amorphous materials. In Na 3-x OH x Cl we remark a decrease of n t Na (from ∼8.0 to ∼6.0) from the value in Na 3 OCl. However, n p Na does the opposite (from ∼4.0 to ∼4.6). The decrease in n t

Na is due to the partial substitution of Na by H, with all the H atoms found coordinated to O atoms, forming in majority hydroxyl OH groups (n t H ∼1.0). This is confirmed by the visual inspection of the snapshots relative to Na 2.5 OH 0.5 Cl (Figure 4e) where H is bonded to O atoms in the form of a hydroxyl group (O-H bonding distance ∼0.98 Å), with no H-Na bond formation. In terms of the cation-anion partial coordination, we observe a slight increase of n p Na , the values being ∼4.0 for Na 3 OCl, ∼4.3 for Na 2.5 OH 0.5 Cl and ∼4.6 Na 2 OHCl. Correspondingly, the distribution of the structural units given in Tab. 2 reveals that the percentage of Na atoms fourfold coordinated decreases (from ∼70.0% to ∼56.9% and ∼43.8%, respectively) with a concomitant increase of the fivefold ones (from ∼13.8% up to ∼30.4% and ∼43.7%).

Table 2 Distribution of the individual cation-anion n α (l) structural units where an atom of species α (Na, Cl or O) is l-fold coordinated to a counter ion computed for glassy Na 3 OCl (see definition 1, Sec. "Total and partial coordination numbers"). In bold are reported the total percentages determined for each l-fold coordination. These quantities have been calculated including neighbours separated by a cutoff corresponding to the first minimum in the g αβ (r). For the present work, the individual pair cutoffs used are 3.30, 3.94, 4.10, 5.06, 5.51, and 5.39 Å for, respectively, the Na-O, Na-Cl, Na-Na, O-O, Cl-Cl and O-Cl distances. A total cutoff of 3.30 Å was defined from the total g(r). Error bars are given in parenthesis.

Proportions nα (l) (%) x = 0 x = 0. OHCl, respectively. The amorphous nature of the network favors relatively short Na-Cl distances (∼2.75 Å) when compared to those found in the crystalline phase, where each Cl atom is surrounded by twelve Na atoms at a distance of ∼3.18 Å.

Structural disorder degree

A deeper insight into the assessment of the structural disorder degree of the three Na 3-x OH x Cl compounds is provided by employing the smooth overlap of atomic positions (SOAP) descriptor. 52,83 SOAP is a well-established mathematical scheme widely used for analyzing the disorder degree in various materials such as amorphous graphene 84 and chalcogenide-based phase-change materials. 85,86 Fig. 7a shows a schematic representation of SOAP descriptor and Fig. 7b presents the distributions of the absolute value of the SOAP p Cl (r) vector computed for Na 3 OCl from 300K to 1200K, along with the comparison to the crystalline phase, and from 300K to 1000K for Na 2.5 OH 0.5 Cl and Na 2 OHCl. Additionally, we provide the average SOAP pCl (r) value and its standard deviation. For Na 3 OCl, the crystalline phase at 300K exhibits a narrow SOAP p Cl (r) distribution with pCl = 0.55, with a standard deviation of 0.04. Conversely, amorphous Na 3 OCl at 300K shows a more spread bimodal distribution with two main peaks (approximately 0.45 and 0.52), an average SOAP pCl (r) value of 0.5, and a standard deviation of 0.07. As the temperature increases from 300K to 450K and up to 800K, the p Cl (r) value distributions transition from a bimodal to a symmetrical distribution, while the average SOAP pCl (r) remains relatively constant up to 800K (around 0.5). However, at higher temperatures, a noticeable shift towards lower values is observed (approximately 0.45 and 0.40 at 1000K and 1200K, respectively), indicating a clear increase in the disordering degree of the system, potentially leading to a phase change (e.g., from solid to liquid). Furthermore, the amorphous Na 2.5 OH 0.5 Cl model also exhibits a bimodal SOAP p Cl (r) distribution at 300K, although less structured compared to amorphous Na 3 OCl, whereas Na 2 OHCl shows a more spread SOAP p Cl (r) distribution. Both Na 2.5 OH 0.5 Cl and Na 2 OHCl models demonstrate a symmetrical p Cl (r) distributions from 450K to 600K, with a clear shift towards lower averaged pCl (r) values at temperatures T≥800K. Capitalizing on these insights, for Na 3 OCl we have developed similarity metrics (on a ξCl scale from 0 to 1) to quantify the degree of disorder with reference to the highly ordered crystalline phase and the highly disordered liquid phase. In Fig. 7c and7d, we report the average pCl (r) as a function of temperature, normalized with respect to the average value of the crystalline phase for Na 3 OCl (Fig. 7c), and normalized with respect to the amorphous phase at 300K for the two hydroxylated phases (Fig. 7d) such as: It is pertinent to highlight that, at 300K, the partial substitution of Na ions with H ions leads to heightened disordering, concurrent with a lower temperature threshold for the transition from a solid to a liquid phase (at T>800K for Na 3 OCl and at T>600K for Na 3-x OH x Cl). Furthermore, at 300K, the disordering degree exhibited by Na 2 OHCl is described as a 'slurry' solid, indicating a significantly lower value when compared to Na 2.5 OH 0.5 Cl.

