Structure of crystalline and amorphous materials in the NASICON system Na_1+xAl_xGe_2−x(PO₄)₃

Homogeneous (or internal) nucleation in bulk glass can be used to create a uniform distribution of crystallites of controllable size and shape throughout a material. This process is crucial in the production of many glass-ceramics that include photosensitive and photochromic materials, along with tough fracture-resistant ceramics that have uses ranging from cookware to missile nose-cones.¹ The creation of a uniform distribution of crystallites also makes homogeneous nucleation an attractive option for optimizing the properties of superionic conducting materials for battery and sensor applications. A case example is provided by the technologically relevant Na_1+xAl_xGe_2−x(PO₄)₃ (0 ≤ x ≤ 1) (NAGP) system in which the structure of the crystalline phase is based on that of the sodium (Na) super-ionic conductor (NASICON) NaGe₂(PO₄)₃.² 

Initial work on these highly conducting materials, which are candidates for the electrolyte in all-solid-state sodium ion batteries,^3,4 focused on pressed powders. However, it was discovered that they can also be prepared via the glass-ceramic route, giving the significant advantage of moldable bulk materials.⁵ It is therefore desirable to know the glass structure and how this structure changes during thermal annealing and the early stages of crystal nucleation and growth. The latter is of general scientific interest for any glass undergoing homogeneous nucleation but has not been explored in non-siliceous systems.⁶ 

In this paper, we investigate the structure of NAGP materials with x = 0, 0.4, and 0.8. Here, the structure of the crystalline phase with x = 0 is based on a negatively charged three-dimensional framework in which tetrahedral PO₄ units are linked via bridging oxygen (BO) atoms to four octahedral GeO₆ units. This framework of corner-sharing polyhedra contains interstitial sites in which the Na⁺ ions reside² and where the size of a connection between sites determines the extent to which it acts as a bottleneck for ion motion.⁷ The ionic conductivity increases when a proportion of the Ge⁴⁺ ions are aliovalently substituted by Al³⁺ ions, and additional Na⁺ ions enter interstitial sites to compensate for the charge deficit.⁸ For glassy NAGP materials, the ionic conductivity also increases with x, and the addition of Al improves the glass-forming ability.⁸ The lithium analogs Li_1+xAl_xGe_2−x(PO₄)₃ (LAGP) of these materials nucleate homogeneously⁹ and have already found commercial usage as separator membranes in lithium–air batteries and related systems.^3,10 Comparatively little is known, however, about the Na-containing materials, although solid-state nuclear magnetic resonance (NMR) experiments on the LAGP¹¹ and NAGP¹² systems indicate that all structural aspects of the glass-to-crystal transition are the same, suggesting the same crystallization mechanisms. Homogeneous nucleation in NAGP glasses is supported by the results obtained from differential scanning calorimetry experiments on monolithic vs powdered samples.⁸ 

The structure of the NAGP materials was probed by combining neutron diffraction (ND) and x-ray diffraction (XRD) with ²⁷Al and ³¹P magic angle spinning (MAS) and ³¹P/²³Na double-resonance NMR spectroscopy. The structure of the glass was investigated in its as-prepared condition and after thermal annealing near the glass-transition temperature T_g in order to investigate the initial stages of thermal relaxation. The structure of the fully crystallized glass was also investigated by applying (i) the Rietveld method^13,14 to refine powder diffraction patterns and (ii) the pair-distribution function method. The results for the crystalline materials provided a starting point for interpreting the pair-distribution functions measured for the vitreous materials. Here, the analysis was helped by the provision of site-specific information on the coordination environments of P and Al from the solid-state NMR experiments. In turn, the pair-distribution functions provided information on the Ge coordination environment, which is largely inaccessible from NMR spectroscopy.

This paper is organized as follows. The diffraction theory for pair-distribution function analysis is outlined in Sec. II, and the experimental methods are described in Sec. III. The results are presented in Sec. IV and are discussed in Sec. V where a model is proposed for the glass structure. Conclusions are drawn in Sec. VI.

In a diffraction experiment, the total structure factor¹⁵ 

is measured, where k denotes the magnitude of the scattering vector; c_α ≡ N_α/N denotes the atomic fraction of chemical species α, where N_α is the number of atoms of type α and N = ∑_αN_α is the total number of atoms; w_α(k) represents the k-dependent x-ray atomic form-factor f_α(k) or the k-independent coherent neutron scattering length b_α of chemical species α; the mean value $w(k)=∑αcαwα(k)$⁠; and S_αβ(k) is a Faber–Ziman partial structure factor. Neutral atom form-factors were used in the XRD data analysis.¹⁶ The neutron scattering lengths of the elements in NAGP materials are b_Na = 3.63(2) fm, b_Al = 3.449(5) fm, b_Ge = 8.185(20) fm, b_P = 5.13(1) fm, and b_O = 5.803(4) fm.¹⁷ 

The corresponding real-space information can be represented by the total pair-distribution function

where r is a distance in real space and ⊗ denotes the one-dimensional convolution operator. M(k) is a window function that originates from the finite k-range that is accessible to a diffractometer, which extends to a maximum value k_max. This accessibility has the effect of applying to S(k) − 1 the step modification function defined by M(k) = 1 for k ≤ k_max and M(k) = 0 for k > k_max. M(r) is the real-space manifestation of M(k) and is a symmetrical function. In a neutron diffraction experiment, where the scattering lengths are k independent,

where ρ is the atomic number density and g_αβ(r) is a partial pair-distribution function. At r-values smaller than the distance of the closest approach between two atoms, g_αβ(r) = 0. It follows that, in the absence of Fourier transform artifacts, such as those originating from M(k), D′(r) = −4πρr, which is often called the density line.

In order to distinguish between features in D′(r) that describe the glass structure from those that are an artifact of M(r), each peak i in rg_αβ(r) can be represented by the Gaussian function

where $rαβi$⁠, $σαβi$⁠, and $n̄αβ(i)$ are the peak position, standard deviation, and coordination number of chemical species β around α, respectively.¹⁸ The contribution of each peak $pαβi(r)$ to S(k) − 1 is given by

where $Wαβi=2−δαβcαicβiwαi(k)wβi(k)/w(k)2$ and δ_αβ is the Kronecker delta.

