Reciprocal Space Imaging of Ionic Correlations in Intercalation Compounds

TL;DR

This paper introduces a novel synchrotron x-ray technique to measure ionic correlations in intercalation compounds, revealing short-range order effects that impact battery performance.

Contribution

It demonstrates a model-independent method using diffuse scattering to analyze ionic correlations in layered materials, advancing understanding of structural ordering.

Findings

Measured temperature dependence of ionic correlation length scales.

Revealed short-range ionic order not detectable by conventional diffraction.

Provided a new approach for probing structural evolution in crystalline materials.

Abstract

The intercalation of alkali ions into layered materials has played an essential role in battery technology since the development of the first lithium-ion electrodes. Coulomb repulsion between the intercalants leads to ordering of the intercalant sublattice, which hinders ionic diffusion and impacts battery performance. While conventional diffraction can identify the long-range order that can occur at discrete intercalant concentrations during the charging cycle, it cannot determine short-range order at other concentrations that also disrupt ionic mobility. In this article, we show that the use of real-space transforms of single crystal diffuse scattering, measured with high-energy synchrotron x-rays, allows a model-independent measurement of the temperature dependence of the length scale of ionic correlations along each of the crystallographic axes in a sodium-intercalated V$_2$O$_5$. The techniques described here provide a new way of probing the evolution of structural ordering in crystalline materials.

Equations

$$I_{tot}(\mathrm{Q})=F(\mathrm{Q})F^{\ast }(\mathrm{Q})=|F_{HKL}(\mathrm{Q}){|}^2+|\Delta F(\mathrm{Q}){|}^2$$

$$\begin{array}{rl}{P}_{tot}(\mathrm{r})& =\mathrm{FT}[|F_{HKL}(\mathrm{Q}){|}^2]+\mathrm{FT}[|\Delta F(\mathrm{Q}){|}^2]\\ & =P_{HKL}(\mathrm{r})+\Delta P(\mathrm{r})\end{array}$$

Full text

1 Introduction

The cathodes of the first lithium-ion batteries developed over forty years ago were materials with van der Waals-bonded layers between which the lithium ions could be rapidly intercalated and de-intercalated [(1), (2)]. Since the first tests on LixTiS2, many other intercalation compounds have been investigated as potential electrodes because of their rapid charging rates and inherent stability over multiple charge/discharge cycles [(3)]. Insertion of alkali ions between such weakly-bonded layers is highly reversible because the weak coupling of the intercalants to the host produces only minor structural modifications during the electrochemical cycle. The potential use of multivalent ions in batteries has renewed interest in such materials [(4), (5), (6)], although weak coupling can compromise a battery’s gravimetric and volumetric energy density. The transition metal oxides most frequently used as cathodes in consumer electronics, such as LiCoO2 [(7)], are also layered materials, albeit with stronger host-intercalant interactions [(8)].

Even if the interactions of the alkali ions with the host lattice are weak, the Coulomb interactions between the intercalants cannot be ignored, especially at high concentrations [(9)]. Voltage anomalies as a function of lithiation in TiS2 were quickly ascribed to correlations between the intercalants induced by Coulomb repulsion [(10)], causing changes in the configurational entropy and consequently the chemical potential for ion insertion [(11)]. Berlinsky showed that order-disorder transitions can be inferred from extrema in the voltage derivatives versus composition [(9)]. Such anomalies have also been observed in transition metal oxide cathodes [(12), (13)], where they have been modeled by both lattice gas [(13), (14), (15)] and first-principles methods [(16), (17), (18)]. Ordering of intercalants impacts battery performance by disrupting ionic mobility, and there is evidence that suppressing ordering can improve charging rates [(19)].

