Alan Stewart Hare‡
3, Marshbarns, Bishop's Stortford, UK. E-mail: alan@alanhare.me.uk; Tel: +44 (0)1279 465527
First published on 24th August 2016
Mass spectral line positions m/z are computed from a general formula for aluminium cations, AlnOp(OH)q(OH2)r(3n−2p−q)+, without treating n, p, q and r as independent variables but instead constraining parameters by tetrahedral or octahedral symmetry, in accordance with each of a number of structural possibilities. In one instance, a ring is distinguishable from a chain; in another, an orthorhombic sheet from a hexagonal prism. Amongst other polyhedra, five distinct icosahedral cases are discussed. Classical constructions are revisited in the light of aperiodic tiling. Successive Fibonacci numbers may describe the nucleation of a structure previously regarded as amorphous. Recent experimental observations are reviewed in the same context, and re-interpreted. A spectral value may be ascribed to a tetrahedral or an octahedral species (ambiguity is rare). The results may be a significant factor in an understanding of crystal nucleation.
Alumina, or aluminium hydroxide or oxide hydroxide, or aluminium oxide, or aluminal species generally in solution, may contain positive ions or ionic complexes that may be generally formulated as AlnOp(OH)q(OH2)rz+, where z = 3n − 2p − q and where {p, q, r} have often been treated as if they were independent variables and were independent of n (p, q, r are each ≥ 0). This formulation provides explicitly for minor variations in ligand, extent of hydration and ionic charge.
In a mass spectral interpretation, m = 27n + 16(p + q + r) + q + 2r. In what follows, I propose an analysis where {p, q, r} are not independently variable, but are constrained by the symmetry of the aluminal species. Limits are placed on the number of ligands which may surround any one Al3+ cation. In broad terms, the symmetry may be regarded as either octahedral, Oh, or tetrahedral, Td; six or four ligands may surround the cation: s = 6 or 4. In this analysis, r = r(s, n, p, q). m/z is then computed for each instance of the formula so constrained.
I have distilled out the essence of the computational method and presented this in Tables 1–5. This enables the reader to make progress through the text without having to dwell on each step of the derivation. Each table contains a text reference column referring the interested reader to a section containing the full derivation. The detailed results of the computation are appended as ESI.† In this context, observations made by Urabe and his group7,8 using Electrospray Ionisation (ESI) mass spectrometry have been reviewed independently and the results summarised in Table 6. Initially at least, negative ions are disregarded (except in the special case of Keggin or, more generally, Biliński: see below, Sections 2.4 and 2.6.6).
Special case | n | p | q | r | s | m | z | m/z | Text reference |
---|---|---|---|---|---|---|---|---|---|
Hexagonal ring | 2.2 | ||||||||
6 | 0 | 2n = 12 | ns − 2q = 12 | 6 | 582 | 3n − q = 6 | m/z = 97 | ||
2-d tiling by hexagons - tessellated hexagon (serrated) | 2.2.3 | ||||||||
6(1 + x)2, where x is the integer part of half the greatest column-height ξmax | 0 | 6(2 + 5x + 3x2) | 12(1 + x) | 6(97 + 175x + 78x2) | 6(1 + x) | m/z = (97 + 175x + 78x2)/(1 + x); e.g., for x = 0, 1, 2 or 3, m/z = 97, 175, 253 or 331 | |||
Chain of hexagons | 2.2.8 | ||||||||
2(2ξ + 1), where a row has ξ adjacent hexagons | 0 | 2(2ξ + 1) | 4(ξ + 2) | 350ξ + 232 | 2(ξ + 2) | m/z = (175ξ + 116)/(ξ + 2); e.g., for ξ = 1, 2, 3 or 4, m/z = 97, 116.5, 128.2 or 136, respectively | |||
Chain (general) | 2.3.1 | ||||||||
n | p | q | (s − 2)n + 2 − (p + q) | 4 or 6 | 27n + 16p + 17q + 18r | 3n − 2p − q | m/z | ||
Keggin cage – may be a special case of Biliński, rD | 2.4, 2.6.6 | ||||||||
4nL + 1, where the tetrahedral centre attaches 4 chains, assumed all to have length L and identical parametric values {nL, pL, qL} | 4pL + y, where y is the number of O2− ligands in the tetrahedron | 4qL + (4 − y) | nL(sL − 2) − (K − 2) − (pL + qL), with 13 − (pL + qL) being the most likely case: K = 1 (apical bonding), n = 3, s = 6 | sL = 6 | 27n + 16p + 17q + 18r | 4c − (y + 1), where in each cationic chain c = 3nL − 2pL − qL | m/z = [4m(nL, pL, qL) + m(y)]/[4c − (y + 1)] | ||