In the case of the ND data, the method of least squares was used to fit D′(r) to a sum of Gaussian functions convoluted with M(r) such that

In the case of the XRD data, each $pαβi(k)$ function was Fourier transformed to real-space using the same M(k) function as used for the experimental data. For a selected region in r-space, the method of least squares was then used to fit an appropriate sum of these Fourier transforms to T′(r) = T(r) ⊗ M(r) using the program PXFIT (Hannon, private communication), where T(r) ≡ D(r) + T⁰(r) and T⁰(r) = 4πρr. For ease of comparison with the neutron diffraction results, the fitted functions are presented as $D′(r)=T(r)−T0(r)⊗M(r)=T′(r)−4πρr$⁠, noting that T⁰(r) ⊗ M(r) = T⁰(r) because M(k = 0) = 1. The goodness-of-fit was assessed by the parameter¹⁹ 

calculated over the fitted range, where “exp” and “fit” denote the measured and fitted functions, respectively.

The use of a step modification function in Eq. (2) not only leads to sharp peaks in D′(r), which is the reason for its application, but can also lead to Fourier transform artifacts that include large excursions below the density line D′(r) = −4πρr on each side of the first peak in D′(r). The latter can be suppressed by using the Lorch^20,21 modification function $M(k)=sinπk/kmax/πk/kmax$ for k ≤ k_max and M(k) = 0 for k > k_max, but at the expense of broadened peaks.

The glasses were made using the method described in Ref. 8. For each composition, part of the glass was kept in its as-prepared condition, part was annealed at a thermal treatment temperature T_TT near to T_g for a time chosen to relax the glass structure while minimizing the formation of crystallites, and part was annealed at T_TT for a time chosen to fully crystallize the material. For the x = 0 composition, two additional parts of the as-prepared glass were taken for further investigation. One part was annealed at T_TT = 876 K for 0.25 h, and the other part was annealed at the same temperature for 0.5 h.

The glass transition temperature T_g and crystallization temperature T_x were measured for ∼20 mg monolithic samples using a differential scanning calorimeter (Netzsch DSC404) with a heating rate of 10 K/min. The mass density of the glasses was measured using a Quantachrome 1200e pycnometer operated with He gas. The density of the crystals was taken from the powder diffraction work. Parameters describing the samples and their thermal history are listed in Table I.

The neutron powder diffraction patterns were measured using the fine resolution powder diffractometer E9 (FIREPOD)²² at the reactor source BER II, Helmholtz-Zentrum Berlin (HZB), Germany. The datasets were collected at ambient temperature (295 K) using an incident wavelength λ = 1.7982(1) Å. The samples were handled in air and held in vanadium cans of 5 mm diameter. The x-ray powder diffraction patterns were measured using a Rigaku SmartLab system with Bragg–Brentano diffraction geometry, monochromatic copper Kα₁ radiation, and a fixed 0.66° divergence slit. The instrument was equipped with a Johansson-type Ge-crystal monochromator and a HyPix-3000 2D single photon counting detector.

Rietveld refinement^13,14 of the crystal structures was performed using the FullProf suite.^23,24 The ND and XRD patterns for each sample were refined simultaneously. The crystal structure data were standardized using STRUCTURE TIDY.²⁵ Crystal structures were drawn using the three-dimensional visualization system for electronic and structural analysis (VESTA) package, version 3.3.1.²⁶ 

Two sets of ND experiments were performed on the NAGP materials to measure the total pair-distribution functions.

The first set of experiments used the D4c diffractometer at the Institut Laue–Langevin²⁷ with λ = 0.4955(1) Å (for the samples annealed at T_TT = 876 K) or λ = 0.4980(1) Å (all other samples). This instrument was chosen because of its stability and high count-rate over a wide k-range, which enables it to probe small differences in structure.²⁸ The powdered samples were held in a cylindrical vanadium container of inner diameter 6.8 mm and wall thickness 0.1 mm. Diffraction patterns were measured at room temperature (≃298 K) for each of the samples in its container, the empty container, the empty instrument, and a cylindrical vanadium rod of diameter 6.08 mm for normalization purposes. The diffraction pattern was also measured for a slab of neutron absorbing ¹⁰B₄C of dimensions similar to the sample in order to estimate the effect of the sample’s attenuation on the background count-rate at small scattering angles.²⁹ The data analysis followed the procedure described in Ref. 30.

The second set of experiments used the diffractometer GEM at the ISIS pulsed neutron source.³¹ This instrument was chosen because it accesses a much larger k_max value than D4c, thus improving the resolution of the peaks in D′(r). The powdered samples were held in a cylindrical vanadium container of inner diameter 6 mm and wall thickness 40 µm. Diffraction patterns were measured at room temperature (≃298 K) for each of the samples in its container, the empty container, the empty instrument, and a cylindrical rod of the alloy V_0.9486Nb_0.0514 (diameter 7.95 mm) for normalization purposes. The datasets were processed using the Gudrun analysis program³² with inelasticity corrections calculated according to Ref. 33.

In each set of experiments, the relative counting times for the sample-in-container and empty container measurements were optimized in order to minimize the statistical error on the container-corrected intensity.³⁴ Self-consistency checks were also made to assess the reliability of the measured functions. For instance, it is necessary that (i) each of the measured total structure factors satisfies the sum-rule relation $∫0∞dkk2S(k)−1=−2π2ρ$⁠; (ii) the measured D′(r) functions oscillate about the density line D′(r) = −4πρr at r-values smaller than the distance of the closest approach between two atoms; and (iii) when these oscillations in D′(r) are set to the density-line values, the back Fourier transform is in good overall agreement with the original total structure factor.