Order-disorder transitions are usually associated with discrete stoichiometries, corresponding to commensurate ordering of the intercalant sublattice. In LixCoO2 and other transition metal oxides, ordering occurs at $x=\frac{1}{2}$ and a number of other rational fractions [(11), (20), (21), (22)]. Although it is possible to infer order-disorder transitions at these values from bulk measurements, direct evidence can only be obtained by the observation of (often weak) superlattice peaks from unit-cell expansion in diffraction measurements. The low cross section of lithium makes this difficult with x-ray scattering, unless associated changes in the host lattice space group are observable. For this reason, electron diffraction provided the first conclusive evidence of lithium-vacancy ordering in Li0.5CoO2 [(23)], showing unit-cell doubling both within and perpendicular to the planes. However, conventional Bragg diffraction is only sensitive to long-range order and cannot be used to probe short-range correlations that will also play a role in decreasing ionic mobility across the whole composition range. Understanding such correlations is important in modeling diffusion kinetics, for example as a function of concentration fluctuations close to the electrolyte interface.

The most powerful technique to probe short-range correlations is single crystal diffuse scattering, using either x-rays or neutrons, which comprises all the scattering that results from nanoscale deviations from the average crystalline structure [(24), (25), (26)]. The average occupancies of all the atomic sites within the translationally invariant unit cell of a crystal can be determined from Bragg peak intensities using standard crystallographic techniques, but, apart from thermal diffuse scattering from lattice vibrations, all the scattering between (and under) the Bragg peaks results from the presence of local disorder, often in the form of point defects, such as interstitials and vacancies, or of short-range order from defect-defect correlations. When measured over a sufficiently large volume of reciprocal space, diffuse scattering contains a true thermodynamic average of disorder scattering from the entire sample and so is complementary to techniques that utilize x-ray coherence to reconstruct real-space images of atomic correlations over a limited coherence volume.

Although it is an extremely powerful technique, single crystal diffuse scattering has not been as widely utilized as, for example, pair-distribution-function (PDF) measurements of polycrystalline materials [(27)], because of the challenge of both measuring and modeling large volumes of reciprocal space (Q-space). By contrast, the PDF technique, which gives a one-dimensional spherical average of interatomic vector probabilities, has become a standard tool in materials science because the information can be interpreted intuitively and modeled effectively with existing software. In recent years, Thomas Weber and colleagues have shown that it is possible to extend the PDF technique to three dimensions by transforming single crystal diffuse scattering into real space [(28)]. The result is a Patterson function, containing the summed probabilities of interatomic vectors in the crystal with contributions from both the average structure and local disorder. By excluding Bragg scattering from the real space transform, the resulting PDF function only includes those interatomic vectors whose probabilities differ from the average (3D-$\Delta$PDF). This is not possible in polycrystalline data because of the substantial overlap of Bragg and diffuse scattering intensity, but in single crystals, it allows a determination of the length scales of short-range order extending over 100 Å or more. By exploiting a new generation of fast area detectors, it is possible to measure sufficiently complete 3D volumes of reciprocal space in under 20 minutes using high-energy x-rays at a synchrotron source, enabling the generation of robust $\Delta$PDFs fast enough to allow detailed studies as a function of temperature.

We have applied this technique to measure ionic correlations in single crystals of V2O5 intercalated with sodium ions. In $\beta$-NaxV2O5, the sodium ions are confined to two-leg ladders [(29)], but the configuration of ions within the partially-occupied ladders cannot be determined by powder diffraction alone. The $\Delta$PDFs reveal both the tendency for sodium ions to form zig-zag configurations within each ladder, which are in phase with those in neighbouring ladders, as well as the temperature dependence of the correlation length of this ordering in all three crystallographic directions, obtained from the exponential decay of the $\Delta$PDF intensities. It has previously been reported that this material undergoes a structural phase transition above 200 K [(30)], but our results show that this temperature marks a crossover from two- to three-dimensional correlations without true long-range order being established down to 30 K. In the following article, we describe the measurement process in more detail, the steps used to generate the 3D-$\Delta$PDF results, and the subsequent analysis to extract ionic correlation lengths, before discussing the implications of this technique for studying battery electrodes and other disordered materials.