ξ NOC Cl = pi -p1200K pcryst -p1200K (

Dynamical properties

Mean square displacements

A first insight into the dynamical properties of amorphous Na 3-x OH x Cl is provided by the mean square displacement (de-fined in Sec. 2.2 and labeled MSD) of the different ions shown in Fig. 8 at T = 300 K. Additional details can be found in Fig. S2 of the ESI. At T = 300 K, Na ions in Na 3 OCl are mobile, while the MSD of both Cl and O is remains very low. This trend holds at higher temperatures (Fig. S2 of the ESI), with Na ions reaching over the span of ∼50 ps MSD values of ∼ 2.5 Å 2 at T= 300 K, ∼ 4 Å 2 at and 450 K, ∼ 15 Å 2 at T = 600 K, ∼ 48 Å 2 at T = 800 K and ∼ 130 Å 2 at T= 1000 K. These results indicate that amorphous Na 3 OCl behaves differently from its corresponding defectfree crystalline phase, where Na/Li diffusion is absent due to the lack of disorder induced by the presence of vacancies, as found in crystalline antiperovskites via Schottky defect pairs. 87,88 Regarding Cl and O ions, the mobility for both ions at T= 300K is lower than ∼ 1 Å 2 , and increases significantly only at very higher temperature (T≥1000K) indicating a motion beyond a mere local rearrangement. Heenen and coworkers 23 obtained analogous results for both Cl and O ions; yet, the Cl higher mobility was interpreted as not-negligible and, later also termed 'sluggish'. 89 Different behaviors are recorded in the case of the hydroxylated systems. For both the hydroxylated phases, we reported the MSD data of individual oxygen atoms and those bonded to H atoms (as OH -anions) separately (corresponding to about ∼50% of oxygen atoms in Na 2.5 OH 0.5 Cl and only to a minor % for Na 2 OHCl). At T = 300 K, we found that Na and individual O ions in Na 2.5 OH 0.5 Cl have MSD values similar to those found in Na 3 OCl. The MSD of Cl reaches ∼ 0.9 Å 2 over ∼40 ps followed by stabilization, O atoms involved in OH groups show a MSD value of about ∼ 1 Å 2 and whereas H atoms show a steady time increase of MSD. The enhanced mobility of H atoms can be ascribed to the high rotational dynamical disorder of H atoms around an O atom of a hydroxyl group (see Fig. S3 in ESI). Individual O atoms show the least mobility of all, with a MSD stable at ≤ 0.5 Å 2 . This trend is clearly supported by the MSD of Na 2.5 OH 0.5 Cl as a function of temperature. Focusing on Na ions at higher temperature, their MSD after ∼50 ps is ∼1.4 and ∼2.0 times larger than the one found in Na 3 OCl at 450k and 600K, respectively. Therefore, it appears that the presence of H atoms promotes an increase in Na ions mobility at temperatures higher than T = 300 K. This occurs also for Cl ions and both O and H involved in OH groups. In contrast, individual O ions (i.e. non bonded to H) are found somewhat less mobile. To complete our case study and extract a global picture from the behaviour of the mean square displacement, we focus on Na 2 OHCl. A first observation from Fig. 8 (rightmost part) reveals that similar trends for the MSD are found. However, OH - anions boost the MSD of the other species significantly already at room temperature. This effect can be fully appreciated by noting that in Na 2 OHCl ∼92% of O atoms binds at least one H atom either as OH -anions or in water molecules. At T = 300K and after ∼50 ps, Na ions in Na 2 OHCl reached a value of MSD ∼3.2 times larger than the one found in Na 3 OCl, and with H equally highly mobile and with Cl and O involved in OH groups showing a certain degree of mobility although lower than for Na ions. Individual O atoms show a stable MSD value of ≤ 3.5 Å 2 , indicating mere local vibrations. At 300K, by considering density plots spanning the 50-picosecond simulation period (Fig. 8b), we observe that atoms other than Na ions only exhibit localized vibrations around central positions. This vibration's extension increases in the order Na 3 OCl < Na 2.5 OH 0.5 Cl < Na 2 OHCl, with Na ions showcasing extended migration paths. To enhance comprehension of atom dynamics throughout the simulation timeframe, we highlighted in Fig. 8b individual local migration paths for Na, Cl, O, and OH in the density plot visualizations. These visualizations further corroborate our MSD-based observations, with Na ions demonstrating extended migration paths. Meanwhile, individual O ions (not involved in OH groups) exhibit highly localized vibrations, and Cl and OH ions show slightly extended vibrations, though consistently localized. We conclude that all three models at 300K exhibit solid-state dynamical characteristics. At higher temperature, in Na 2 OHCl Na MSD after ∼50 ps is about ∼4.5 times larger than the one found in Na 3 OCl at 450k and 600K. These results indicate, by all means, that the mobility of Na ions is further boosted by the presence of H atoms. Overall, it is evident that Na 3 OCl exhibits a consistent profile from 300K to 800K, wherein Na ions remain highly mobile with their MSD progressively increasing. In contrast, Cl and O ions stay localized around their initial positions. Notably, these latter two ions begin steady migration only from T=1000K. In the hydroxylated phases, Na 2.5 OH 0.5 Cl and Na 2 OHCl, the profiles remain akin to those observed at 300K up to 600K. Both Na and ions experience escalating MSD values, whereas individual O ions (not involved in OH groups) maintain a constant position with minimal vibrations. Remarkably, as Na 2 OHCl features only a minor concentration of individual O atoms, we can assert that only a small portion of the model exhibits solid-like behavior up to 600K. Beyond this temperature, no individual O atoms 'live', as all O atoms are either bonded or interact with H atoms, and the systems behave as liquid phase. These results are fully consistent with the previous structural analysis based on the SOAP descriptor. The Na MSD value obtained for Na 2 OHCl at 450K (∼ 21 Å 2 ) is remarkably larger than the one reported by Dawson et al. 12 for crystalline Li 2 OHCl (∼ 6 Å 2 after ∼50 ps T = 800 K). What we found for Na 2 OHCl differs drastically from what has been reported in crystalline Li 2 OHCl antiperovskites, where only rotational dynamics were highlighted for the hydroxyl OH groups and no long-range mobility. 12 In this crystalline phase, H atoms are characterized by a constant MSD of ∼ 2 Å 2 at T = 600 K on a time scale of ∼50 ps, whereas the present Na 2 OHCl model features an increase of the MSD to a value of ∼ 60 Å 2 after ∼50 ps at T = 600 K (Fig. S2 in ESI).