The XRD experiment was performed on beamline 6-ID-D at the Advanced Photon Source with an incident photon energy of 80.019 keV. The scattered x-rays were counted using a Varex 4343CT amorphous silicon flat panel detector, which was placed at a distance of 271.932 mm from the sample position as deduced from the diffraction pattern measured for crystalline CeO₂. The powdered glass samples were held in cylindrical Kapton polyimide tubes of 1.00(1) mm internal diameter and 0.051(6) mm wall thickness. Diffraction patterns were collected for each sample in its container, an empty container, and the empty instrument. The data were converted to one-dimensional diffraction patterns using FIT2D,³⁵ and the corrections for background scattering, beam polarization, attenuation, and Compton scattering were made using PDFgetX2.³⁶ 

Single-pulse ²⁷Al MAS NMR spectra were measured using a Bruker AVANCE 600 spectrometer with a 2.5 mm probe operated at 20.0 kHz. A pulse length of 0.34 µs, corresponding to a flip angle of π/6 and relaxation delays in the range 0.1–2 s, was used. Chemical shifts are reported against a 1M solution of Al(NO₃)₃ using Al(PO₃)₃ (isotropic chemical shift δ_iso = −20.8 ppm) as a secondary reference. Line shapes were fitted to the Czjzek model³⁷ simulating a wide distribution of quadrupolar coupling constants, as implemented in DMFit.³⁸ 

Solid-state ³¹P NMR experiments were performed using an Agilent DD2 spectrometer interfaced with a 5.64 T magnet and a commercial 3.2 mm probe operated at a spinning speed of 10.0 kHz. Chemical shifts were referenced using powdered BPO₄ (δ_iso = −29.27 ppm against H₃PO₄ 85%) as a secondary reference. Experimental spectra were simulated using the DMFit software.^38 23Na{³¹P} rotational echo double resonance (REDOR) experiments³⁹ were performed using a Bruker AVANCE 600 spectrometer with a 2.5 mm triple resonance probe and spinning frequency of 10 kHz, operated with the following conditions: ²³Na detection using a rotor synchronized Hahn spin-echo sequence with 180° refocusing pulses of 3.25 µs length and a ³¹P dipolar re-coupling π-pulse of 3.5 µs length, at a spinning speed of 15.0 kHz and a relaxation delay of 1 s. ³¹P{²³Na} rotational echo adiabatic passage double resonance (REAPDOR) spectra were measured with the same apparatus used for the REDOR experiments and the pulse sequence proposed by Gullion,⁴⁰ applying all the 180° pulses on the ³¹P detection channel and the adiabatic passage pulse (1/3 of the length of the rotor period) to the ²³Na nuclei. Spinning and nutation frequencies were 10.0 and 80 kHz, respectively.

The NASICON aristotype crystallizes into space group-type R$3̄$c, as originally determined by Hagman and Kierkegaard⁴¹ for NaZr₂(PO₄)₃. In the solid solution series Na_1+xTi_xGe_2−x(PO₄)₃ at room temperature, the boundary phases NaTi₂(PO₄)₃ (NTP) and NaGe₂(PO₄)₃ (NGP) crystallize into this structure-type and the lower symmetry form R$3̄$⁠, respectively.^2,42 The R$3̄$c and R$3̄$ space groups have a group–subgroup relation that leads to a splitting of crystallographic sites. According to the literature, the solid solutions crystallize into the NGP-type phase for x ≤ 1 or the NTP-type phase for x > 1.^2,42,43

The crystal structures of the NAGP compounds were refined using the Rietveld method. The x = 0 and x = 0.4 compositions were assigned to the NGP-type structure, whereas the x = 0.8 composition was assigned to the higher symmetry NTP-type structure. The Rietveld refinements are shown in Figs. 1–3 where, for each experiment, the intensity is plotted vs sin θ/λ = k/4π and 2θ is the scattering angle.¹⁵ The corresponding crystal structure data are summarized in Tables II–IV where SOF represents a site occupation factor; B_iso represents a Debye–Waller factor; and R_exp, R_p, and R_wp represent the expected, profile, and weighted profile R-factors,⁴⁴ respectively.

The crystalline NAGP compounds were found to be almost phase pure. For the x = 0 sample, the XRD pattern shows some peaks from a non-determined impurity phase, but according to the peak intensities, the phase fraction is <1%. For the x = 0.4 sample, no impurities could be identified. For the x = 0.8 sample, some impurity peaks and shoulders are present and the intensities suggest a phase fraction of a few percent. The diffraction background for the x = 0 and x = 0.4 samples is flat, which suggests full crystallization. The background for the x = 0.8 sample is increased slightly, suggesting some amorphous residue.

Maldonado-Manso et al.⁴³ reported differences between the nominal and actual compositions of NAGP and related compounds prepared via a solid-state reaction. To check for non-stoichiometry, different refinement strategies were tested. In the initial model, only the phosphorus SOF was set to unity. Refinement did not show any indication for non-stoichiometry at the oxygen sites, so the occupancies of these sites were also set to unity. The germanium site occupancies for the x = 0 sample were refined and showed negligible deviation from full occupancy. Hence, for the aluminum substituted samples, the combined occupation factor SOF(Ge) + SOF(Al) for each site was set to unity, but the Al/Ge-ratio was allowed to vary. The diffraction patterns for the x = 0 sample can be reproduced with two fully occupied sodium sites Na(1) and Na(2). Nevertheless, the aluminum substituted samples show occupancy of an additional Na(3) site. Accordingly, this site was included in the refinement of the x = 0 data with coordinates fixed at those obtained for the x = 0.4 sample. The refinement showed the Na(3) site to be unoccupied within the estimated standard deviation. For the NAGP samples with nominal compositions x = 0, 0.4, and 0.8, the refined compositions were found to be Na_0.99(2)Ge_1.96(2)(PO₄)₃, Na_1.42(2)Al_0.43Ge_1.57(2)(PO₄)₃, and Na_1.72(2)Al_0.804Ge_1.196(12)(PO₄)₃, respectively.

The crystal structures obtained from powder diffraction are illustrated in Fig. 4. They serve as a starting point for interpreting the pair-distribution functions. A summary of the coordination numbers and bond lengths is given in Table V.

The structures have a single phosphorus site in which a phosphorus atom is coordinated tetrahedrally by four symmetry independent oxygen atoms. The tetrahedra are quite regular with a narrow distribution of P–O distances ranging from 1.50 to 1.54 Å. The phosphorus site occupation was set to unity in all the Rietveld refinements.