2 Results

The orthorhombic structure of pure V2O5 consists of both edge- and corner-shared square pyramids that form van der Waals-bonded planes, between which a variety of cations may be intercalated [(31)]. However, for some intercalants above a critical concentration, the structure transforms to a monoclinic $\beta$-phase (space group $C2/m)$ with a more complex network of vanadium oxide octahedra and square pyramids that contain one-dimensional channels, into which the intercalated cations are inserted (Fig. 1) [(29), (32)]. In $\beta$-NaxV2O5 ($0.2\lesssim x\lesssim 0.5$), the intercalants partially occupy a sublattice of two-leg ladders, whose legs are parallel to the crystal’s b-axis and whose rungs are parallel to the c-axis. The ladder sites would be filled with a value of $x=\frac{2}{3}$ , but stereochemical constraints should prevent occupations greater than 50% since the ladder rungs are too short ($<$2 Å). It has been postulated that the excess sodium ions would occupy additional octahedral and tetrahedral sites for $x>0.33$ in a $\beta^{\prime}$-phase [(33)], but these have never been conclusively observed. In the Supplementary Information, we discuss features in our data that provide evidence in favour of the octahedral site occupation preferred in Ref. 33.

We measured diffuse x-ray scattering on single crystals of $\beta^{\prime}$-NaxV2O5 with x = 0.45, corresponding to 68% occupation, using high-energy monochromatic beams at two synchrotron x-ray sources, the Advanced Photon Source (APS) and the Cornell High Energy Synchrotron Source (CHESS). Advances in the dynamic range and speed of x-ray area detectors now allow both the Bragg peaks and diffuse scattering to be measured efficiently, with three-dimensional volumes of scattering in reciprocal space, S(Q), collected in under 20 minutes at each temperature. More details are given in the Methods section and the Supplementary Information.

The diffuse scattering is mostly confined to rods that are parallel to $H$, i.e., orthogonal to the ladders, occurring at half-integer values of $K$ (Fig. 2), from which we can infer that there is predominantly two-dimensional short-range order within the ladder planes generated by ionic correlations within the sodium sublattice that tend to double the unit cell along b. A sinusoidal modulation of the rod intensities as a function of $L$, i.e., parallel to the direction of the ladder rungs, whose periodicity is the inverse of the rung length, indicates that site occupations across each ladder rung are strongly correlated. At room temperature, there are no sharp peaks along the rods (Fig. 2c), which are broad laterally ($\sim$0.1 Å-1 along $K$ and $\sim$0.13 Å-1 along $L$), indicating that correlation lengths are of the order of 10-20 Å. However, below $\sim$230 K, peaks start to appear at all integer values of $H$ (Fig. 2b), growing steadily in intensity down to the lowest measured temperature of 30 K (Fig. 2a). These peaks are evidence of the development of longer-range three-dimensional correlations within the sodium sublattice below $\sim$230 K, in agreement with previous x-ray measurements Kanai:1982fx ; Yamaura:2002ia .

Most analyses of diffuse scattering require a calculation of the Q-variation of the data using atomistic models that parametrize the disorder, but an alternative approach is to Fourier transform the data to generate real-space PDFs. This has the advantage of converting complicated intensity distributions in reciprocal space into discrete peaks in real space, whose positions and intensities are given by the interatomic vectors present in the disordered structure and their weighted probabilities, respectively. A further simplification is to include in the generated PDFs only those peaks whose probabilities deviate from the average structure. This is possible using the “Punch and Fill” method pioneered by Weber et al Weber:2012en , which utilizes the fact that the total scattering can be separated into two components: one representing the average crystal, i.e., the Bragg peaks, and the other representing scattering from defects.

[TABLE]

where $I_{tot}(\boldsymbol{\mathrm{Q}})$ is the measured total scattering and $F(\boldsymbol{\mathrm{Q}})$ is proportional to the Fourier transform of the electron density. The Fourier transform of this scattering function is also separable.

[TABLE]

$P_{H\!K\!L}$ is the Patterson function of the average structure while $\Delta P(\boldsymbol{\mathrm{r}})$, or the 3D-$\Delta$PDF, is the difference Patterson function due to the disorder. The “Punch and Fill” method isolates $\Delta P(\boldsymbol{\mathrm{r}})$ by eliminating the Bragg peaks, i.e., by removing the scattering in a small sphere around each Bragg peak and interpolating over the missing data, before performing the Fourier transform. The resulting $\Delta$PDF contains both positive and negative values for interatomic vectors that are, respectively, more or less probable than in the average structure.