The paddlewheel effect

We have found that the mobility of Na ions in Na 3-x OH x Cl is strictly related to the presence of hydroxyl OH -anions. An important relationship between the Na-ion transport mechanism and O-H rotation effects in these systems has been reported by Song et al. 90 and Howard et al.. 91 These studies demonstrate that the rapid rotation of O-H bonds leads to the creation of empty space promoting the formation of defects, which, in turn, are crucial for achieving fast, correlated Na-ion transport. The rotational disorder due to the hydroxyl OH -anions, along with other anions such as BH - 4 and NH - 2 , is a feature common to other solid electrolytes, such as crystalline Na 3 OCl 13 and Na 3-x O 1-x (NH 2 ) x (BH 4 ). 92 Ions migration occurs via a mechanism that combines the concerted motion of ions with large quasipermanent reorientations of the surrounding hydroxyl groups. This latter effect, known as the paddlewheel mechanism, 28,93,94 is typically observed in high-temperature crystalline polymorphs. 95 Unlike in crystalline materials, the paddlewheel dynamics in glassy matrices can contribute to the ion mobility also at room temperature, 28 via the strong coupling between the rotational motion of the anions and the translational motion of the alkaline cations. 28,93,95 Based on the above analyses, we propose that H-containing amorphous Na 3-x OH x Cl show three distinct features that facilitate the occurrence of paddlewheel dynamics. The degree to which these dynamics manifest themselves is temperature-dependent. First, these systems contain hydroxyl groups which are free to rotate, impacting the neighbouring cations. Second, the amorphous matrix has a lower density than its crystalline counterpart that makes available additional free volume for O-H rotations, facilitating anion reorientations. Third, while it is typical for most glasses to exhibit a lower density compared to their crystalline form, amorphous Na OH x Cl systems have a potentially unusual characteristic -the absence of covalent bonding in their atomic structure. These features are found in strong accordance with the features of paddlewheel dynamics reported for 75Li 2 S-25P 2 S 5 glass. 28 A more detailed view of the dynamical mechanism is shown in Figure 9a-b. These panels illustrate the rotation of two hydroxyl (OH) groups nearest neighbor to one sodium (Na) ion during Na migration. Throughout the entire process, O(OH) and Na maintain their coordination of six cations and four-to-five anions, re-spectively. The coordination of Na is preserved by means of rotational displacements of the anion through a paddlewheel-type mechanism. Another salient feature observed in these migration events is the dissociation or undocking of Na from a subset of its neighboring anions at the onset of the process, followed by its association or docking to new anions towards the end. 28 Former experimental and computational results have also stirred a debate about the formation of water molecules in antiperovskites. Based on analysis presented above, Na OH 0.5 Cl appears not to show any water formation since all H atoms are bound in OH groups, while our Na 2 OHCl model system does, with ∼9.3 % of O atoms involved in H 2 O molecules. The formation of these water molecules can be elucidated by following, as done in Fig. 10a,a 10c-d). Its transition rate, defined as the number of inter-unit transitions per ps observed within a given process (Hbond or water molecules formation), is of about 1.0 before the formation of a stable new H 2 O molecule. In Fig. 10 panels e-f exemplify the dynamical evolution of the previous environment toward H transfer to a third hydroxyl group (from O 2 to O 3 ), resulting in the formation of a new water molecule. The presented mechanism is very similar to the known Grotthuss one [77][78][79][80][81][82] in which the H atom switches between a σbond and a hydrogen bond to propagate and eventually stabilize in a newly formed water molecule, as in proton-conducting solid-oxide materials.

Ions diffusion

The long-range mobility of alkaline ions in a solid-state electrolyte can be assessed by considering the log-log plot of the MSD (Eq. ( 3) vs t for the temperatures of interest (see Fig. 11). Several regimes are noticeable as a function of time, especially at T= 300 K in our case. The initial part consists of a ballistic regime with the MSD taking a quadratic dependence with time. With increasing t, a plateau is attained corresponding to the so-called caging regime with Na ions trapped in a very limited portion of space by the nearest neighbors. In Na 3 OCl, the cage regime clearly persists up to a ∼1 ps at T = 300 K and ∼0.8 ps at T = 450 K, while it does not manifest itself at higher temperatures. Then, the diffusive regime for Na atoms sets in at longer times with Na ions moving along preferential paths. Under these conditions, the MSD takes a linear dependence on time and the ion self-diffusion coefficients (D * ) can be obtained via the Einstein equation (Eq. ( 5)) at the different temperatures. The onset of the diffusive regime can be monitored via the time behaviour of β (Eq. ( 4)), given in Fig. 11 for Na 3 OCl. As expected, the higher the temperature the faster β approaches ∼1, i.e. at T = 450K, 600K, 800K, and 100K this target value is reached after ∼33 ps, ∼23 ps, ∼15 ps, and ∼8 ps respectively. At 300K, the diffusive regime is not reached within ∼50 ps , the behaviour of β being affected by statistical noise for increasing time. Analogous information on the MSD of the two hydroxylated systems is given in Fig. S2, ESI (linear plot) and Fig. S4, ESI (log-log plot, as in Fig. 11). Remarkably, for T≤450K, the cage regime disappears for Na 2.5 OH 0.5 Cl and Na 2 OHCl. This is consistent with the boosted Na ions dynamics due to OH anions rotational mechanism discussed before. Also, the MSD trend of H atoms features high values at short times (<∼1 ps) for any given temperature, again due to the rotational disordered of OH anions. At longer times MSD profiles of H and Na atoms do superpose as a result of a fully enhanced diffusive behavior concerning all Na Table 3 Na tracer diffusion coefficients (D * ), Na ions tracer (σ * ), charge (σ σ ) and total conductivities (σ tot ), activation energies (E a ) and Haven ratio (H J,tot R ) extrapolated at T = 300 K and directly computed at T = 450 K on the basis of FPMD simulations. We report the values for the three Na 3-x OH x Cl (x= 0, x=0.5, x=1) models.