In the lower symmetry R$3̄$ structure, there are two sixfold coordinated germanium sites. The Ge(1) site has three O(1) and three O(3) neighbors, while the Ge(2) site has three O(2) and three O(4) neighbors. In the higher symmetry R$3̄$c structure, the germanium sites are equivalent, i.e., Ge(1) = Ge(2) and O(1) = O(2) with O(3) = O(4). The octahedra are fairly regular with a narrow distribution of Ge–O distances ranging from 1.83 to 1.90 Å for all compounds. Sixfold coordinated Ge⁴⁺ and Al³⁺ ions have similar ionic radii of 0.67 and 0.675 Å,⁴⁵ respectively, so no change in bond lengths is expected as Ge is substituted by Al in the NAGP compounds.

In the lower symmetry R$3̄$ structure, there are two sixfold coordinated sodium sites Na(1) and Na(2). The Na(1) site is at the center of a trigonal prism formed by six symmetry equivalent oxygen atoms O(3), and the Na(2) site is similarly coordinated by O(4) oxygen atoms. In the higher symmetry R$3̄$c structure, the Na(1) and Na(2) sites are equivalent. In the x = 0 and x = 0.4 compounds, the Na(1) site is fully occupied and the SOF was set to unity during the last refinement cycle. The Na(2) site is almost fully occupied and was refined. In the x = 0.8 compound, the single Na(1)/Na(2) site is fully occupied and its SOF was set to unity. The Na(1)–O and Na(2)–O distances are similar in all three compounds, ranging from 2.39 to 2.52 Å. The crystal structures also have a low-symmetry Na(3) site at the center of an irregular polyhedron of eight oxygen atoms with a broad distribution of Na(3)–O distances ranging from 2.04 to 2.78 Å (Table V). This Na(3) site also has two phosphorus atoms at a similar distance (2.6–2.8 Å) to several of the oxygen atoms, and there is a third Na(3)–P distance of about 3 Å, which is slightly shorter than the Na–Na and the Na–Ge/Al distances. In the x = 0 compound, the Na(3) sites are unoccupied. In the x = 0.4 and x = 0.8 compounds, however, the Na(3) site occupancy is 14% vs 24%, respectively. According to the review by Guin and Tietz,⁷ the sodium ions move from Na(1) to Na(2) sites via Na(3) sites. The high mobility leads to large Debye–Waller factors for the sodium atoms.

The ³¹P MAS NMR spectra taken for both the glassy and crystallized NAGP samples with x = 0, 0.4, and 0.8 confirm the results obtained in previous work.¹² The datasets for the x = 0 composition are shown in Fig. 5. The spectrum for the as-prepared glass is about 25 ppm wide and somewhat asymmetric. Its center of gravity [−28.9 ppm, Fig. 5(a)] suggests that P⁽³⁾ units are the majority phosphate species,¹² where the superscript denotes the number of BO atoms per P atom. The spectrum differs significantly from that measured for the crystalline state [−37.4 ppm, Fig. 5(d)], indicating that the local structures in the glassy and crystalline materials are quite different. For the crystal, the chemical shift of −37.4 ppm characterizes a $P4Ge(4)$ unit, in which phosphorus is connected to four sixfold coordinated Ge atoms. The spectrum of the as-prepared glass can be deconvoluted into two Gaussian components at −25.3 and −36.5 ppm having an area ratio of roughly 2:1 (Table VI), consistent with the prediction of the structural model to be introduced in Sec. V C.

There are, however, many other possibilities for deconvoluting the ³¹P MAS NMR spectrum into two Gaussians. Independent evidence in favor of this particular deconvolution is therefore required. Such evidence is provided by a REAPDOR experiment⁴⁰ where, in the ³¹P{²³Na} double resonance experiment conducted here, the magnetic dipole–dipole coupling strength between the observed ³¹P nuclei and the surrounding ²³Na nuclei was probed. This magnetic dipole coupling, which is averaged out by magic-angle spinning, can be re-introduced by irradiating the ²³Na nuclei on resonance during the rotor period, resulting in a decrease in signal intensity. A comparison is then made between the intensities S₀ of the regular MAS NMR signal and S of the MAS NMR signal observed with simultaneous irradiation of the ²³Na nuclei for a fixed dipolar re-coupling time. The corresponding difference in signal intensity ΔS = S₀ − S depends on the strength of the dipolar coupling between the ³¹P and ²³Na sites and thus has the potential to distinguish between different local environments in a glass.

Figure 6 shows the Fourier transforms of the echoes obtained after a 1.4 ms mixing time in NAGP glass with x = 0. The data indicate that the dipolar dephasing is indeed non-uniform as a function of the isotropic chemical shift: The ³¹P nuclei on the high-frequency side of the resonance display stronger dipolar dephasing than those on the low-frequency side. Based on these data, it was possible to achieve a consistent deconvolution of the echo signal with dipolar dephasing (which emphasizes the less-strongly coupled ³¹P nuclei) and the difference signal ΔS (which emphasizes the more-strongly coupled ³¹P nuclei), in terms of two line shape components near −27(2) and −37(1) ppm (see Fig. 6). Table VI summarizes the line shape parameters for the two-component fit for the single-pulse spectra and the REAPDOR data. In the latter experiments, the ratio of areas of the high-frequency component to the low-frequency component was found to be lower than 2:1 for the S signal and higher than 2:1 for the ΔS signal, consistent with the difference in dipolar coupling strengths between the two sites. The component near −27 ppm is identified with P⁽³⁾ units, and the component near −37 ppm is identified with P⁽⁴⁾ units. The chemical shift of the latter is close to that found for the P⁽⁴⁾ units in the crystalline state.

Figures 5(b) and 5(c) show the ³¹P MAS NMR spectra for the annealed glasses with composition x = 0. In the case of the sample annealed at 873 K for 0.25 h, the data clearly show that the material remains completely amorphous. Some subtle line shape changes are, however, evident (Fig. S1), which may be attributed to a structural re-arrangement in the glass caused by the onset of the nucleation process. For example, the full-width at half-maximum (FWHM) of the line shape increases from 21.7(5) ppm for the as-prepared glass to 22.8(5) ppm for the annealed glass. Table VI indicates that the change to the principal line shape involves a change to the ratio of the relative areas of the two line shape components, indicating a moderate increase in the fraction of P⁽⁴⁾ units having the same local coordination environment as the crystalline phase. This finding was confirmed by fixing the chemical shifts and linewidths for the P⁽³⁾ and P⁽⁴⁾ sites at the average of the values listed in Table VI for the as-prepared and annealed glasses and fitting the spectra for both glassy samples with the peak areas as the only adjustable parameters. The peak areas showed a change in the ratio of P⁽⁴⁾:P⁽³⁾ sites from 33.5(1.0):66.5(1.0) for the as-prepared glass to 36.0(1.0):64.0(1.0) for the annealed glass. In the case of the sample annealed at 876 K for 0.5 h, the change in peak areas is more dramatic, with the P⁽⁴⁾:P⁽³⁾ ratio increasing to ∼1:1. For this sample, a small amount of crystalline material is also detected.