We have used this technique to produce $\Delta$PDFs, which eliminate any contribution from the framework V2O5 lattice to first order and leave only the sodium-sodium pair correlations. In the average structure determined by powder diffraction Hughes:1983ux , the sodium sites are randomly occupied but our data shows that there are significant short-range correlations. In Fig. 2d-f, the $\Delta$PDF in the $b-c$ plane is displayed as a symmetric log plot consisting of triplet patterns of red and blue dots corresponding to positive and negative probabilities, respectively (see the inset to Fig. 2d). These triplet motifs are consistent with the allowed sodium-sodium interatomic vectors; the two-leg ladders of the real-space structure produce three-leg ladders in the PDFs, since they include interatomic vectors that connect sites on the left leg to sites on the right leg and vice versa.

There are weaker $\Delta$PDF peaks in other planes that could either be associated with additional sodium sites, such as those proposed by Galy et al Galy:1970dy , or with relaxations of the vanadium and oxygen ions. In the scattering data, additional superlattice peaks are also observed at $K=\pm\frac{1}{6}$, below $\sim$130 K, which is the temperature of a metal-insulator transition ascribed to V4+/V5+ charge ordering Yamada:1999kn ; Yamaura:2002ia . These give rise to a weak modulation of the $\Delta$PDF intensity along the $y$-axis, but does not appear to affect the sodium correlations significantly. These additional features are discussed in the Supplementary Information.

Some immediate conclusions can be drawn by inspection of the $\Delta$PDF data. Figure 2 shows that the nearest-neighbour sites within the ladder are less likely to be occupied, but next-nearest-neighbour sites are more likely to be occupied, leading to a zig-zag configuration of occupied sites. This is expected from considerations of both stereochemistry and Coulomb repulsion. Furthermore, these zig-zag configurations are in phase with neighbouring ladders, since the vectors connecting sites on the same leg (left or right), i.e., with $\pm ma\pm nc$, where $m$ and $n$ are integers, have positive probability whereas those connecting the opposite legs are negative. At 250 K (Fig. 2f), these correlations extend over only a few neigbouring sodium ladders, but as the temperature is lowered, the correlation length increases significantly, as seen at 200 K (Fig. 2e). At 50 K, the correlations extend over more than 100 Å in all directions (Fig. 2d). Since we have not removed the low-temperature peaks from the $k=\frac{1}{2}$ rods, it is evident that the $\Delta$PDFs are able to track continuously the growth in correlations from above to below the transition temperature, with the $\Delta$PDF peak intensities providing a measure of the incipient order parameter.

To quantify the growth in correlations, we have modeled the dependence of the $\Delta$PDF peak intensities on interatomic distance. Figure 3 shows that these intensities follow an exponential decay at high temperature, consistent with a one-dimensional Ornstein-Zernike function Collins:1989th . At lower temperature, it is necessary to include the effect of the finite resolution in reciprocal space, which produces in an envelope function in real space. Its Gaussian width can be estimated by fitting the total PDF, i.e., the data transformed without eliminating the Bragg peaks, which represents long-range crystalline order. The $\Delta$PDF intensities have then been fit to the product of this Gaussian envelope function with a temperature-dependent exponential decay, whose decay constant is a direct measure of the correlation length, $\xi$. More details are provided in the Supplementary Information.

The temperature dependence of these fits is shown in Fig. 3 along all three crystallographic directions. As the correlation length approaches 200 Å, the effect of the finite resolution leads to an increase in the the error bars although $\xi$ and the $r=0$ amplitude of the $\Delta$PDF correlations is measurable down to 30 K.