T (K)

x = 0 x = 0.5 x = 1.0 D * (cm Having identified the temperatures at which Na ions reached the diffusive regime (range 450K-800K for Na 3 OCl and 300K-600K for Na 2.5 OH 0.5 Cl and Na 2 OHCl), we calculated for the three systems the self-diffusion coefficients (tracer, D * ) for Na by using the Einstein equation (Eq. ( 5)) and then converted it into Naion conductivities with Eq. (7). By plotting these data according to the relationship logD * vs 1000/T (Fig. 12) and fitting it in the Arrhenius form, we obtain by extrapolation the uncorrelated tracer conductivity σ * of Na ions at T = 300 K, as customarily done for crystalline solid-state electrolytes. 7,12 D * and σ * values are reported in Table 3, together with the calculated activation energy (E a ) of Na ions transport. We have also computed the tracer diffusion coefficient (D * ) for the least mobile element, namely individual O atoms, at 300K for all three models. The calculated values are 1.7•10 -7 cm 2 .s -1 , 1.5•10 -7 cm 2 .s -1 , and 3.9•10 -7 cm 2 .s -1 for the three respective models. Notably, these values are lower by two orders of magnitude than the typical diffusion coefficients observed in liquids (∼ 10 -5 cm 2 .s -1 ), thus providing additional reinforcement to the previous observations that underline the solid-like nature of the three models at 300K.

Ionic conductivity and correlation effects

In terms of conductivity σ * , we obtained at 300K a value of ∼16 mS.cm -1 for Na 3 OCl and a E a value of Na ions transport of 0.21 eV. The σ * vaue obtained is two order of magnitude larger of the ∼0.2 mS.cm -1 obtained by Dawson et al. at 500 K 7 and showing a similar E a calculated for defected crystalline Na 3 OCl. 1,17 For this model, we also computed the energy barrier E a of Na ions during migration through distinct anion pairs, namely OO, ClCl, and OCl, within the Na 3 OCl model through static transition calculations, as commonly performed for crystalline systems. 15 The outcomes of our computations reveal energy barrier values of 0.23 eV, 0.41 eV, and 0.73 eV, respectively, for the aforementioned anion pairs (Fig. S5 within the ESI). This analysis underscores a significantly higher favorability for migration via the OO pair, in contrast to the ClCl and OCl pairs. It is noteworthy that the minimum energy barrier obtained, which corresponds closely to the value of 0.21 eV obtained from Arrhe-nius fitting of self-diffusion and conductivity data, supports the consistency and validity of our findings. While we acknowledge that the static calculations do not take into account the influence of thermal motion and collective dynamics, their alignment with the Arrhenius fitting outcomes adds weight to the support they provide. It is worth mentioning that, due to the intricate nature of OH rotational dynamics as the main promoter of Na migration in the hydroxylated models, similar static calculations have not been performed for these two compounds as these aspects are better assessed by following the Arrhenius fitting of self-diffusion and conductivity processes extracted from FPMD trajectories. For Na 2.5 OH 0.5 Cl and Na 2 OHCl, we obtained at 300K E a = 0.15 eV, E a = 0.11 eV and a Na tracer conductivity equal to ∼80 mS.cm -1 and ∼280 mS.cm -1 , respectively. These data demonstrate that Na ions transport is promoted by the addition of H atoms. These values correspond to the ideal case where no collective effects are taken into account, as those due to alike-Na ions or different ions. However, since these effects are present in experiments, it is appropriate to account for them by considering the so-called Haven ratio (H R ) which can be obtained via Eqs. (9-10) in order to quantify the contribution of alike-Na ions (H σ R ) [START_REF] Der Ven | [END_REF][64][65]68 or a total contribution coming from all positive and negative ions (H tot R , Eq. 11). 58 Often, previous studies on crystalline Li/Na antiperovskites assumed a Haven ratio equal to 1 disregarding collective effects. 7,12 While single-ion conducting glasses typically exhibit Haven ratios of ∼0.1-0.6 indicative of positive correlating effects, [START_REF] Habasaki | [END_REF][START_REF]Molecular Dynamics of Nanostructures and Nanoionics, Simulations in Complex Systems[END_REF] ionic liquids 58 and molten salts 59 behave oppositely with H R values larger than 1. For comparison, in Tab. 3 we report the values of H σ R , H tot R and the corresponding Na and total conductivities obtained at T = 450 K. We obtained H σ R values larger than 1, indicating a strong negative effect of the collective correlations of alike-Na ions. Therefore, at this temperature, the behaviour of amorphous Na 3-x OH x Cl systems in terms of conductivity is closer to ionic liquids and molten salts than to a glassy matrix. We also remark that this specific effect is more important when Na is partially replaced by H (from 2.0 for Na 3 OCl to 2.7 and 3.5 for the two hydroxylated phases). Finally, the resulting Na charge conductivities (σ ion ) are lower than the uncorrelated Na tracer conductivity σ * for the three systems. In terms of H tot R , we obtained for Na 3 OCl a much lower value than H σ R (0.89 vs 2.03) which indicates significant positive collective correlated effects when the mobility of all ions is taken into consideration. The value obtained for Na 3 OCl is found close to the typical values found for single-ion conducting glasses, [START_REF] Habasaki | [END_REF][START_REF]Molecular Dynamics of Nanostructures and Nanoionics, Simulations in Complex Systems[END_REF] This is not the case for the two hydroxylated models, as shown values always larger than 1 for H tot R meaning that in the presence of H (OH hydroxyl groups) negative effects are more important. The corresponding final total ionic conductivities (σ tot ion ) are moderately larger to the Na ions uncorrelated σ * estimation for the three models Na 3 OCl (0.67 vs 0.17 S.cm -1 ), Na 2.5 OH 0.5 Cl (0.75 vs 0.27 S.cm -1 ), and Na 2 OHCl (0.99 vs 0.77 S.cm -1 ). The σ tot ion obtained here at 450K is finally found 1.5 times higher in Na-deficient Na 2 OHCl than Na 2 OHCl, whereas for the case of Na 2 OHI/Na 2 OHI it was reported an increase of 10 2 times at about 550K. [START_REF] Wang | [END_REF] Therefore, Na ionic conductivity of disordered Na 3-x OH x Cl systems can be a viable, effective alternative to the crystalline counterparts, the partial replacement of Na with H enhancing ion transport. However, the mobility of the other ions (Cl, O, and H) is concomitantly boosted which might be detrimental to practical electrolyte applications.