Finally, the ²³Na/³¹P double resonance experiments also allow for a comparison between the ³¹P/²³Na dipolar coupling strengths of glassy and crystalline materials by plotting the normalized difference signal as a function of the dipolar mixing time NT_r (= number of rotor cycles × rotor period). The corresponding ³¹P{²³Na} REAPDOR curves are compared in Fig. 7(a) and qualitatively illustrate that the internuclear ³¹P/²³Na dipole–dipole interactions are significantly stronger in the crystalline rather than in the glassy material. A more quantitative estimate can be found on the basis of the ²³Na{³¹P} REDOR results³⁹ shown in Fig. 7(b). In this experiment, the dephasing of the ²³Na nuclei in the dipolar field created by the ³¹P spins is being observed. The simpler spin dynamics arising from the spin-1/2 character of the recoupled ³¹P nuclei permits an approximate analysis in terms of dipolar second moments $M2Na–P$⁠, using the formula

in the limit of short mixing times when ΔS/S₀ ≤ 0.2–0.3,⁴⁶ where f is a calibration factor on the order of unity. The quantity $M2Na–P$ can be calculated from the internuclear distances using Van Vleck theory;⁴⁷ it is proportional to the sum of the inverse sixth powers of all Na–P internuclear distances present and can thus be computed from any crystal structure. For crystalline NAGP with x = 0, a theoretical value $M2Na–P$ = 4.1 × 10⁶ rad² s⁻² is expected, and the experimental data obtained by us on this compound indicate an f-value of 0.72. Applying this calibration factor to the REDOR data obtained on the glass gives $M2Na–P$ = 3.3 × 10⁶ rad² s⁻², i.e., a 20% lower value as compared to the crystalline compound. It follows that the mean Na–P⁽⁴⁾ distance back-calculated from $M2Na–P$ in the NAGP crystal is shorter than the average of the Na–P⁽³⁾ and Na–P⁽⁴⁾ distances back-calculated from $M2Na–P$ in the glass.

The ²⁷Al MAS NMR spectra taken for the glassy and crystallized NAGP samples with x = 0.4 and 0.8 are shown in Fig. 8. The line shapes were fitted using a Czjzek model³⁷ that accounts for a distribution of quadrupolar coupling constants originating from a distribution of local field gradients. The fitted parameters are listed in Table S1.

For the glasses, the spectra feature broad resonances at isotropic chemical shifts around 46(1), 15(1), and −12(1) ppm that can be attributed to aluminum in fourfold, fivefold, and sixfold coordination environments, respectively. For some of the glassy samples, a sharp peak near −20 ppm is observed, which indicates a small amount of a yet unidentified crystalline impurity with aluminum in sixfold coordination. For the x = 0.8 composition, the presence of this site in the as-prepared glass, but its absence in the annealed material, suggests a non-uniform distribution of more ordered regions when the glass is first formed.

For the crystallized materials prepared by thermally annealing the glass for an extended period of time, the ²⁷Al MAS NMR spectra show that crystalline NASICON is the dominant phase. The dataset for the x = 0.8 composition shows a small amount of fourfold coordinated aluminum in an AlPO₄ impurity phase. The results for both compositions indicate that a small fraction of aluminum atoms remain in a glass-like domain where they occur in all three coordination environments, which contributes toward an average Al–O coordination number less than six (Table VII).

In Table VII, the results for the aluminum speciation are compared to those obtained for a different set of NAGP samples in previous work.¹² For the as-prepared glass, there is some variation in the speciation that may be influenced by the sample preparation procedure although the NMR spectra in Ref. 12 were measured at a lower field strength of 5.7 T (cf. 14.1 T in the present work) where the resolution of the individual ²⁷Al resonances is substantially lower. The results show an increase in the mean Al–O coordination number when the as-prepared glass is thermally annealed.

The S(k) functions measured by ND (using GEM or D4c) and XRD are shown in Figs. 9–12. For the crystalline materials, there are discrepancies between the intensities of Bragg peaks measured using GEM vs D4c that originate from (i) differences between the k-space resolution function of the diffractometers and (ii) the possibility of some preferred orientation of the crystallites (Fig. 9). For the glassy materials, the GEM and D4c datasets for the as-prepared glasses are in overall agreement (Fig. 10). The XRD patterns indicate a small amount of crystallinity for the annealed x = 0 and x = 0.8 samples (Fig. 12) that is not observed using ND with D4c, which may reflect the enhanced k-space resolution of the x-ray diffractometer. For all the glassy materials, the S(k) functions display a shoulder at k₁ ∼ 0.85 Å⁻¹ that is indicative of ordering on a real-space length scale of 2π/k₁ ∼7.4 Å. In Na₂O–GeO₂–P₂O₅ glasses, the appearance of a similar feature has been linked to an inhomogeneous distribution of Na and Ge atoms.⁴⁸ 

For crystalline NAGP with x = 0, the D′(r) functions measured using GEM and XRD are shown in Fig. 13. By comparison with the powder diffraction work (Table V), the first peak at 1.52 Å and the second peak at 1.86 Å originate from P–O and Ge–O correlations, respectively, and the third peak will have a contribution from both Na–O and O–O correlations. The D′(r) functions were therefore fitted by using the following protocol.

