Thermal Expansion in Layered NaxMO2

Layered oxide NaxMO2 (M: transition metal) is a promising cathode material for sodium-ion secondary battery. Crystal structure of O3- and P2-type NaxMO2 with various M against temperature (T) was systematically investigated by synchrotron x-ray diffraction mainly focusing on the T-dependences of a- and c-axis lattice constants (a and c) and z coordinate (z) of oxygen. Using a hard-sphere model with minimum Madelung energy, we confirmed that c/a and z values in O3-type NaxMO2 were reproduced. We further evaluated the thermal expansion coefficients (αa and αc) along a- and c-axis at 300 K. The anisotropy of the thermal expansion was quantitatively reproduced without adjustable parameters for O3-type NaxMO2. Deviations of z from the model for P2-type NaxMO2 are ascribed to Na vacancies characteristic to the structure.

constant] and z well reproduced the experimental values for O3-type Na x MO 2 . By introducing T-linear expansion of the hard sphere, the anisotropy of the thermal expansion was quantitatively reproduced without adjustable parameter for O3-type Na x MO 2 .

Results
Temperature dependence of a(c)-axis lattice constants and z coordinate of oxygen. Figure 2 Table 1, the values of a, c at 300 K, α a , and α c of O3-and P2-type Na x MO 2 at 300 K were listed. Figure 3(a) shows z coordinate of oxygen for O3-and P2-type structure (z O3 and z P2 ) against a at 300 and 700 K. Blue and red circles represent z O3 at 300 and 700 K, respectively. Light green and pink triangles represent z P2 at 300 and 700 K, respectively. The values were almost independent of T. Figure 3(b) shows a ratio of interlayer distance (d) to a against a at 300 K, where d is c/3 for O3-type and c/2 for P2-type structure. Light blue and purple circles represent d/a of O3-type compounds at 300 and 700 K, respectively. Green and yellow triangles represent d/a of P2-type compounds at 300 and 700 K, respectively. d/a slightly decreases with an increase in a. Figure 4 shows (a) α a , (b) α c , and (c) α c /α a against a. α a and α c of O3-type Na 1.00 CoO 2 were 0.98 × 10 −5 K −1 and 1.71 × 10 −5 K −1 , respectively. These values are comparable to those of inorganic compounds; α a = 1.44 × 10 −5 K −1 for LiMn 2 O 4 72 , and α a (α c ) = 0.85(2.5) × 10 −5 K −1 for layered BaFe 1.84 Co 0.16 As 2 73 . α a (α c ) in Fig. 4 is rather scattered against a around the averaged value of 0.92(1.96) × 10 −5 K −1 . The ratio α c /α a is also scattered around the averaged value (=2.30). However, the data point for P2-Na 0.59 CoO 2 are seriously deviated from the average value. This is probably due to the Na ordering 67 .