Conclusions

X 3 OA-based antiperovskites have shown promising potential as solid-state electrolytes for ion batteries due to their structural and dynamical properties. However, while the structure and ion dynamics mechanisms of crystalline X 3 OA compounds are well-known, there is a lack of quantitative structural characterizations for amorphous antiperovskites. In this study, we used first-principles molecular dynamics to quantitatively assess the structure and ion dynamics mechanisms of three amorphous Na 3-x OH x Cl (with x=0, 0.5, 1) Na-rich antiperovskites. We found that amorphous Na 3 OCl shows a substantially different structure compared to its crystalline counterpart, with a disordered structure exhibiting limited intermediate range order and short-range order driven by a majority of four-fold Na atoms (∼70%), in terms of counter-ions coordination by means of ionic bonds. Na-O bond distances are close to those reported in the crystalline phase, while Na-Cl distances are significantly shorter. As a result, a reduced number of neighboring counter-ions and Na atoms are favored in the local surrounding of Na atoms in the amorphous phase. Na is found primarily coordinated to both anions (O and Cl), with only a small percentage of Na atoms coordinated exclusively to O or Cl atoms. These results unveil contrasting differences in the structure of amorphous Na 3 OCl compared to previous models conjectured for glassy Li 3 OCl, which was based on the segregation of Li 2 O and LiCl-rich regions. The partial replacement of Na atoms with H in the two hydroxylated phases studied (Na 2.5 OH 0.5 Cl and Na 2 OHCl) gives rise to a lower degree of structural organization when compared to Hfree Na 3 OCl, with all H atoms bonded to O atoms. At 300K, the partial substitution of Na ions with H precipitates heightened disordering degree concurrent with a lower temperature threshold for the solid-to-liquid transition. Particularly, Na 2 OHCl exhibits a disordering degree we characterize as 'slurry', owing to a degree to be significantly greater with respect to Na 2.5 OH 0.5 Cl. Amorphous Na 3 OCl has a unique microscopic structure that enables remarkable Na ion dynamics and ionic conductivity, challenging that of defective crystalline phases. The presence of hydroxyl OH -anions in the hydroxylated models is crucial for the mobility of Na ions, which is enhanced by rapid rotation of O-H bonds and paddlewheel-type mechanisms. This promotes the formation of available space, which is essential for achieving fast Na-ion transport. Interestingly, the paddlewheel dynamics in the hydroxylated models glassy matrices can contribute to ion mobility even at room temperature, owing to the strong coupling between the rotational motion of the anions and the translational motion of the alkaline cations. In terms of correlated dynamical effects, the behaviour of amorphous Na 3 OCl is found closer to single-ion conducting glasses whereas the two hydroxylated models are found closer to ionic liquids and molten salts than to a glassy matrix.

We also remark a balance in favor of positive correlated effects for Na 3 OCl, whereas for the two hydroxilated phases negative effects are more important. Concerning the presence of water molecules in antiperovskites, our study proves that the presence of H atoms promotes the formation of H-bonds in Na 2.5 OH 0.5 Cl while only in the case of Na 2 OHCl formation of a small amount of water is observed. Overall, the results obtained in this work adds to the growing evidence that disordered Na-rich antiperovskytes provide a viable and effective alternative to their crystalline counterparts as solid state electrolytes. Our results underscore the significance of conducting quantitative structural assessments to better comprehend the impact of structure on the performances of these systems.

a

  Multiscale Materials Modeling Laboratory, Department of Physics, University of Ulsan, Ulsan 44610, Republic of Korea; E-mail: xxxx@aaa.bbb.ccc b Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67034 Strasbourg, Franc;E-mail: guido.ori@ipcms.unistra.fr c Institut de Recherche sur les Céramiques, UMR 7315 CNRS-Univesité de Limoges, Centre Européen de la Céramique, 12 rue Atlantis 87068 Limoges Cedex, France. † Electronic Supplementary Information (ESI) available at DOI: 00.0000/00000000.

Fig. 1 (

 1 Fig. 1 (Color online) Temperatures along the thermal cycle for amorphous Na 3 OCl: light blue line, instantaneous T values; blue line, T values sliding average with a window of 0.1 ps. The dashed vertical grey line sets the time at which optimization of the volume at T = 300K and 0 GPa was implemented. Inset: final density values optimized at T = 300 K for the three Na 3-x OH x Cl models. Error bars are estimated with tests at different volume performed on several final configurations (300 K) to obtain a final stress tensor close to 0 GPa (see Figure S1 in ESI).

Fig. 2 (

 2 Fig. 2 (Color online) Total X-rays (top) and neutrons (bottom) structure factors for amorphous Na 3-x OH x Cl (x= 0, x=0.5, x=1) obtained through Fourier transform (FFT) of the total pair correlation functions at T = 300 K. Insets: zoom-in in the 1.8-4.0 Å -1 and 1.5-4.5 Å -1 ranges, respectively.

Fig. 3 (

 3 Fig. 3 (Color online) Left: Total pair correlation functions for amorphous Na 3-x OH x Cl (x= 0, x=0.5, x=1) obtained at T = 300 K. Right: Zoomin into the 0.8-1.2 (top) and 1.8-3.2 (bottom) ranges.

Fig. 4 (

 4 Fig. 4 (Color online) Partial pair correlation functions g αβ (r) for amorphous Na 3-x OH x Cl (x= 0, x=0.5, x=1) at T = 300 K relative to the pairs Na-O, Na-Cl, Na-Na, O-O, Cl-Cl, O-Cl, O-H, and Cl-H. The running integrals over the interatomic distance are shown.

Fig. 5 (

 5 Fig. 5 (Color online) a) Representation of the unit cell of crystalline Na 3 OCl. The red polyhedra identify the octahedron O site. b) Crystalline supercell of 135 atoms (unit cell replicated 3×3x×3). c-e) Coordination units for Na, O, and Cl found in amorphous Na 3 OCl at T = 300 K: a fourfold Na atom (O 2 Cl 2 ) (c), a sixfold Cl atom (d), a sixfold O atom (e). f) A Na atom found in amorphous Na 2.5 OH 0.5 Cl coordinated by at least an O atom as a hydroxyl group. Counter-ions interatomic bonds are identified by a dashed line with the corresponding values. We show the atoms within a radial cutoff of 3.3 Å. Color legend: Na, yellow; O, red; Cl, green; H, white.

Fig. 7 (

 7 Fig. 7 (Color online) a) Schematic representation of the SOAP descriptor. b) Na 3 OCl, Na 2.5 OH 0.5 Cl, and Na 2 OHCl SOAP p Cl distributions computed at different temperatures. The data computed for crystalline Na 3 OCl is also reported. Structural disorder degree of the three models framed as SOAP similarity to crystalline Na 3 OCl ( ξ NOC Cl , c)) and to amorphous Na 2.5 OH 0.5 Cl phase (, ξ NxOHC Cl , d)), respectively.

  pNa 2.5 OH 0.5 Cl300K -p1000K(17) respectively. For the hydroxilated phases the upper bound 1 of the ξCl metric is defined by the average pCl (r) value of Na 2.5 OH 0.5 Cl at 300K, not having at disposal any information related their crystalline phases and consistently on the basis of the same number of elements and total number of descriptors computed (see Sec. 2.1). These metrics have enabled us to identify ranges of structural disorder corresponding to its crystalline ( ξ NOC Cl =1), amorphous solid (

Fig. 8 (

 8 Fig. 8 (Color online) a) MSD versus time for each element in amorphous Na 3-x OH x Cl (x= 0, x=0.5, x=1) at T = 300 K. Colors legend: Na, yellow; O non-bonded to H atoms, red (solid line); O bonded to H atoms, red (dashed line); H, grey; Cl, green. Note that the MSD calculated at the temperatures T= 300 K, T= 450 K, T = 600 K, T = 800 K, T = 1000 K are reported in the ESI, (Fig. S1). b) Density plots ∼50 ps trajectories of all elements are shown as dots for the three models simulated at 300 K. Na, O, and Cl atoms positions are given in yellow, red, and green, respectively. H atoms are omitted for clarity. Examples of an individual local Na migration path and Cl, O and OH local vibrations are highlighted in spheres.

Fig. 9 (

 9 Fig. 9 (Color online) Proposed paddlewheel dynamics monitored in amorphous Na 2.5 OH 0.5 Cl at 300K. We show the temporal evolutions of two distinct scenarios involving different atoms. a,d) The interatomic distance between a O atom (O 1 and O 2 ) of one hydroxyl group and a coordinated Na atom (Na 1 and Na 2 ) and the HONa angles (H 1 O 1 Na 1 and H 2 O 2 Na 2 ) along the ∼50 ps trajectory. Light orange and purple lines correspond to the instantaneous interatomic distance d OxNax and θ HxOxNax angle, wheres the strong lines correspond to the the sliding average with a window of 0.1 ps. b,d) Atomistic-views of the temporal frames Ty (y = 1-6) indicated in a) and b). In b), between 0 and 5 ps, a sixfold coordinated O 1 atom of one hydroxyl group shows a bonding distance of 2.26 Å with the Na atom labelled Na 1 and an HONa angle of ∼150 • (T1). At 5 ps, the hydroxyl OH bond is rotating with respect to Na 1 (from ∼150 • to ∼100 • ) while maintaining a distance O 1 -Na 1 equal to 2.38 Å. This scenario last up to 17 ps (T2). After 17.5 ps and up to 25 ps a further rotation of the OH group bringing the corresponding angle from ∼100 • to ≤50 • (∼6 • at T3) induces the displacement of Na 1 atom to a larger distance (>4.5 Å) escaping the first coordination shell of O 1 atom. In d) Between 4 and 23 ps, a sixfold coordinated O 2 atom of one hydroxyl group shows a bonding distance of 2.29 Å with the Na atom labelled Na 2 and an HONa angle of 90 • -120 • (T4). At 26 ps, the hydroxyl OH bond is rotating with respect to Na 2 (to ≤50 • ) inducing the displacement of Na 2 atom to a larger distance (>3.6 Å) the first coordination shell of O atom (T5). A further rotation to 3 • is accompanied to a displacement of Na 2 atom to a larger distance (>5 Å, T6). The surrounding atoms of O and O 2 at a distance lower than >5 Åare shown in transparent color. Dashed lines indicate the atoms coordinated within the first coordination shell of O 1 and O 2 (<3.3 Å). The concerted migration pathways of Na 1 and Na 2 atoms along the ∼50 ps are shown in transparent dark yellow. As a guide, an arrow is added to indicate the direction of the pathway. Color legend: Na atoms, yellow; O atoms, red; Cl atoms, green; H atoms, white; Na 1 and Na 2 atoms at Ty points are shown in blue.

Fig. 10 (

 10 Fig.10(Color online) For the Na 2 OHCl system: proton hopping process by which H atoms migrate through a dynamical switch between σand hydrogen-bonds giving rise to the formation of metastable water molecules (i.e. Grotthuss-like mechanism). 77-82 a) Temporal evolution of the interatomic distance between one H atom labelled H 1 atom, belonging to a water molecule initially formed, and its bonded O atom labelled O 1 . Also shown is the temporal evolution of the distance between H 1 and a second O atom labelled O 2 that is part of a neighboring hydroxyl group. Inset: zoom-in on the temporal window along which H 1 is shared between two O 1 and O 2 , leading to the transient formation and breaking of OH bond and water molecules. b-f) Inter-unit transitions between structural units in amorphous Na 2 OHCl at 300 K. Indicatively, the mean lifetimes of the transient H-bonds and water molecules are reported on top of the panels whereas the transition rates are reported between the panels. Here, the transition rate is defined as the number of transitions per ps observed within a given process (H-bond or water molecules formation). Color legend: Na, yellow; O, red, and Cl green.

  typical H hopping process involving OH groups in Na 2 OHCl at T = 300 K. We first show the temporal evolution of two representative distances. The first is the one between the H atom termed H 1 belonging to a water molecule, previously formed, and the O atom termed O 1 atom to which it is bonded (d O 1 H 1 , initially at ∼0.98 Å). The second one concerns the same H 1 atom and a second O atom termed O 2 atom belonging to a neighbouring OH group (d O 2 H 1 , initially at ∼3.8 Å; Figure 9b). After about ∼15 ps, H 1 starts forming a strong hydrogen bond (H-bond) with the neighbouring O 2 atom of one OH hydroxyl

Fig. 11 (

 11 Fig. 11 (Color online) Top: Log-log plot of MSD vs time for Na in Na 3 OCl at temperatures in between T = 300 K and 1200 K. Bottom: corresponding values of β . Inset: β value as a function of time at T = 800 K. Note that the log-log plots MSD vs time data for the Na 3-x OH x Cl (x= 0, x=0.5, x=1) systems are reported in the ESI, (Fig. S2).

Fig. 12 (

 12 Fig. 12 (Color online) Na-ion self-diffusion coefficients (inset) and conductivities of amorphous Na 3-x OH x Cl (x= 0, x=0.5, x=1). We also report the activation energies of Na ionic conduction obtained via a linear Arrhenius fit. Closed circles correspond to FPMD data and open circles correspond to data at T = 300 K extrapolated by using the linear Arrhenius fit.

3

  OCl Na 2.5 OH 0.5 Cl Na 2 OHCl r

	NaO	2.25	2.28(2)	2.27(2)	2.32(2)
	FWHM		0.28	0.31	0.33
	r NaCl	3.18	2.75(2)	2.75(3)	2.74(4)
	FWHM		0.47	0.46	0.52
	r NaNa	3.18	3.14(3)	3.22(4)	3.34(4)
	FWHM		0.92	1.01	1.02
	r OO	4.50	3.85(1)	3.72(4)	3.61(4)
	FWHM		0.86	0.77	0.72
	r OCl	3.89	4.00(4)	3.91(2)	3.78(4)
	FWHM		0.79	1.00	0.93
	r ClCl	4.50	4.06(3)	3.87(3)	4.05(3)
	FWHM		0.82	1.04	1.01
	r OH	-	-	0.98(2)	0.97(2)
	FWHM	-	-	0.02	0.02
	n t Na n p Na n t O n p O n t Cl	14 6 6 6 12	8.01(3) 4.0(1) 6.30 (2) 6.29 (3) 5.73(3)	7.26(2) 4.30(1) 6.44(2) 5.94(2) 5.45(2)	6.04(2) 4.58(2) 6.09(4) 5.40(2) 4.93(3)
	n

p Cl

3

  OCl, the coordination units given in Tab. 2 are indicative of a first coordination shell for Na essentially fourfold (∼70.0%) in terms of cation-anions (i.e. counterions) according to definition 1 (see Sec. 3.1.3). The decomposition in terms of chemical species gives in decreasing order of importance the following units: O 2 Cl 2 (∼36.6%), O 3 Cl 1 (∼18.1%), O 1 Cl 3 (∼9.9%) and O 4 (∼3.7%). Threefold and fivefold units correspond to ∼14.6% and ∼13.8%, respectively, with a larger presence of two or one oxygen atoms (O 2 Cl 1 ∼10.4%, O 2 Cl 3 ∼5.4% and O 1 Cl 4 ∼4.4%). Our results demonstrate that the structure of amorphous Na 3 OCl is significantly different from the molecular dynamics model conjectured for glassy Li 3 OCl, based on nanophase segregation of regions rich in Li 2 O and LiCl. 23 Indeed, by looking at Na atoms fourfold or even fivefold coordinated, in amorphous Na 3 OCl only ∼7.3% of them are exclusively coordinated to O atoms (O 3 and O 4 ) and ∼1.5% exclusively coordinated to Cl atoms (Cl 5