A single peak was used to represent the P–O correlations. A single peak was also used to represent the Ge–O correlations in the ND work, whereas two peaks were used to represent these correlations in the XRD work, where the large atomic number of Ge (Z = 32) makes the results particularly sensitive to the germanium coordination environment. A single peak was used to represent the Na–O correlations with a coordination number fixed at the value found from the Rietveld refinement. Two peaks were used to represent the O–O correlations because each BO atom is shared between a PO₄ tetrahedron and a GeO₆ octahedron, i.e., it has three nearest-neighbor O atoms within a tetrahedron and four nearest-neighbor O atoms at a longer distance within an octahedron (the fifth O atom within an octahedron is at an even longer next nearest-neighbor distance). The coordination numbers for these peaks were fixed at $n̄OO=3$ and $n̄OO=4$⁠, respectively. The fitted D′(r) functions gave $n̄PO$ = 3.58(5) with $n̄GeO$ = 5.98(5) for GEM [Fig. 13(b)] vs $n̄PO$ = 3.78(3) with $n̄GeO$ = 5.90(1) for XRD [Fig. 13(c)]. In comparison, for the x = 0 glass, a similar fit to D′(r) using a single Gaussian peak for the P–O correlations gave $n̄PO$ = 3.32(5) (GEM) vs $n̄PO$ = 3.61(3) (XRD).

The P–O coordination number found from both ND and XRD is smaller than the value $n̄PO$ = 4 expected from Rietveld refinement and from ³¹P MAS NMR experiments (Secs. IV A and IV B). In the latter, the measured chemical shifts are typical of phosphorus in a fourfold coordination environment and are quite different to the chemical shift of phosphorus in a PO₃ coordination environment, which is observed near 100 ppm in the crystalline molecular systems P₄O₇ and P₄O₆S, as referenced to an aqueous solution of H₃PO₄.⁴⁹ A measured P–O coordination number $n̄PO<4$ is typical of diffraction work on phosphate materials,^48,50,51 where the shortfall is often attributed to the k-space resolution of the diffractometer.^19,52

Several steps were taken in order to investigate the origin of the reduced $n̄PO$ values. In this analysis, the GEM dataset for crystalline NAGP with x = 0 was used as an exemplar because (i) the large k_max value of ≃39 Å⁻¹ leads to well-resolved peaks in D′(r), (ii) the analysis is simplified by the absence of Al–O correlations, and (iii) the crystal structure is known.

1. The coherent neutron scattering length of P is reported to be in the range 5.0–5.3 fm^53–55 with a recommended value b_P = 5.13(1) fm.¹⁷ The data analysis was repeated using b_P values that extend over this range, but a notable increase in $n̄PO$ was not observed.

2. The dataset was reanalyzed using the density ρ = 0.0843(1) Å⁻³ measured by He pycnometry. A fit to the D′(r) function using the strategy reported above led, however, to a reduced coordination number $n̄PO$ = 3.44(5) with $n̄GeO$ = 5.71(5).

3. The Lorch²⁰ modification function was used in Eq. (2) to suppress Fourier transform artifacts, which facilitates the extraction of coordination numbers by direct integration over the peaks in D′(r). Here, the mean coordination number of atoms of type β, contained in a volume defined by two concentric spheres of radii r_i and r_j centered on an atom of type α, is given by

   $n̄αβ=4πρcβ∫rirjdrr2gαβ(r).$

   (9)

   The resultant D′(r) function is shown in Fig. 13(a). Integration to the first minimum at 1.67 Å gives $n̄PO$ = 3.7(1), whereas integration to the second minimum at 2.19 Å gives an increased value $n̄PO$ = 4.3(2) if the Ge–O coordination number is set at $n̄GeO=6$⁠. The measured D′(r) function therefore has sufficient area to accommodate a P–O coordination number of 4, which suggests a broader distribution of P–O distances than found by fitting a single peak.

4. In view of the findings in step (3), the fitting procedure was adapted to include a second P–O Gaussian peak. This requirement may originate from different coordination environments for the BO atoms in a phosphate tetrahedron. All these BO atoms are shared between P and Ge atoms but only two [O(3) and O(4)] have nearest-neighbor sodium atoms, where the Na–O distance is 2.39–2.52 Å (Table V). The fitted GEM and XRD D′(r) functions are shown in Figs. 14(a) and 15(a), respectively, and the peak parameters are summarized in Tables S2 and S3. Inclusion of the second P–O peak increases the P–O coordination number to $n̄PO≃$ 3.9 and leads to a Ge–O coordination number $n̄GeO$ = 5.90(4)–5.95(5), which compares to an expected value of $n̄GeO=6$⁠. The Ge–O distance of 1.857(2) Å obtained from ND is in agreement with the mean Rietveld value of 1.857 Å (Table V), and the Ge–O distances of 1.847(1) and 1.871(1) Å obtained from XRD compare to average short and long Rietveld Ge–O distances of 1.841 and 1.873 Å, respectively (Table V).

The GEM results for the crystalline material (Table S2) provided starting parameters for fitting the D4c dataset. The fitted D′(r) function is shown in Fig. 16(a), where the peaks are broadened relative to GEM because of the smaller value k_max = 22.25 Å⁻¹ of D4c and the peak parameters are summarized in Table S4. The fit gives $n̄PO$ = 4.00(4) with $n̄GeO$ = 5.97(5).

For the as-prepared glass, the GEM and XRD datasets were fitted using two Gaussian peaks representing the nearest-neighbor P–O correlations and one Gaussian peak representing the nearest-neighbor Ge–O correlations. Additional peaks at larger-r values were used in order to constrain the fitted functions at lower r. The fitted D′(r) functions are shown in Fig. 14(b) (ND) and Fig. 15(b) (XRD), and the P–O and Ge–O peak parameters are summarized in Tables S5 and S6. The GEM results provided starting parameters for fitting the D4c dataset for the as-prepared glass, where the fitted D′(r) function is shown in Fig. 16(b) and the P–O and Ge–O peak parameters are summarized in Table S7. The results give a P–O coordination number of $n̄PO$ = 4 within the experimental error and an average Ge–O coordination number $n̄GeO∼$ 5 that is significantly smaller than found for the crystalline phase.

For the annealed glass, the fits were initiated using starting parameters obtained for the as-prepared glass. The fitted D′(r) functions are shown in Fig. 15(c) (XRD) and in Fig. 16(c) and Fig. S2 (ND). The P–O and Ge–O peak parameters are summarized in Tables S8 and S9. The results do not indicate a change to $n̄PO$ but show that, relative to the as-prepared glass, $n̄GeO$ increases with the time of thermal annealing.