Discussion
A hard-sphere model with minimum Madelung energy. We have constructed a hard-sphere structural model for O3-and P2-type NaMO 2 to reproduce the experimental results (d/a, z, α c /α a ). Firstly, imagine a sheet consists of hard spheres that were arrayed on triangular lattice, and then the sheet is alternately stacked as shown in Fig. 1. In the model, hard spheres of Na, M, and O were assumed to have Shannon's ionic radius 74 ; R Na = 1.02 Å, R O = 1.40 Å, and R M is a variable parameter that takes 0.58-0.7 Å, respectively. Since the ionic radius of the hard sphere is different from one another, the structure can not be the hexagonal close-packed structure, and several ] is shown. The bottom panels show top views of O-Na-O stackings in the O3-and P2-type structures, respectively. The BAC stacking in O3 forms NaO 6 octahedron, and the BAB stacking in P2 forms NaO 6 triangular prism.
SCienTifiC REpORTS | (2018) 8:3988 | DOI:10.1038/s41598-018-22279-9 structures with different a are possible for the unique value of R M . Here, we adopted a constraint that M, upper, and lower oxygens coroneted each other, because the constraint minimizes the Madelung energy against a (vide infra). For a calculation of thermal expansion coefficient, the hard sphere is assumed to expand in proportion to the temperature difference (ΔT). ΔT dependence of R Na is expressed as Now, let us derive the expression (a calc ) of a-axis lattice constant as a function of ΔT. Note that the in-plane nearest-neighbor oxygen distance is a for both the P2-and O3-structure. Considering the above-mentioned constraint, a calc (ΔT) for both O3-and P2-structures is expressed as This equation is easily derived using Pythagorean theorem. As shown in Eq. 1, a calc strongly depends on R M value. Due to the finite ionic radius of oxygen (R O = 1.40 Å), minimum value of a calc is 2.80 Å. At a calc = 2.80 Å, R M is evaluated to be ≈0.5799 Å using Eq. 1. With an increase in R M , the oxygen triangular lattice expands in order to keep the connection between M, upper and lower oxygens.  Table 1. a-axis, c-axis lattice constants (a and c), z coordinate of oxygen (z) at 300 K, the linear thermal expansion coefficient [α a (α c )] along a(c)-axis, and α c /α a of O3-and P2-type Na x MO 2 at 300 K. † The original data were referred from previous reports 56,57 . * α a and α c were reevaluated in a T-range of 300-800 K. Blue and red circles represent z O3 at 300 and 700 K, respectively. Light green and pink triangles represent z P2 at 300 and 700 K, respectively. The broken lines represent z calculated by the hard sphere model with minimum Madelung energy for O3 and P2-type compounds (z O3 calc and z P2 calc ) against a, respectively. (b) The ratio of interlayer distance (d) to a against a. Light blue and purple circles represent d/a of O3-type compounds at 300 and 700 K, respectively. Green and yellow triangles represent d/a of P2-type compounds at 300 and 700 K, respectively. The broken line represents d/a calculated by the hard sphere model with minimum Madelung energy (d calc /a calc ) against a. The inset of Fig. 3 Na sheet is sandwiched by the MO 2 layers as BAC (BAB) in the O3-type (P2-type) structure. We noted that the expression (d calc ) of interlayer distance is independent of the stacking manner, and the ΔT-dependence of d calc for O3-and P2-type structure is expressed as, for O3-and P2-type structures. By using Pythagorean theorem, Eq. 2 is easily derived. Expressions of z (z calc ) for O3-and P2-type structures are derived as  We show that this model exhibits minimum Madelung energy (E ME ). Figure 5 shows E ME of O3-and P2-type NaMO 2 against a. Our constraint gives a calc ≈ 2.995 Å [Eq. 1] at R M = 0.65 Å. With the O3 structure, the a calc value corresponds to the minimum position of E ME (−8.67 eV). With an increase in a from 2.995 Å, M becomes isolated from the surrounding oxygens (the right-side inset of Fig. 5), and E ME increases (Note that the oxygen positions were controlled by Eqs 3 and 4). With an decrease in a from 2.995 Å, the upper and lower oxygens are separated (the left-side inset of Fig. 5), and E ME increases as well (We used for the calculation below a = 2.995 Å).
Similar results are obtained for the P2 structure. Thus, our model is energetically stable against the variation of a. Our constraint that M, upper and lower oxygens connected to each other causes the compact layered structure and minimized the long-range Coulomb energy between the layers. We call our model "hard-sphere model with minimum Madelung energy". Companion of the structural parameters and the thermal expansion coefficients with the model. The broken line in Fig. 3

(b) is the calculated d/a based on the hard-sphere model with minimum
Madelung energy (see Eqs 1 and 2). d calc /a calc decreases with an increase in a calc , which reproduces experimental results. The decrease in d calc /a calc is schematically depicted in the inset of Fig. 3(b). When O-O distance along in-plane direction elongates due to increase in R M , Na atoms relatively sink down along out-of-plane direction.
(Decrease in d a /(2 ) NaO 2 against a is displayed in Fig. S5). We further calculated z and plotted them in Fig. 3(a). In the O3-type compounds, z calc well reproduces the experimental data. In the P2-type compounds, however, z calc is slightly smaller than z P2 calc . We ascribed the smaller z P2 to the Na vacancies characteristic to the P2 structure. With the vacancies, the nominal valence of M became higher, and hence R M becomes smaller. Our model tells us that z P2 calc becomes smaller if R M becomes smaller. The broken lines in Fig. 4 represent the calculated α a , α c , and α c /α a (α a calc , α c calc , and α c calc /α a calc ), respectively. α a calc (α c calc ) was evaluated by using the equation, The only fitting parameter A (=2.56 × 10 −4 K −1 ) was chosen to fit the average value of α a (=0.92 × 10 −5 K −1 ). Using the same value of A, α c calc was found to reproduce the magnitude of experimental value for the O3 materials. On the other hand, larger α c for the P2 materials is possibly due to Na vacancies except for the data point of Na 0.59 CoO 2 . We note that α α c a calc calc is expressed without the adjustable parameter A, The hard-sphere model examined in this paper gives intuitive and easy comprehension of the thermal expansion behavior of the layered oxides. The density-functional-thery (DFT) calculation successfully reproduces the linear thermal expansion coefficients of several materials such as Al 75 , S 76 , 4d transition metals 77,78 , Os 78 , MgO 79 , CaO 79 , and ZnO 80 , which is beyond the scope of this paper.