3 Discussion

The elimination of the interatomic vector probabilities of the average structure from the PDF allows the range of short-range order to be determined over length scales that exceed 100 Å, which means that the correlation of ions undergoing an order-disorder transition can be monitored as a function of temperature from well-above to well-below the transition (Fig. 4). In $\beta$-Na0.45V2O5, there is a gradual increase in the correlation lengths along the $y$ and $z$ directions, $\xi_{y}$ and $\xi_{z}$, from 275 K to 230 K, which is easier to see in plots of $\xi^{-1}$ vs temperature (Fig. 4c). At $\sim$230 K, this induces a sharp increase in correlation length of the sodium pair correlations along the $x$-axis, $\xi_{x}$, i.e., perpendicular to the sodium ladder planes, marking a cross-over from two- to three-dimensional correlations (Fig. 4b). However, it does not appear to be a true second-order phase transition, since $\xi_{x}$ saturates at 190 K with a value of only 170 Å, corresponding to approximately 10 unit cells or 20 ladder planes. The increase in $\xi_{x}$ in turn generates a more rapid increase in $\xi_{y}$ and $\xi_{z}$, again without leading to a true divergence. The maximum correlation length at 30 K is still under 200 Å. Instead of a discontinuity in the slope of the inverse correlation lengths, $\xi_{y}^{-1}$ and $\xi_{z}^{-1}$, defining a unique transition temperature, there is a crossover between two temperature regimes at $\sim$200 K close to where $\xi_{x}$ saturates (Fig. 4c).

We propose that the saturation in $\xi_{x}$ is a consequence of frustration. Due to the body-centring of the monoclinic structure, the ladder sites in the $y$-$z$ planes at $x=0$ and $x=\frac{1}{2}a$ are shifted by $\pm\frac{1}{2}b$, which makes the strength of the Coulomb interactions independent of the phase of the zig-zag site occupations between such neighbouring planes. This degeneracy implies that the phase must be stabilized by next-nearest-neighbour interactions, which may be weak enough to allow stacking faults along the $x$-axis, preventing true long-range order. Disorder of the excess sodium ions that cannot be accommodated on the ladder sites would contribute to this frustration providing local pinning of the zig-zag phase. The fall in $\xi_{x}$ below 190 K indicates that any such frustration mechanism is increased by the growth in intraplanar correlations. This could be tested by further measurements on $\beta$-Na0.33V2O5, corresponding to precisely 50% occupation of the ladder sites.

The intensities of the $\Delta$PDF peaks increase monotonically with decreasing temperature (Fig. 4a). Along the $x$-axis, the temperature dependence of the amplitude resembles an incipient order parameter, with a value close to 0 above the transition (see Fig. S5 in the Supplementary Information). However, along the $y$ and $z$ directions, there is a much weaker temperature dependence reflecting the strong nearest-neighbour correlations at all temperatures.

4 Conclusion

Transforming x-ray diffuse scattering data into 3D-$\Delta$PDFs produces a remarkable simplification in how the data are represented. It is possible to interpret the short-range order in the crystal structure without a detailed simulation of the disorder, which has in the past required the optimization of a large number of parameters over a substantial volume of reciprocal space. In fact, the PDF intensities directly determined from the transformed data are related by simple analytic functions to the Warren-Cowley parameters that are frequently used to parametrize diffuse scattering models Welberry:1995wm . Even without a well-defined model of the underlying disorder, our results show that the spatial dependence of the PDF intensities provides a method of extracting correlation lengths as a function of temperature or other parametric variable.

The ability to generate a real-space “image” from reciprocal space data makes this a powerful tool in the investigation of intercalation compounds, as it is especially suited to the measurement of ionic correlations on a sublattice that is distinct from the host structure. Although we have not yet tested this technique on lithiated compounds, we believe that the elimination of the average structure of heavier elements will allow lithium-lithium correlations to be measured in spite of their low cross section. In this article, the method has allowed us to show that the apparent order-disorder transition reported in NaxV2O5 is actually a crossover from two-dimensional to three-dimensional correlations, with interplanar correlations saturating at only $\sim$150 Å, probably because of frustration. Such detailed insight into intercalant concentrations is not possible by any other method.