The GEM and XRD D′(r) functions for crystalline and glassy NAGP with x = 0.4 and x = 0.8 are shown in Figs. 17–20. By comparison with the powder diffraction results (Table V), the first peak in each D′(r) function at ≃1.52 Å is attributed to P–O correlations, the second peak at ≃1.86 Å will have contributions from both Ge–O and Al–O correlations, and the third peak will have contributions from both Na–O and O–O correlations. In view of the results obtained in Sec. IV C 1, the nearest-neighbor P–O correlations in D′(r) were represented by two Gaussian peaks. The parameters describing the Gaussian peaks for the nearest-neighbor Al–O correlations were constrained using the information obtained from other techniques.

For the crystalline materials, the Al–O coordination number was fixed at $n̄AlO$ = 6 and the position of the Gaussian peak was set at the mean nearest-neighbor Al–O distance of 1.86 Å found from powder diffraction (Table V). The third peak was treated in the same way as D′(r) for crystalline NAGP with x = 0. The fitted GEM D′(r) functions for the x = 0.4 and x = 0.8 compositions are shown in Figs. 17(a) and 18(a), respectively, and the peak parameters are summarized in Table S2. The fitted XRD D′(r) functions for the x = 0.4 and x = 0.8 compositions are shown in Figs. 19(a) and 20(a), respectively, and the peak parameters are summarized in Table S3. The P–O coordination numbers are in the range 3.85(3)–3.99(3), and within the experimental error, the Ge–O coordination number is consistent with the powder diffraction value $n̄GeO$ = 6.

The GEM results (Table S2) provided starting parameters for fitting the D4c datasets for the crystalline materials. The fitted D′(r) functions for the x = 0.4 and x = 0.8 compositions are shown in Figs. 21(a) and 22(a), respectively, and the fitted P–O and Ge–O parameters are summarized in Table S4. The fitted coordination numbers are $n̄PO=4$ and $n̄GeO=6$ within the experimental error.

For the glassy materials, three Gaussian peaks were used to represent the nearest-neighbor Al–O correlations for fourfold, fivefold, and sixfold coordinated aluminum atoms. The relative abundance of these species was fixed in accordance with the aluminum speciation found from ²⁷Al MAS NMR experiments (Table VII). The peak positions for fourfold and fivefold coordinated aluminum atoms were set at 1.76 and 1.84 Å, respectively, which are the Al–O bond lengths calculated from the bond valence method.⁵⁶ The position of the peak for sixfold coordinated aluminum atoms was set at the average Al–O bond length of 1.86 Å found from the powder diffraction work (Table V). The width of the Gaussian peak describing AlO₄ units was set at 0.05 Å, which is a typical value for AlO₄ units in alumino–phosphate glasses,⁵⁰ and the width of the Gaussian peak describing AlO₆ units was set at 0.07 Å, which is representative of the value found in the crystalline NAGP materials with x = 0.4 and x = 0.8 (Table S2). The width of the Gaussian peak describing AlO₅ units was set in between these values at 0.06 Å. Additional peaks at larger-r values were used in order to constrain the fitted functions at lower r.

For the as-prepared glasses with x = 0.4 and x = 0.8, the fitted GEM D′(r) functions are shown in Figs. 17(b) and 18(b), respectively, and the fitted P–O and Ge–O peak parameters are summarized in Table S5. The results provided starting parameters for fitting the D4c datasets for the as-prepared glasses. The fitted D4c D′(r) functions for the x = 0.4 and x = 0.8 compositions are shown in Figs. 21(b) and 22(b), respectively, and the fitted peak parameters are summarized in Table S7. The fitted XRD D′(r) functions for the as-prepared glasses with x = 0.4 and x = 0.8 are shown in Figs. 19(b) and 20(b), respectively, and the fitted P–O and Ge–O peak parameters are summarized in Table S6. The results give a P–O coordination number of $n̄PO$ = 4 within the experimental error and indicate a reduction in the average Ge–O coordination number as more aluminum is added to the glass structure.

The XRD and D4c results obtained for the as-prepared glasses provided starting parameters for fitting the datasets for the annealed glasses, where the aluminum speciation was taken from ²⁷Al MAS NMR experiments (Table VII). The fitted XRD D′(r) functions for the x = 0.4 and x = 0.8 compositions are shown in Figs. 19(c) and 20(c), respectively, and the fitted P–O and Ge–O peak parameters are summarized in Table S8. The fitted D4c D′(r) functions for the x = 0.4 and x = 0.8 compositions are shown in Figs. 21(c) and 22(c), respectively, and the fitted P–O and Ge–O peak parameters are summarized in Table S9. The results do not show a change to $n̄PO$ but, for each composition, indicate an increase in $n̄GeO$ relative to the as-prepared glass.

The P–O and Ge–O bond distances and coordination numbers obtained from the ND and XRD pair-distribution function analyses are summarized in Table VIII. The diffraction results for the glasses show Ge–O coordination numbers that are markedly smaller than $n̄GeO=6$⁠, which indicates the presence of polyhedral GeO₅ and/or GeO₄ units. This observation of a substantial difference between the crystal and glass structures is supported by solid-state NMR experiments.¹² For example, the ³¹P chemical shifts differ by more than 10 ppm; P–O–P linkages are absent in the crystal but may be present at low concentrations in the glass; and P⁽³⁾ phosphate units dominate the structure of the glasses but are absent in the isochemical crystals where all the phosphate units are P⁽⁴⁾. In the case of NAGP with x = 0, ²³Na spin-echo decay spectroscopy finds that the second moment quantifying the strength of Na–Na interactions is about twice as large in the glass as compared to the isochemical crystal, which suggests that some Na–Na distances are shorter in the glass than in the crystal. Furthermore, Al is exclusively six-coordinated in the aliovalently (Al + Na) substituted crystalline materials but is found in a mixture of fourfold, fivefold, and sixfold coordination environments in the glass (Table VII).