(6.6) 63.6(4.9) 44.7(6.9)

					5	x = 1
	Na
	l=3		14.6(1.9)	8.4(1.0)	3.5(1.0)
		O 2 Cl 1	10.4	4.4	1.3
		O 3	3.6	2.5	0.6
	l=4		70.0(2.9) 56.9(2.2) 43.8(2.1)
		O 2 Cl 2	36.6	23.8	13.1
		O 3 Cl 1	18.1	13.9	14.5
		O 1 Cl 3	9.9	11.8	11.0
		O 4	3.7	5.7	2.6
	l=5		13.8(1.5) 30.4(1.6) 43.7(2.3)
		O 2 Cl 3	5.4	9.7	10.7
		O 1 Cl 4	4.4	6.7	11.6
		O 3 Cl 2	0.4	7.1	13.2
		Cl 5	1.5	2.5	1.5
	l=6		3.3(0.5)	5.5(0.7)	11.1(1.2)
		O 4 Cl 2		-	0.3	2.3
		O 3 Cl 3		-	1.3	2.3
		O 2 Cl 4	0.5	1.0	1.8
		O 1 Cl 5	0.2	1.1	1.7
	O
	l=4			-	<0.5	8.6(1.8)
		Na 2 H 2		-	-	2.0
		Na 3 H 1		-	0.4	6.6
	l=5		2.1(2.0)	20.8(3.9) 44.4(6.6)
		Na 3 H 2		-	-	5.7
		Na 4 H 1		-	16.5	38.6
		Na 5	1.9	4.3	0.1
	l=6	69.1Na 5 H 1	-	24.9	36.0
		Na 6	69.1	38.6	7.3
	l=7		27.6

(6.1) 15.0(3.8) 2.1(1.9)

  OHCl to ∼5.9 and ∼5.4 for Na 2.5 OH 0.5 Cl and Na 2 OHCl, respectively. This decrease is due to a clear increase of lower coordinated units of O at the expenses of the higher-coordinated units. Na 2 OHCl shows a minimal content of l = 7 units and a comparable content of l = 5 and l = 6 units (∼44% versus ∼45%). The trend in the local environment of O in the hydroxilated phases results from the fact that in Na 2.5 OH 0.5 Cl ∼52% of O atoms are coordinated only by Na atoms with the remaining having, in addition to Na, one H atom as neighbor within hydroxyl OH groups. These percentages change in Na 2 OHCl to accommodate more H atoms. Regarding Cl atoms the behavior is similar to the one of O atoms after partial replacement of Na by H atoms. In particular, n Cl decreases from ∼5.7 in Na 3 OHCl to ∼5.4 and ∼4.8, for Na 2.5 OH 0.5 Cl and Na 2

				p
	Na 6 H 1	-	6.1	2.0
	Na 7	27.6	8.9	-
	Cl
	l=3 Na 3 H 0	-	1.2(1.0)	4.8(3.8)
	l=4 Na 4 H 0	8.2(4.6)	14.2(8.7) 31.5(7.3)
	l=5 Na 5 H 0 31.6(8.0) 40.0(8.6) 45.4(7.4)
	l=6 Na 6 H 0 39.8(9.0) 36.4(7.7) 16.9(6.1)
	l=7 Na 7 H 0 17.1(6.4)	7.7(4.4)	-
	l=8 Na 8 H 0	2.4(1.9)	-	-

The larger amount of fivefold Na-centered units at the expenses of fourfold ones can be traced back to a concomitant decrease of the number of O 2 Cl 2 units (from ∼36.6% to ∼23.8% and ∼13.1%) and a net increase of the O 2 Cl 3 , O 1 Cl 4 and O 3 Cl 2 units (the latter from ∼0.4% to ∼7.1% and ∼13.2%). Worth of notice is the behavior of the total and partial coordination numbers of O atoms. p O decreases from ∼6.3 in Na 3

Journal of Materials Chemistry A

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Acknowledgements

This work was supported by National Research Foundation of Korea (NRF) grants funded by the Ministry of Science and ICT (MSIT), government of South Korea (Grant No. 2021R1F1A1048555). We acknowledge for financial support the Agence National de la Recherche (ANR) within the framework of the project AMSES No. ANR-20-CE08-0021, the Seed Money program (project MEDIA) of Eucor-The European Campus, the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS and Inserm. This project has received also funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 847471. Calculations were performed by using resources from GENCI (Grand Equipement National de Calcul Intensif, grant No. A0100807670, A0120807670, and A0140807670) and the Pole HPC Équipe@Meso of the Université de Strasbourg.

Notes and references

Data availability

Representative trajectory files of the three Na 3-x OH x Cl models are available at the European Center of Excellence Novel Materials Discovery (CoE-NOMAD) repository. 98

Author Contributions

Tan-Lien Pham: Investigation, Writing, Formal analysis. Mohammed Guerboub: Investigation, Writing, Formal analysis. Assil Bouzid: Methodology, Writing. Carlo Massobrio: Methodology, Writing. Mauro Boero: Methodology, Writing. Young-Han Shin: Conceptualization, Supervision, Investigation, Writing. Guido Ori: Conceptualization, Supervision, Investigation, Writing.

Conflicts of interest

There are no conflicts to declare.