Although we have focused on layered materials, the technique can be utilized in more three-dimensional insertion compounds such as the oxide spinels Kim:2001jg , as well as many other disordered materials. The principal limitation of this technique is the need for single crystals, which makes in situ experiments within functioning battery cells challenging, although not impossible. Sample thicknesses can be 100 $\mu$m or less, and provided the rest of the cell is polycrystalline, the real-space transform will only contain discrete peaks that correspond to the crystalline electrode. Measuring the phase diagram of order-disorder transitions as a function of both temperature and intercalant concentration will allow the strength of inter-ionic interactions to be determined and improve our understanding of the how they limit ionic mobility. Such measurements could also be used to evaluate the effectiveness of strategies for mitigating these limitations, for example by co-intercalation with aliovalent cations Meethong:2009bj or substitution of transition metals on the vanadium sites Jovanovic:2018vj , in order to disrupt ordering phenomena.

5 Methods

5.1 Synthesis

NaxV2O5 crystals were grown using a self-flux technique. Five grams of vanadium pentoxide (V2O5, Aldrich, $>98$%) was placed into a nickel crucible and placed in an oven inside an Ar-atmosphere glovebox. The sample was heated to 720°C. While liquid, the crucible was removed from the furnace and a stoichiometric amount of sodium iodide (NaI, Aldrich 99.5%) was added to the liquid. The sample was then put back in the furnace and cooled to 650°C over a one hour period, then radiatively cooled to room temperature. Single crystals were isolated from the flux. EDX analysis showed that the sodium concentration, when normalized to the nominal vanadium stoichiometry, was $x=0.45\pm 0.02$.

5.2 X-ray Scattering

Three-dimensional volumes of diffuse x-ray scattering were collected at the Advanced Photon Source (APS) and the Cornell High Energy Synchrotron Source (CHESS). The APS data was measured on Sector 6-ID-D using an incident energy of 87.1 keV and a Dectris Pilatus 2M with a 1 mm-thick CdTe sensor layer. The CHESS data were measured on beamline A2 using an incident beam energy of 27.3 keV and a Dectris Pilatus 6M detector with a 1 mm-thick Si sensor layer. The data were collected from 30 K to 300 K, with samples cooled by flowing He gas below 100 K and N2 gas above 100 K. During the measurements, the samples were continuously rotated about an axis perpendicular to the beam at 1° per second over 370°, with images read out every 0.1 s. Three sets of rotation images were collected for each sample at each temperature to fill in gaps between the detector chips. The resulting images were stacked into a three-dimensional array, oriented using an automated peak search algorithm, and transformed in reciprocal space coordinates using the software package CCTW (Crystal Coordinate Transformation Workflow) Jennings:CCTW , allowing S(Q) to be determined over a range of \sim$$\pm15 Å*-1* in all directions. Further details are given in the Supplementary Information.

6 References

7 End Notes

7.1 Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division and Scientific User Facilities Division. X-ray experiments were performed at the Advanced Photon Source, which is supported by the Office of Basic Energy Sciences under Contract No. DE-AC02-06CH11357 and the Cornell High Energy Synchrotron Source (CHESS), which is supported by the NSF and NIH/NIGMS via NSF award DMR-1332208. We thank Douglas Robinson and Xiaoyi Zhang for technical support during the experiments, Alexander Rettie for performing the EDX analysis, Thomas Weber and Arkadiy Simonov for discussions about the $\Delta$PDF technique, Branton Campbell for help with the formalism of transforming the data to reciprocal space, and Peter Zapol and Charlotte Haley for discussions about interpreting the results. Crystal structure images were generated using CrystalMaker®, CrystalMaker Software Ltd, http://www.crystalmaker.com.

7.2 Author contributions

Samples were prepared by J.T.V. and prepared for measurement by M.K. The experiments were devised by M.K., S.R., and R.O. The x-ray experiments were performed by M.K., S.R., J. R., J.M.W., and R.O. The data were analyzed by M.K., R.O., J.M.W., and G.J., using software written by G.J., M.K., R.O., and J.M.W. The manuscript and supplementary information were written by R.O. with input from all the authors.