Tracking Sodium-Antimonide Phase Transformations in Sodium-Ion Anodes: Insights from Operando Pair Distribution Function Analysis and Solid-State NMR Spectroscopy

Operando pair distribution function (PDF) analysis and ex situ 23Na magic-angle spinning solid-state nuclear magnetic resonance (MAS ssNMR) spectroscopy are used to gain insight into the alloying mechanism of high-capacity antimony anodes for sodium-ion batteries. Subtraction of the PDF of crystalline NaxSb phases from the total PDF, an approach constrained by chemical phase information gained from 23Na ssNMR in reference to relevant model compounds, identifies two previously uncharacterized intermediate species formed electrochemically; a-Na3–xSb (x ≈ 0.4–0.5), a structure locally similar to crystalline Na3Sb (c-Na3Sb) but with significant numbers of sodium vacancies and a limited correlation length, and a-Na1.7Sb, a highly amorphous structure featuring some Sb–Sb bonding. The first sodiation breaks down the crystalline antimony to form first a-Na3–xSb and, finally, crystalline Na3Sb. Desodiation results in the formation of an electrode formed of a composite of crystalline and amorphous antimony networks. We link the different reactivity of these networks to a series of sequential sodiation reactions manifesting as a cascade of processes observed in the electrochemical profile of subsequent cycles. The amorphous network reacts at higher voltages reforming a-Na1.7Sb, then a-Na3–xSb, whereas lower potentials are required for the sodiation of crystalline antimony, which reacts to form a-Na3–xSb without the formation of a-Na1.7Sb. a-Na3–xSb is converted to crystalline Na3Sb at the end of the second discharge. We find no evidence of formation of NaSb. Variable temperature 23Na NMR experiments reveal significant sodium mobility within c-Na3Sb; this is a possible contributing factor to the excellent rate performance of Sb anodes.

c-Na 3 Sb = 0.2181/0.9682 = 0.2252 therefore 22.5% of the total c-Na 3 Sb has been formed (NB: this accounts for approx. 0.68 Na per Sb). This represents 0.2181/0.85 = 0.26 = 26% of the sodiated phase (accounting for the fact that only 85% of the Sb is in sodiated phases). If we don't account for any sodium being present in the SEI, the calculation gives x = -0.17 (this would imply that it is over-sodiated). This is unlikely to be the case, given that a surplus of 0.375 Na per Sb are added throughout the sodiation, and the 23 Na NMR indicate that these are formed as surface species. The least-squares refinement of the Na 3 Sb structure against PDF data for this structure implies a contracted c-parameter, suggesting that there are vacancies in the structure which cause the layers to sit closer together, consistent with a undersodiated version.

At the end of D-1a
At the end of sodiation, Na 3 Sb is formed, but 3.375 Na per Sb have been inserted. We assume that the 0.375 Na per Sb used in side reactions and are no longer active in the electrode. Therefore, the starting electrode stoichiometry is 3 Na per Sb. During D1-a, 1.125 Na per Sb are removed from the electrode, resulting in a whole electrode stoichiometry Na 1.875 Sb. The electrode is a mixture of c-Na 3 Sb and the new amorphous phase, a-Na x Sb.
From high-r (20 -50 Å) PDF refinements, the scale factor of the c-Na 3 Sb phase is 0.140, compared to 0.9682 at the end of sodiation. Therefore, 0.140/0.9682 = 0.144 (14%) of the Sb is still present as c-Na 3 Sb, 0.8556 as the a-Na x Sb phase. Therefore, 1.875 = 0.1444(#Na per Sb in Na 3 Sb) + 0.8556(x in a-Na x Sb phase) x = (1.875 -0.144(3) )/ 0.8556, x = 1.68 Therefore, the stoichiometry of the a-Na x Sb phase is approximately Na 1.7 Sb At the end of D1-b We use the same assumption about SEI as above. During D1-b, 0.9 Na per Sb are removed, making the overall stoichiometry Na 0.975 Sb. No c-Na 3 Sb remains, but a very small amount of c-Sb appears (Scale factor = 0.036=> 0.036/0.988 = 0.037 is present as c-Sb. 0.963 is present as the amorphous antimony network. Therefore, the stoichiometry of the amorphous Sb network: 0.975 = 0.963(x), x = 1.01.
At the end of D1-c 0.3375 Na per Sb are removed during D1-c, making the overall electrode stoichiometry Na 0.6375 Sb. The electrode is a mixture of c-Sb and an amorphous Sb phase. From high-r (20 -50 Å) PDF refinements, the scale factor for the c-Sb is 0.404 (compared to 0.988 for the pristine electrode). Therefore, 0.395/0.988 = 0.3998 (40%) of Sb present as c-Sb, 0.6002, as amorphous phase, Na x Sb. x = 0.6375/0.6002, x = 1.0425, so the amorphous phase stoichiometry is approximately Na 1.0 Sb.
Note, that comparing the area of the amorphous-component PDF extracted from the residual from high-r refinements against c-Sb, and the crystalline component PDF first peaks (assuming that both represent 3 x Sb-Sb bonds -no significant defects in the a-Sb) gives 33% c-Sb, 67% amorphous a-Na 1.0 Sb phase: a similar result to the previous calculation. Figure S1: Comparison of the first peaks of the amorphous (left) and crystalline (right) PDF components at the top of charge. Open circles show experimental data, red line is the fit to experimental data. Grey lines give Gaussian peaks fitted for first (solid line) and second (dashed line) peaks in the experimental PDF. Relative areas of the first peaks to each other labelled.
At the end of S2-a During S1-a, 0.5625 Na per Sb are added to the electrode. Total electrode stoichiometry is Na 1.2 Sb The electrode is a mixture of c-Sb and an amorphous Sb phase. From high-r (20 -50 Å) PDF refinements, the scale factor for the c-Sb is 0.357 meaning 0.375/0.988 = 0.3613: 36% is as c-Sb, 63% as a-Na x Sb. The stoichiometry of the amorphous phase is: 1.2 = 0.63(x) = 1.90, the amorphous phase is Na 1.90 Sb.
At the end of S2-b During S2-b, 0.675 are added into the electrode, making the total electrode stoichiometry Na 1.875 Sb. Scale factor for Sb is 0.0501, meaning 0.0501/0.988 = 5.1% of the c-Sb remains. Therefore, total stoichiometry of the amorphous phase(s) is Na 2.0 Sb. At this point it is a mixture of the Na 1.7 Sb and the Na 3-x Sb. The amount of each amorphous phase can be estimated by using a linear combination of Na 1.7 Sb (amorphous PDF from end of D1-a -this was scaled by 0.821818 to account for the fact only 86% of the Sb was in this phase at the end of D1a) and Na 3-x Sb (from end of S2-c). Linear combination = x( PDF for a-Na 1.7 Sb) + (1-x)(PDF for Na 3-x Sb). The sum of squared differences between the linear combination of these phases and the experimental data is minimised by varying the relative amounts of each phase and an overall scale factor: Sum of differences = ∑ ሺGሺrሻexpt -Gሺrሻlinear combinationሻ 2 This gives a reasonable match to the experimental PDF (below) using a phase distribution of 53% as Na 1.7 Sb, 47% as a-Na 3-x Sb. If this is the case, then the total electrode stoichiometry should be: y = 0.53(1.7) + 0.47 (2.43) = 2.04 => Na 2 Sb (compared to Na 1.90 from electrochemical measurements) Figure S 2: Linear combination fit to PDF at the end of S2-b using the PDFs for Na 1.7 Sb (from the end of D-1a) and the Na 3-x Sb (from the end of S2-c). The difference between the linear combination and the experimental data is shown offset for clarity.

The end of S2-c:
During D1-c 0.5625 per Sb are added to electrode, making stoichiometry Na 2.4625 Sb. At this stage, there is no c-Na or c-Na 3 Sb in the electrode, so all sodiation is present in as an amorphous phase. There is no peak left at 2.85 Å, indicating that all the Na 1.7 Sb has been sodiated further. Therefore, all sodium is present as a-Na 3-x Sb. In this case, x is estimated to be 0.54 (compared to 0.4 from 1 st sodiation).
The total charge capacity of the electrode is 79% of the theoretical value. In the above calculations, we assume that this is due to sodium being trapped within Na x Sb phases in the electrode, and that the whole electrode remains in electrical contact during the desodiation and 2nd sodiation processes, for the reasons discussed in the text. However, for completeness we also consider the sodiation level of the amorphous phases, were loss of electrical contact responsible for the lower desodiation capacity, below: Comparison of stoichiometries calculated using 100% "active" electrode and 79% "active" electrode: a-Na 3-x Sb: S1-a and S2-c (after 450 mV process): if we consider 100% of the electrode to be "active": S1-a: x = 0.31-0.36, S2-c: x = 0.53 if we consider 79% of the electrode to be "active": S1-a x = 0. 31-0.36, S2-c: x = 0.89 Dumbbell phase: 100% active electrode D1-a = Na 1.7 Sb; S2-a = Na 1.9 Sb 79% active electode: D1-a = a-Na 1.2 Sb; S2-a = Na 1.2 Sb

SI.2 PDF data for first sodiaton, first desodiation and second sodiation.
Data are offset for clarity. The colour of the curve corresponds to colours of the points on the electrochemical curve. Figure S3: PDFs for the first sodiation S8 Figure S4: PDFs for the first desodiation S9 Figure S5: PDFs for the second sodiation S10

SI.3 PDFs for amorphous and crystalline components of electrode
PDFs for the amorphous (left) and crystalline phases (right) as a function of sodiation. Amorphous phases are extracted from the residual of a two-phase (c-Sb and c-Na 3 Sb) refinement over the full r-range (2 -50 Å) with structural parameters fixed to those values determined from refinements at high-r (20 -50 Å). These have been r-averaged over the period of the termination ripples. The PDFs for the crystalline phases were calculated from structural parameters in the same two-phase refinements using PDFGui. Where the crystalline and amorphous component data are shown on different scales, this is indicated on the y-axis. Figure S6: Amorphous (left) and crystalline (right) components of the electrode for the first sodiation S11 Figure S7: Amorphous (left) and crystalline (right) components of the electrode for the first desodiation S12 Figure S8: Amorphous (left) and crystalline (right) components of the electrode for the second sodiation S13

SI.4 Density-functional theory calculations
A simple species-swapping procedure was used to generate likely candidate structures of Na x Sb, this method has been used to successfully generate low-energy structures in the Li-Si, Li-Ge 4 and Li-S 5 systems. All of the known stoichiometric crystal structures of Li-P, Li-As, Li-Sb, Na-P, Na-As, Na-Sb, K-P, K-As and K-Sb were obtained from the International Crystallographic Structure Database (ICSD). For each structure the anions were replaced by Na and the cations by Sb. All of the structures were then relaxed using forces calculated using the CASTEP 6 density-functional theory (DFT) code. The Perdew-Burke-Ernzeof (PBE) exchange-correlation functional was used with CASTEP on-the-fly Vanderbilt ultrasoft pseudopotentials, (Na 2|1.3|1.3|1.0|16|19|21|20U:30U:21(qc=8) and Sb 2|2.0|2.0|1.6|4|7|8|50:51). The Kohn-Sham eigenvalues were represented by a basis set containing plane waves with energies of up to 700 eV and Fourier Transform grids were set to represent without aliasing frequencies up twice the size of the basis set. A Monkhorst−Pack grid corresponding to a Brillouin zone sampling grid finer than 2π×0.03 Å −1 was used.
A convex hull was generated between Na (Im-3m) and Sb (R-3m) see Figure S9. Average voltages were calculated from ground state energies 7 .  For these, and all subsequent refinements shown, black circles show experimental data, green lines shows the calculated PDF. The difference between the experimental and model PDFs (defined as G(r) experiment -G(r) calculated ) is shown by the black line, offset for clarity.   Fits were performed using the SOLA function in Topspin. All parameters were refined during the fit. The expected ratio of the two peaks is 2:1. Due to the larger C Q of site 2, some intensity is lost to spinning sidebands, resulting in a slightly larger ratio of integrals.  Variable-temperature 23 Na NMR spectra were recorded at MAS rate of 10 kHz in an external field of 16.4 T. The temperature was determined using a previous calibration of gas-flow rates performed using Pb(NO 3 ) 2 . Simulations were carried out using the EXPRESS simulation package within Matlab. For the simulations, the 23 Na quadrupolar and chemical shift parameters extracted from the experimental spectrum for Na 3 Sb at 268 K were used. Powder averaging was performed using a ZCW1597 tiling scheme and 500 Hz Gaussian line broadening was applied prior to Fourier transformation of the simulated free induction decays. Two sites are observed at 22 ppm and 17 ppm which integrate to approx. 1:1 intensity, with very small quadrupoles, corresponding to the two Na-sites in the structure. The MQMAS spectrum was recorded using a z-filtered pulse. Asterisks denote spinning sidebands. The amount of sodium residual in the electrode grows during the desodiation process due to plating of the removed sodium onto the counter electrode resulting in texture which could not be removed from the PDF. The amount of sodium shows little change during the second sodiation. At all times, the contribution to the PDF of the sodium is very small compared to the peaks from the Na x Sb phases at low-r.   S23 Figure S17: Refinement of the Sb structure against PDF data for the pristine electrode extracted from the in situ electrochemical cell. R w = 11.2 %

Figure S 18:
Left axis (red crosses): change to the antimony phase scale factor during the first sodiation in one-phase refinements (normalised with respect to the scale factor for the pristine electrode). Right axis (green circles): change to R w for one-phase least-squared refinement of Sb versus PDF data in the distance range 20 -50 Å. Very little change to the scale factor is observed until approximately 0.8 Na per Sb has been added to the electrode.

Figure S 19:
Contributions of c-Sb, c-Na 3 Sb and a-Na 3-x Sb to the PDF obtained at a total electrode stoichiometry of Na 2.7 Sb on the first sodiation. Contribution of phases is calculated using the parameters obtained during a three-phase least-squares refinement against PDF data.

Figure S 20:
Comparison of the PDF a-Na 3-x Sb obtained at the end of S2-c with that obtained during the S1-a. Calculated PDFs for NaSb, NaSb-helix and parallel dumbbell connectivities were calculated using PDFGui. For NaSb, the PDF was calculated from the structure of Cromer et al. 5 For the NaSb helix, a supercell containing only a single helix with no additional correlations in the a and c directions within 10 Å. For the PDF of dumbbells, a subset of the NaSb structure containing pairs of dumbbells with the correct orientation was chosen and all other atoms removed. Linear combinations of the PDFs for c-Na 3 Sb, a-Na 1.0 Sb and a-Na 1.7 Sb were used to model the intermediate datasets during D1-a and D-1b using both two and three phase models. The PDF for c-Na 3 Sb was taken from the end of S1-b, the PDF for a-Na 1.0 Sb from the end of D1-b and the PDF for a-Na 1.7 Sb was taken from the amorphous component of the PDF obtained at the end of D1-a.
The sum of the squared differences between the data and the linear combination (G(r) experiment -[xG(r) a + yG(r) b ]) 2 (where a and b are the PDFs for the two phases) over the whole r-range, was minimised by varying the phase fraction of the two phases. A three phase fit was achieved by using three phase scale factors and a global scale factor.
When only c-Na 3 Sb and a-Na 1.0 Sb were used, the fit to the data at the end of D1-a was very poor, as shown in the figures below and is reflected in the large value for the sum of least-squares. The fit after incorporation of the additional a-Na 1.7 Sb phase was at least as good as the two phase fit in all regions, with a significant improvement shown in the intermediate region.
It is on the basis of these results and from the visual change in amorphous component of the electrode during desodiation (Figure S8b), as well as the 23 Na NMR results presented in the text, that we propose the formation of the additional phase, a-Na 1.7 Sb, during D1-a.  Bottom left: phase fractions determined by minimisation of squared differences by varying phase fraction using two phase (c-Na 3 Sb, a-N 1.7 Sb crosses, c-Na 3 Sb, a-Na 1.0 Sb open circles) and three phase (triangles) fits. Red = c-Na 3 Sb, blue = a-Na 1.7 Sb, green = a-Na 1.0 Sb. Bottom right: sum of the squared difference for two-phase c-Na 3 Sb and a-Na 1.0 Sb (orange circles) and three-phase (blue squares) combinations. The data indicate a two-phase region between c-Na 3 Sb and a-Na 1.7 Sb during D1-a, and conversion of a-Na 1.7 Sb to a-Na 1.0 Sb and the breakdown of remaining c-Na 3 Sb during D1-b. c-Na 3 Sb is still present at the end of D1-a, and is broken down further during D1-b.   The difference curve is shown offset in grey for clarity. Below the structure refinement data are the calculated contributions to the PDF of sodium-metal (which is modelled as an additional phase a discussed in the Experimental Methods section) and the a-Na 3-x Sb calculated from the refined structures in PDFGui.

SI.7 Calculation of error bars for sodiation level
Errors on the sodiation level were considered by considering the highest and lowest sodiation which could be achieved within the errors for phase fraction obtained from PDFGui refinements The least Na per Sb would be achieved when the maximum amount of Sb was present as c-Sb: • The amount of c-Sb = F 1 + ∆ • The amount of a-Na 3-x Sb = F 2 + ∆, or in the case that 1 -(F 1 + ∆) > F 2 + ∆, the amount of a-Na 3-x Sb was set to 1 -(F 1 + ∆). • The amount of c-Na 3-x Sb = 1 -(F 1 + ∆) -(F 2 + ∆), or in the case that 1 -(F 1 + ∆) -(F 2 + ∆) = 0, the amount of c-Na 3 Sb was set to 0.
where F 1 , F 2 and F 3 are the phase fractions for c-Sb, a-Na 3-x Sb and c-Na 3-x Sb, and ∆ the error given by PDFGui.
The amount of Na per Sb was calculated in the same manner as outlined in section 2 of the main text. The difference between the these values and the values calculated for the phase fractions were used as error bars in Figure 4. A similar procedure was used for the two-phase refinement.