Conclusion
We systematically determined the temperature dependent lattice constant and z-coordinates of P2-and O3-type NaMO 2 . We proposed a simple hard-sphere model with constraint that M, upper and lower oxygens are coroneted to each others. The model quantitatively reproduced a, c, z, α a and α c for O3-type Na x MO 2 . On the other hand, z coordinate of P2-type Na x MO 2 deviates from the hard-sphere model possibly due to Na vacancies. This simple model can be easily applied for the other layered compounds to intuitively understand and design the thermal expansion behaviors.  O3-Na 0.99 Fe 0.5 Ni 0.5 O 2 , Na 2 O 2 , Fe 2 O 3 and NiO were mixed under the molar ratio of Na:Fe:Ni = 1.2:0.5:0.5, and calcined at 650°C in O 2 for 15 h. Then the product was finely ground and again calcined in the same condition. For O3-Na 0.94 Ti 0.5 Ni 0.5 O 2 , Na 2 CO 3 , TiO 2 and NiO were mixed under the molar ratio of Na:Ti:Ni = 1.05:0.5:0.5, and calcined at 900°C in air for 15 h. Then the product was finely ground and again calcined in the same condition.

Methods
For P2-Na 0.50 Mn 0.5 Co 0.5 O 2 , Na 2 CO 3 , MnCO 3 and Co 3 O 4 were mixed under the molar ratio of Na:Mn:Co = 0.7:0.5:0.5, and calcined at 900°C in air for 12 h, Then, the product was finely ground and again calcined in the same condition. For P2-Na 0. 70 O 3 were mixed under the molar ratio of Na:Mn:Fe = 0.7:0.5:0.5, and calcined at 900°C in air for 12 h. Then, the product was finely ground and again calcined in the same condition. All the samples were taken out from the hot furnace (>200°C), and then immediately transferred into a vacuum desiccator to avoid moisture in air.
X-ray diffraction. The synchrotron radiation x-ray diffraction (XRD) patterns were measured at BL02B2 beamline 81 at SPring-8. The capillary was placed on the Debye-Scherrer camera at the beamline. The sample temperature was controlled by blowing a hot N 2 in the temperature range of 300 K ≤ T ≤ 800 K. The XRD patterns were detected with an imaging plate (IP). The exposure time was 5 min. The wavelength of the x-ray was 0.499420 Å for P2-Na 0.48 Mn 0.5 Fe 0.5 O 2 and P2-Na 0.50 Mn 0.5 Co 0.5 O 2 , and 0.499892 Å for O3-Na 0.98 Fe 0.5 Co 0.5 O 2 and Na 0.94 Ti 0.5 Ni 0.5 O 2 , and 0.499838 Å for the others. The wavelengths are calibrated by the cell parameter of standard CeO 2 powders. Crystal structure was analyzed by RIETAN-FP program 82 . Schematic figure of the crystal structure were drawn by VESTA program 83 . All the reflections can be indexed with the O3-type (R m 3 ) or P2-type (P6 3 /mmc) structures except for a tiny amount of impurity of O3-type Fe-rich phase for Na 0.99 Fe 0.5 Ni 0.5 O 2 and NiO for Na 0.94 Ti 0.5 Ni 0.5 O 2 . All the structural parameters against T (300 K ≤ T ≤ 800 K) were listed in Tables S1-S9. During heating process, any extra impurity peaks were not appeared. We observed no tendency of Na deintercalation due to heating (Fig. S4).
The actual Na concentrations in the compound were determined by the Rietveld refinement based on the synchrotron XRD patterns at 300 K. We note that ref. 45 reported a consistency of Na contents determined by ICP-AES (Inductively Coupled Plasma Atomic Emission Spectroscopy) and Rietveld refinement using synchrotron x-ray diffraction for P2-Na x Mn 1/2 Fe 1/2 O 2 phase.
Madelung energy calculation. Madelung energy (E ME ) was computed by the MADEL program in the VESTA software using the Fourier method 83  where g j is the occupancy of the j th ion, Z i is the valence of the j th ion, ε 0 is the vacuum permittivity, and l ij is the distance between ions i and j. E ME is calculated by using the formula φ = ∑ E Z W i i i i ME 1 2 , where W i is a factor depending on g i and the number of equivalent atomic positions at the site i in the unit cell. For O3-type structure (space group: R m 3 ), we put +1, +3, and −2 charges on 3a Na (0,0,0), 3b M (0,0, 1 2 ), and 6c O (0,0, z) sites in stoichiometric NaMO 2 . In the calculation of the P2-type structure (P6 3 /mmc), we assume a stoichiometric NaMO 2 with fully occupied 2d Na site. We put +e, +3e, and −2e charges on 2d Na ( ) , 2a M (0,0,0), and 4f O ( ) z , , . A radius (s) of the hard sphere was set to 0.3 Å, and Fourier coefficients are summed up to 10 Å −1 in the reciprocal space.