In order to investigate the network connectivity via the BO vs non-bridging oxygen (NBO) content of the NAGP materials, it is instructive to consider a simple model based on the oxygen coordination number.⁴⁸ Let us assume that all oxygen atoms are part of a network and occupy either NBO or BO sites with coordination numbers of one or two, respectively. Then, the fraction of NBO atoms is given by

where N_NBO is the number of NBO atoms and N_O is the total number of oxygen atoms. For a fully polymerized network, $n̄OP+n̄OGe+n̄OAl=2$ and f_NBO = 0. For a fully depolymerized network, $n̄OP+n̄OGe+n̄OAl=1$ and f_NBO = 1. The model does not allow for the formation of threefold coordinated oxygen atoms as supported, for example, by ¹⁷O NMR experiments on GeO₂–P₂O₅ glass.⁶⁴ The number of O-β bonds between oxygen and atoms of chemical species β is equal to the number of β-O bonds, i.e., $NOn̄Oβ=Nβn̄βO$⁠, where N_β is the number of β atoms, such that $cOn̄Oβ=cβn̄βO$⁠. Hence, the fraction of NBO atoms can be re-written as

In the following, it will be assumed that all the nearest-neighbor oxygen atoms to Ge and Al are BO atoms, in accordance with the crystal structures where each of these oxygen atoms is shared with a phosphorus atom. The number of NBO atoms per phosphorus atom is given by $NNBO/NP=cO/cPfNBO$⁠, where N_P is the number of phosphorus atoms. Hence, if the P–O coordination number $n̄PO=4$ and NBO atoms reside on phosphorus atoms alone, the number of BO atoms per phosphorus atom

In the case of glassy (GeO₂)_x(NaPO₃)_1−x, for example, O-1s x-ray photoemission spectroscopy (XPS) experiments indicate that most of the NBO atoms are bound to P and not Ge, e.g., 100% for x = 0 vs 82.5% for x = 0.5.⁶⁵ For M₂O–P₂O₅ materials, Eq. (12) will also give the nearest-neighbor P–P coordination number $n̄PP$⁠. The number of NBO atoms per M atom $NNBO/NM=cO/cMfNBO$⁠, where N_M is the number of M atoms. The number of BO atoms per M atom $NBO/NM=cO/cM1−fNBO$⁠.

The nearest-neighbor O–O coordination number is given by

where the fraction of BO atoms f_BO = 1 − f_NBO. The coordination numbers $n̄BOO$ and $n̄NBOO$ for the BO and NBO atoms will depend on the network connectivity. For example, the structure of crystalline NAGP with x = 0 is based on a network of corner-sharing tetrahedral PO₄ and octahedral GeO₆ units in which each BO atom is shared between a tetrahedron and an octahedron. Each BO atom therefore has three nearest-neighbor O atoms within a tetrahedron and four nearest-neighbor O atoms within an octahedron such that $n̄OO=n̄BOO=7$⁠.

The parameters predicted by the structural model for a variety of crystalline M₂O–Al₂O₃–GeO₂–P₂O₅ materials (Table IX) match the measured values. For the cases of crystalline NAGP^2,43 and LAGP,⁶³ the Al³⁺ ions substitute at the sites of the Ge⁴⁺ ions such that $n̄AlO$ = $n̄GeO$ = 6, so there is no change to N_NBO/N_P or N_NBO/N_M and f_NBO = 0. By contrast, in glassy NAGP, the Ge–O and Al–O coordination numbers take values smaller than 6, so NBO atoms appear. The fraction of NBO atoms increases as the composition is changed from x = 0 to x = 0.8, leading to an increase in N_NBO/N_P and a decrease in N_NBO/N_Na [Figs. 23(c)–23(e)]. The N_NBO/N_P ratios are consistent with a mixture of P⁽³⁾ and P⁽⁴⁾ species, as found from ³¹P solid-state NMR experiments (Sec. IV B). If all the phosphate groups are either P⁽³⁾ or P⁽⁴⁾, it follows that the fraction of P⁽³⁾ species for the as-prepared glass is 0.618(44) at x = 0 vs 0.770(52) at x = 0.8. The annealing procedure leads to a reduction in f_NBO for each of the glasses, corresponding to a small increase in the Ge–O and Al–O coordination numbers, which leads to a decrease in both N_NBO/N_P and N_NBO/N_Na.

A challenge is to provide a structural model for NAGP glasses in which the phosphate species are constrained to be either P⁽³⁾ or P⁽⁴⁾ and the mean Ge–O and Al–O coordination numbers are both substantially larger than four. A starting point is provided by the model of Ren and Eckert⁶⁶ for sodium phosphosilicate glasses, which invokes super-structural units containing sixfold coordinated silicon atoms (Fig. 24). For simplicity, the focus in the following will be on the NAGP glass with x = 0. BO atoms will be denoted by Ø, and as mentioned before, the number of BO atoms per Ge or P atom will be denoted by a superscript.

The NAGP glass composition for x = 0 can be written as (Na₂P₆GeO₁₈)_1/2(GeO₂)_3/2. In the model of Ren and Eckert, the Na₂P₆GeO₁₈ part of the glass is built from [Ge⁽⁶⁾$P6(3)$Ø₁₀Ø_4/2O₆]²⁻ super-structural units, where the negative charge on a unit is compensated by two Na⁺ ions, and the GeO₂ part of the glass is built from charge-neutral corner-sharing tetrahedral Ge⁽⁴⁾ units. The model therefore contains only P⁽³⁾, Ge⁽⁴⁾, and Ge⁽⁶⁾ species. All NBO atoms reside within the P⁽³⁾ motifs of a super-structural unit, where charge-shuttling ensures that they carry a partial charge of −(2/6)e = −0.33e and e is the elementary charge. The model delivers a coordination number $n̄GeO$ = 4.5 for NAGP glass with x = 0, which is smaller than the value found from the diffraction experiments (Table VIII).

A larger coordination number can be obtained, however, by using the NBO atoms within the P⁽³⁾ motifs of a super-structural unit to convert Ge⁽⁴⁾ to Ge⁽⁶⁾ units according to the following scheme:

In the crystal structures (Sec. IV A), P⁽⁴⁾–Ø–P⁽⁴⁾ linkages are absent, i.e., the phosphate chains in the model of Ren and Eckert must dissociate as Ge⁽⁶⁾ species join to expand a super-structural unit. It is therefore proposed that a proportion y of the initial super-structural units are converted to [$Ge(6)P6(4)$Ø₆Ø_18/2]⁴⁺ units according to the following scheme: