Fibonacci Stoichiometry and Superb Performance of Nb₁₆W₅O₅₅ and Related Super-Battery Materials

Hans Hermann Otto

Materialwissenschaftliche Kristallographie, Clausthal University of Technology, Clausthal-Zellerfeld, Lower Saxony, Germany.

DOI: 10.4236/jamp.2022.106133                

Abstract

In this contribution, two important crystallographic concepts for the formation of a series of block structures associated with channeling have been compared: chemical twinning and crystallographic shear. Twin planes respectively shear planes besides formed channels serve as a sink for charge carriers or, when the oxidation state of metal ions can be reduced, as a reservoir for intercalated lithium. In this way, Wadsley-Roth shear phases such as niobium tungsten oxide exhibit channels for ultra-fast lithium-ion diffusion. They are in focus as anode material for super-batteries, superb in terms of energy respectively power density, charging time, cycle life and safety. It should be noted that the transition metal to oxygen ratio TM/O = 21/55 of the title compound is a Fibonacci number quotient. Also, the crystal lattice can be traced back to Fibonacci geometry. When replacing only 0.0213 tungsten atoms in the formula with less expensive titanium, a TM/O ratio of 0.381966 =ϕ² can be adapted besides an average valence electron concentration of 2⋅ϕ^-2, where represents the most irrational number of the golden mean. The additional disorder caused by even such small titanium replacement and accompanied oxygen vacancies could fasten up the already high lithium diffusion further. Ultrasonic treatment may be applied besides thermal cycling to prepare phase-pure of the highest electrochemical performance. A replacement of oxygen by some fluorine atoms is an obvious synthesis possibility, but the higher binding energy expected between lithium and fluorine in contrast to oxygen may rather hinder than promote lithium diffusion.

Keywords

Super Battery, Niobium Tungsten Oxide, Crystallographic Shear, Chemical Twinning, Fibonacci Stoichiometry, Ti Substitution

1. Introduction

In the last years, excellent research was done to overcome the uneconomically long charging times and other odds of lithium-ion batteries, caused by the low performance of lithium intercalated graphite anodes. The best performance was now reached with the new anode material ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ that can uptake more than 8 lithium ions per formula unit in its channel structure [1].

In the year 1965, two different mechanisms were described independently, by which homologous series of crystal structures can be formed. Chemical twinning was first described by Otto in the PbS-Bi₂S₃ binary system [2] [3] [4], and the crystallographic shear mechanism has been introduced by Roth and Wadsley [5]. The metal to anion ratio decreases in the case of chemical twinning, whereas in the case of crystallographic shear the ratio increases. We compare both systems of phase formation and their property to form channels near the twin planes respectively shear planes. These planes and channels can act as a sink for charge carriers or as housing for intercalated ions, if the metallic ions of the host lattice are able to undergo an alteration of the oxidation state. The top performance of the anode material ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ with ultra-fast mobility of intercalated lithium ions may be coupled with a suggested disorder caused by its interesting transition metal to oxygen Fibonacci number ratio TM/O = 21/55. Super-ionic conduction depends on the “surface” defect density, which is evident since the work of Lehovec in 1953 [6] and is coupled to stoichiometry. We first highlight in Chapter 2 the Fibonacci number series and the golden mean, then compare chemical twinning in Chapter 3 with crystallographic shear in Chapter 4, and then perform a Fibonacci analysis of known niobium tungsten oxide shear phases, and finally suggest chemical alteration in Chapter 5. In addition, some examples were quoted, where Fibonacci nets are used by the present author to determine the position of heavy atoms in crystal structures.

2. Fibonacci Number Sequence and Golden Mean

As nature’s effective-evolutionary pre-calculator, the golden ratio dominates all areas of science, life and cosmos. In physics, phase transitions are often governed by this number or its fifth power [7]. Stoichiometry as the numerical skeleton beyond chemistry should also not be excluded from a golden mean based explanation. So we begin with a short essay of number theory and the reciprocity property of this fundamental number and its connection with the Fibonacci number series [8]. The number series $F_n=\left\{0,1,1,2,3,5,8,13,21,34,\mathrm{55...}\right\}$ was named after the Italian mathematician Leonardo Pisano named Fibonacci (1170-1240), where the sum of two succeeding numbers defines the next following number of the series: $F_n+F_{n+1}=F_{n+2}$ [7], while the quotient of $F_n/F_{n+1}$ for $n\to \infty$ determines the golden mean or golden ratio φ. The golden ratio is the most irrational number with the simplest infinite continued fraction representation at all and a very adaptable number-theoretical chameleon. Special attention is paid to the reciprocity property of the golden ratio as effective pre-calculator of natures creativness. We use the definition

$\phi =\frac{\sqrt{5}-1}{2}=\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}=0.6180339887\cdots$ (1)

However, the golden ratio is frequently used by others as the reciprocal of this value

${\phi }^{-1}=1+\phi =\frac{\sqrt{5}+1}{2}=1.6180339887\cdots$ (2)

Many examples for golden mean geometry are found in regular Platonic solids such as icosahedron or pentagonal dodecahedron [9], in the C₆₀ bucky-ball [10] and in quasicrystals [11]. Also the structure of the elementary particle electron can be traced back to chiral icosahedral symmetry [12]. The stoichiometry of Hume-Rothery phases such as the Cu₅Zn₈ alloy indicates Fibonacci behavior with an average electron concentration equal to 21/13, fulfilling the Hume-Rothery electron concentration rule [13]. This leads directly to the explanations that follow. The golden ratio omnipresence in nature and life has been described in the highly recommended monograph by Olsen [14] and in the book “Grand Unification of the Sciences, Arts and Consciousness” by a well-known quartet of authors [15]. Fibonacci numbers in stoichiometry of chemical compounds have been also treated by Vasyntinskij [16].

3. Homologous Block Structures Formed by Chemical Twinning

The present author explained as first the mechanism of formation of block structures by chemical twinning [2] [3] [4]. Cationic vacancies, created by the substitution of Pb²⁺ by the smaller Bi³⁺ ion in the crystal structure of PbS (galena), are shown to be reduced by chemical twinning along the (311) twin plan. The first member of the series has the formula 6PbS∙Bi₂S₃ and was found in nature as the mineral heyrovskyite. Figure 1 shows its projected orthorhombic crystal structure with the typical PbS-like block structure, where Pb²⁺ on the twin plane has an increased sulfur coordination of 8 or 9 instead of 6. The crystallographic data are: space group Bbmm (No 63), a = 13.695 (3) Å, b = 31.358 (5) Å, c = 4.135 (1) Å, Z = 4, D_x = 7.29 Mg∙m⁻³.

By reducing the block width one gets 3PbS·Bi₂S₃ composition (lillianite structure) [4]. As end member the CaIrO₃ structure type would result with Ca [8] and Ir [6] coordination. From this structure, the structure of Bi₂S₃ (bismuthinite) can be deduced by a distortion adapting sevenfold coordination for both Bi sites according to Paulings fifth crystal-chemical rule [17].

Figure 1. Projection of the orthorhombic crystal structure of 6PbS∙Bi₂S₃ (heyrovskyite) down [001], exhibiting a galena-like block structure. Lead brownish, bismuth blue, sulfur yellow. Formed channels along [001] can be clearly seen on the vertical (311) twin plane at the height b = ¼.

4. The Wadsley-Roth Crystallographic Shear Phase $N{b}_{16}^{5+}{W}_5^{6+}{O}_{55}$

In the Nb₂O₅-WO₃ binary system, Roth and Wadsley described a series of structures formed by crystallographic shear (see Table 1), exerted on the ReO₃ basis crystal structure (Figure 2), when the transition metal to oxygen ratio increases by replacement of some TM⁶⁺ by TM⁵⁺ or transition metal ions of even lower valence. In this way, anion vacancies are depleted, which have been enriched on certain planes, by edge-sharing of some oxygen octahedra. Crystallographic shear and shear propagation has been comprehensively described by van Landuyt [18].

The empty cage of the ReO₃ structure is surrounded by 12 oxygen atoms. In case of perovskites it is occypied by a large cation. Even in the perovskite SrTiO₃ crystallographic shear could be observed near dislocation [19].

Figure 3 shows the block of 4 by 5 ideal oxygen coordination octahedra and illustrates the 4-side windows, through which lithium ions can effectively jump.

The 4 by 5 octahedron extended single block structure of monoclinic ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ as an example of shear structures has been displayed in Figure 4 as projection down [010]. The crystallographic data are: monoclinic space group C2 (No 5), lattice parameters a = 29.657 (4) Å, b = 3.8225 (3) Å, c = 23.106 (4) Å, ß = 126.50 (1)˚, Z = 2, D_x = 5.182 Mg∙m⁻³. Because the block contains only 20 metal atoms, a supplementary W metal site is located at the shear plane boundaries at the cell origin respectively centering of the ab-plane. The process of lithianation causes only small lattice alterations. The lithium diffusion mechanisms have been investigated thoroughly by Koçer et al. and can be attributed to the entire family of CS phases [20]. Jumps between 4-sided windows have lowest activation energy and are considered as one-dimensional.

5. Proposal for Chemical Alterations and Preparation

We pose the question, whether the extreme super-ionic Li conduction of ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ is (partly) caused by its Fibonacci number governed metal to oxygen ration of TM/O = 21/55 respectively the average valence electron concentration, and whether it is possible to further optimize the electrochemical

Figure 2. Prototypic crystal structure of cubic ReO₃, space group $Pm\bar{3}m$, a = 3.748 Å. Left: unit cell with yellow Re⁶⁺ and red O²⁻. Right: projected corner-shared ReO₆ octahedra showing the large 4-sides window.

Figure 3. Checkerboard block of 4 by 5 ideal oxygen coordination octahedra around Re⁶⁺. The block consists of 12 unit-cells. In the actual shear strucures, the outer octahedra are tilted slightly outwards to enable their edge linking to the next block.

Figure 4. Crystal structure of Nb₁₆W₅O₅₅ projected down [010] [5]. Unit-cell magenta colored, tungsten yellow, niobium (tungsten) octahedra blue.

Table 1. Fibonacci anlysis of Nb₂O₅-WO₃ shear phases. In the proposed golden mean formulas the TM/O is adapted to φ² = 0.381966, whereas the average valence electron concentration <c_ve> yields 2·φ⁻² = 5.236007.

performance by adapting a golden mean stoichiometry? For the TM/O ratio one finds

$\frac{16+5}{55}=\frac{21}{55}=0.38181818\approx 0.3819660={\phi }^2$ (3)

For the average valence electron concentration it yields

$\frac{16\times 5+5\times 6}{16+5}=\frac{110}{21}=5.238095\approx 5.2360679=2\cdot {\phi }^{-2}$ (4)

Now we want to adapt exact golden mean values in the above relations by replacement of W⁶⁺ respectively Nb⁵⁺ by a small amount of Ti⁴⁺. The wanted result is a replacement of only $\Delta =0.02128657{\text{Ti}}^{4+}$ for W⁶⁺ respectively $2\Delta =0.04257{\text{Ti}}^{4+}$ for Nb⁵⁺, which is accompanied by an oxygen loss of Δ

$\frac{21}{55-0.021286}={\phi }^2$ (5)

$\frac{110-0.04257}{21}=2\cdot {\phi }^{-2}$ (6)

Due to the very similar ionic radii of both W⁶⁺ and Ti⁴⁺ of r_ion = 0.42 Å a replacement seems to be possible in this limit. It can be suggested that the W site at the origin is affected. The crystal lattice would tolerate such a small replacement. However, a slightly altered cation to anion ratio towards M/O = 0.381966 = φ² could already be realized on the crystallite grain boundaries. But the question is, whether one could really expect any additional performance effect caused by such a small cationic replacement, introducing more disorder and some anion vacancies besides reducing stress for optimal lithium uptake? Only an experiment can verify it. With respect to very small differences in the free energy between adjacent phases of the Nb₂O₅-WO₃ binary system [21], it is recommended to try ultrasonic treatment besides thermal cycling during the preparation of nearly phase-pure samples of yellow-green ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ nanomaterial. In this way an optimal electrochemical performance should be guaranteed. However, particle size and the architecture of the electrode with a large surface area are also important. Recently, Jang and Zhao proposed the fabrication of hollow nanotubes by electro-spinning methods [22].

6. Fibonacci Variation of the Shear Phase Nb₁₈W₁₆O₉₃

Another example is the shear phase Nb₁₈W₁₆O₉₃ [1] [21]. Again we can proceed as in Chapter 5 and replace W by Ti, but in an larger amount to propose the formula Nb₁₈W₁₂Ti₄O₈₉. Of course, the crystallographic data will change significantly compared to the starting phase. Nevertheless, a synthesis attempt could be interesting. For the TM/O ratio we found now

$\frac{34}{89}=0.382022\approx {\phi }^2$ (7)

and for the average valence electron concentration

$\frac{178}{34}=5.235294\approx 2{\phi }^{-2}$ (8)

Therewith we have applied the next higher Fibonacci number set. Again, a further little variation shifts the values in the exact golden mean power ratio.

$\frac{34-0.005025}{89}=0.381966={\phi }^2$ (9)

$\frac{178}{34-0.005025}=5.2360679=2{\phi }^{-2}$ (10)

As an overall result we present in Table 1 new golden mean-based formulas of crystallographic shear phases of the system Nb₂O₅–WO₆ for future synthesis routes to improve (by fortune) the electrochemical performance of battery materials. However, an initial TM/O ratio higher than φ² = 0.381966 means that additional oxygen would have to be accommodated in the lattice, and this should be ruled out.

The best adapted values to golden mean symmetry has ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ and this phase should get our full attention. The phase Nb₁₈W₈O₆₉ [23] as oxygen reduced variant Nb₁₈W_7.07Ti_0.93O_68.07 is also interesting.

The Fibonacci analysis summarized in this Table 1 indicates that consequent variation of Nb₁₈W₁₆O₉₃ would lead to a Nb₁₄W₃O₄₄ type phase. One could also add Nb₂O₅ as an end member of the niobium-tungsten-oxide homologous series with the Fibonacci number ratio TM/O = 2/5 = 0.4 > φ² to the table, but the crystal structure with fivefold coordinated polyhedra and without pronounced block formation behaves something differently to the tungsten-bearing compounds [24]. However, a colossal insulator-metal transition was recently observed in single-crystalline T-Nb₂O₅ films accessed by lithium [25] [26], where Li migration channels are oriented perpendicular to the film surface. Although the effect is different, it remembers the present author to the colossal conductance observed on oriented multiphase Y-Ba-Cu-O thin films [27] [28] [29], but also confirmed on a simple CuO-Cu interface [30] or on a Ag₅Pb₂O₆/CuO composite [31]. As the first researchers, we used at that time a (110) SrTiO₃ substrate for thin film deposition [27], now also used successfully for T-Nb₂O₅ deposition [25]. However, also a hexagonal graphene sheet could serve as substrate for T-Nb₂O₅ thin film deposition, when calculating between both surfaces the index of overall atomic distance deviation.

7. Fibonacci Nets Compared to Projected Shear Phase Lattices

The crystal structure of hexagonal Bi_6.3JS₉ served as example to develop the Fibonacci net concept [2] [32]. In Figure 3 such a net and its mirror image is displayed showing 13 sub-cells. If the sub-cells would be decorated with heavy atoms, then the strongest X-ray diffraction intensities could be observed for reflections having Miller indices $31\bar{4}0$ and $52\bar{7}0$, giving index sums of h² + k² + hk = 13 respectively 3 × 13. The angle α between sub-cell direction and a-axis can be calculated as

${\delta }_{13}=\arctan \left(\frac{5}{3\sqrt{3}}\right)-30=13.898˚$. (11)

Such net, helically twisted to a tubule, has been connected with tubulin microtubules that show strongest reinforcement of the ordered pattern when the protofilament number equals even PF = 13 [33]. Such dominant architecture may be also the basis for quantum computer architecture and can be shared with considerations regarding human consciousness [7].

On can proceed with a further hexagonal Fibonacci net having 21 sub-cells with strongest $41\bar{5}0$ lattice planes and the index sum of 4² + 1 + 4 = 21, which is again a Fibonacci number.

The offset angle is calculated to be

${\delta }_{21}=\arctan \left(\frac{2}{\sqrt{3}}\right)-30=10.893˚$. (12)

Coordinates of both nets are given in the Appendix. Although the lattices of niobium tungsten oxides deviate strongly from hexagonal symmetry, their monoclinic unit-cells with monoclinic angles near 120˚ show interesting similarities to the Fibonacci nets.

When comparing the simplest 3 by 4 block shear structure of Nb₁₂WO₃₃ = TM₁₃O₃₃ (Figure 5) with the 13 sub-cell Fibonacci net, one can identify similar atomic arrangements, marked by red lines starting from the tungsten site at the “origin”. In the 4 by 5 block structure of ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ (Figure 4), one can identify details of a 21 sub-cell Fibonacci net (right side of Figure 6). We have to consider in the different crystal structures an atom at the “origin” and a block of atoms, giving 1 + 3 × 4 = 13 respectively 1 + 4 × 5 = 21, and 1 + 5 × 5 = 2 × 13. However, we have two formula units each and therefore the double number of atoms. In Figure 7 a crystallographic interpretation is given by comparing the metal sites with positions of a Fibonacci net, using a transformed unit-cell with exact 120˚ angle and lattice parameters of a' = 2c' to adapt C-centering of the

Figure 5. Crystal structure of Nb₁₂WO₃₃ projected down [010]. Space group C2, a = 22.37 Å, b = 3.825 Å, c = 17.87 Å, ß = 123.6˚, Z = 2, D_x = 4.76 Mg∙m⁻³ [5]. Tungsten (yellow tetrahedra, brownish sphere) joins four neighboring 3 × 4 blocks. The atomic arrangement highlighted by red lines can be compared with that on the Fibonacci net in Figure 6.

Figure 6. Fibonacci arrangement of hexagonal nets [2] [32] [34]. Unit-cells blue outlined, left: 13 sub-cells (green), offset by an angle of α = 13.9˚, right: 21 sub-cells, offset angle 10.9˚.

Figure 7. Transformed unit-cell of Nb₁₂WO₃₃ indicating a Fibonacci-like net of TM sites. In the left “hexagonal” half-cell, limited by the green line, exactly 13 transition metal sites can be identified.

original cell. Coordinates of the Nb₁₂WO₅₅ structure have been exemplarily compared with these of the 13 sub-cell Fibonacci net. In the middle part of Figure 7, representing the inner TM block and suffering not of atomic shifts by crystallographic shear, an almost hexagonal net is located. The sufficient agreement between both sets of coordinates suggests using such Fibonacci approach to crystal structure determination, because the phases of a large number of X-ray reflections are in this way correctly fixed (TableA2). The Fibonacci analysis suggests furthermore that strong reflections of the X-ray powder patterns of the shear phases should show a Fibonacci number indexing. Years ago, the present author has successfully applied Fibonacci lattice decoration with heavy atoms to determine crystal structures [32] [35]. A Fibonacci net with 21 sub-cells was used to determine the heavy atomic positions of zeolitic Pb₆[Ge₆O₁₈]·2H₂O respectively of the crystal water free compound, space group $R\bar{3}$. From the hexagonal lattice parameter a we could find the positions of the sub-cell lattice with

$a_{sub}=a/\sqrt{21}$ and offset angle ${\delta }_{21}=60-\arccos \left(\frac{3}{\sqrt{21}}\right)=10.89˚$ (see also equation 12) [35].

8. Fluorination Synthesis Approach

A replacement of oxygen by some fluorine atoms is an obvious synthesis possibility, but the higher binding energy expected between lithium and fluorine in contrast to oxygen may rather hinder than promote lithium diffusion to obtain optimum performance. However, the example of lithium insertion in titanium oxyfluoride, TiOF₂, teaches us not to rule out such a synthesis route [36]. When replacing for example maximum oxygen by fluorine in ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$ by maintaining the Fibonacci stoichiometry_, one would get a tungsten-free hypothetical compound of formula ${\text{Nb}}_{11}^{5+}{\text{Ti}}_{10}^{4+}{\text{O}}_{40}{\text{F}}_{15}$ as a cheap alternative. The smaller radii of substituted ions would influence the channel diameters a little bit, assuming such a phase would exist.

9. The Mysterious Niobium

The special properties of niobium leading to many high-tech applications point towards a possible raw material competition situation in near future. This hardly fully understood metal with 5 electrons in its outer valence shells and oxidation states, ranging from −1 to +5, was in its history frequently in the focus of science. As pure metal it shows among all metals the highest superconducting transition of 9.25 K [37] and a transition of 23.2 K, when alloyed with germanium to A15-type Nb₃Ge [38] [39]. If one supposes that holes are essentially involved in the superconductivity of conventional materials as well [40] [41], especially niobium should be investigated in this direction anew. Niobium is also used as steel strengthener in super-alloys. Other interesting properties have been found in compounds of niobium such as pyrochlore, (Na, Ca)₂Nb₂O₆(OH, F), which is the main niobium-bearing mineral raw material. The world’s largest pyrochlore deposit, worked in the open pit, is located in Brazil near Araxá, Minas Gerais [42]. The carbonatitic complex contains bariopyrochlore with 16.5 wt% BaO and 63.5 wt% Nb₂O₅ [43] [44]. The pyrochlore structure, space group $Fd\bar{3}m$, shows as one of the simplest interacting systems (Ising ice model) frustrated magnetism and magnetic monopole formation [45].

Turning again to niobium tungsten oxides and their hidden properties, researchers look always at the oxidation states of niobium, but should also investigate the state of oxygen and conceivable holes located at oxygen, frequently being unobserved. The reader may study the work of Gippius et al. to get some ideas [46]. Catalytic properties of niobium oxides are due to the ability to dissociate hydrogen and its absorption in the crystal lattice [47]. Another aspect is the acentricity of the phases, which can significantly determine their physical properties. If we stay with Fibonacci number 13 again, the electron number of Nb²⁺ is 3 × 13. The chemistry of a compound is determined by the orchestra of electrons involved, but we have not yet dealt with the higher octaves.

10. Conclusion

Vacant atomic sites generated by chemical substitution in simple prototypic structures can be cut back by different mechanisms depending on whether they are cation vacancies or anion ones. Whereas cation vacancies can be removed by chemical twinning, anion vacancies can be cut back by crystallographic shear. Thereby structural channels will be generated allowing the intercalation of hydrogen respectively lithium and their super-ionic exchange as evidenced by the crystallographic shear phases of the Nb₂O₅-WO₃ system. From a Fibonacci stoichiometry analysis, some suggestions for the alteration of chemistry and preparation of crystallographic shear phases of this system are derived with special attention to super-ionic ${\text{Nb}}_{16}^{5+}{\text{W}}_5^{6+}{\text{O}}_{55}$. These proposals include the replacement of a small amount of W⁶⁺ by Ti⁴⁺ to adapt exact golden mean stoichiometry to gain disorder and the highest electrochemical performance. The substitution would cause some oxygen vacancies, which could reduce stress and support lithium diffusion and mobility further. Phase-pure material may be synthesized by ultrasonic treatment besides thermal cycling. The complex disorder landscape of phases of the Nb₂O₅-WO₃ system may be understood rather well by a Fibonacci stoichiometry analysis besides Fibonacci lattice arguments. The secrets of nature can only be unraveled with a more holistic approach, bringing different aspects of one object together.

Appendix

Table A1. Coordinates for the fibonacci nets (two-dimensional group p6, see also Figure A1).

m = site multiplicity.

Figure A1. Illustration of the 2D Fibonacci nets with differently colored atomic sites (coordinates according to Table A1).

Table A2. Comparison between coordinates of the Nb₁₂WO₃₃ crystal structure and the Fibonacci net.

*) x coordinates have been halved in comparison to the original net (see Figure 7). The same approach can be applied successfully to the case of Nb₁₆W₅O₅₅ using the 21 sub-cell Fibonacci net.

Conflicts of Interest

The author declares no conflict of interests regarding the publication of this paper.

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