Prismatic species | 2.5 | ||||||||
including | |||||||||
Prism (general) | 2.5.1 | ||||||||
[n(x)]j, where the prism is of length j ≥ 2 | [p(x, h)]j | [q(x, h) − v[1 − (1/j)]]j + (j − 1)v | [r(x, h)]j | 27n + 16p + 17q + 18r | 3n − 2p − q + v[1 − (1/j)] | m/z | |||
Sheet of octahedra | 2.5.2 | ||||||||
x(2h + 1) − h, comprising 2h + 1 rows of octahedra | 2h(x − 1) | 2h(2x − 3) | 4(1 + 2h + x) | 6 | 3x + 7h − 2hx | Cannot equal an m/z above, e.g., for a 2-d tiling | |||
DoubleSheet – a special case of the polygonal prism, where the polygon is a rectangle and j = 2 | 2.5.3 | ||||||||
2[x(2h + 1) − h] | h(6x − 7) | 2h(2x − 3) | 8(1 + 2h + x) | 6 | 2(3x + 7h − 2hx) | ||||
Extended DoubleSheet – a more general case of the DoubleSheet, in which j ≥ 2 | 2.5.4 | ||||||||
2j[x(2h + 1) − h], where αvj hydroxyl groups are shared between adjacent DoubleSheets | j[h(6x − 7)] + h(j − 1) (x − 3) | j[3hx − 3h + (h/j)(x − 3)] | j[8(1 + 2h + x)] | 6 | [6x + 11h − 3hx − (h/j)(x − 3)]j − 2h(j − 1)(x − 3) | ||||
Polyhedra (non-prismatic) | Table 2 |
Special case | n | p | q | r | s | m | z | m/z | Text reference |
---|---|---|---|---|---|---|---|---|---|
Polyhedron (non-prismatic) | 2.6.1 | ||||||||
including | |||||||||
Platonic solid (including the regular icosahedron) | 2.6.2 | ||||||||
C + V = V | αD = 0, because D = 0 | 2βD + 2εE = 2εE, where E = F + V − 2 | (s − (4εE/V))V | 4 or 6 | 27n + 16p + 17q + 18r | 3V − 2εE | |||
of which the icosahedron is the special case, V = 2(1 + 5) | 2.6.2 | ||||||||
V = 12 | E = 30; q = 2V, for example (in the case where ε = 2/5) | r = V(6 − 10ε) = 2V, for example (where s = 6) | 4 or 6 | V(3 − 5ε), neutral if ε = 3/5; equal to V if ε = 2/5 | m/z = 61 + 18(s − 4) (if ε = 2/5, for example) | ||||
Archimedean solid | 2.6.3 | ||||||||
C + V = V | αD = 0 | 2εE, where E = F + V − χ (if convex, χ = 2) | (s − (4εE/V))V | 4 or 6 | 27n + 16p + 17q + 18r | 3V − 2εE | |||
of which Goldberg or the truncated icosahedron is a special case, V = 60 | 2.6.3 | ||||||||
V = 60 (“buckminstalumina” – were it to exist) | F = 32, χ = 2; q = 2E = 3V = 180, for example (in the case where ε = 1) | r = 0, for example | 4 or 6 | 3V(1 − ε), neutral if ε = 1 | |||||
3-d space-filler (non-prismatic) | Table 3 | ||||||||
C + V | αD | 2βD + 2εE | (s − (4εE/V))V | 4 or 6 | 27n + 16p + 17q + 18r | 3V − 2εE | |||
Stochastic nucleation product | 2.6.9 | ||||||||
cV = 23 or ck + 2(23 − ck) initially, in the classical case for example; in which ck = 2k, where k = 0, 1, 2 | aαD = αk2k, in the example – see text | 2bβD + 2eεE = βk2k + 2eεE, in the same example | A cV(s − 6ε) component is to be expected, its r = 23(s − 6ε) or 2(23 − ck)(s − 6ε) initially | 4 or 6 | 27n + 16p + 17q + 18r | 3cV − 2(aα + bβ)D − 2eεE, in general (a = 2b, in the example) | |||
N-dimensional species | 2.6.10 | ||||||||
C(n) + V(n) = 2n, on whose exterior V(n) = 23 | αD(n) = a(n − 1)2n−2, approx. | 2β(D(n) + E(n)) = 2βn2n−1 | sV(n) − 4β(12 + V(n)) | 4 or 6 | 27n + 16p + 17q + 18r | z = 2n3 − [α(n − 1) + 2βn]2n−1, or 2n−1[6 + α − (α + 2β)n], approximately. Charge-neutrality would arise when β = 3/n (if α = 0); or, more generally, when n = (6 + α)/(α + 2β); e.g., in the hexeract (α = 0, β = 1/2, s = 6) |
Given that aluminium salt solutions are amenable to scientific study,6 there is now the opportunity to examine the possibility of linear, hexagonal or polyhedral crystal nucleation. This could be of fundamental importance. A bigger question still might be whether the salt solution is really nothing more than “a crystal waiting to happen” (i.e., comprises a pre-determined set of nuclei of given geometry) … or whether the introduction of a chemical reagent – or impurity such as a metal ion, or other electric field or perturbation – must first radically alter the structure of the solution to induce its own characteristic nucleus.
The imperfect octahedron (symmetry lower than Oh) may be conceptualised as a sphere of radius R that encloses all the atoms, regarded as if they were point-positions, within its geometric span dgeo. From Jung's theorem in 3 dimensions, R ≤ dgeo sqrt(3/8). Since all six octahedra are indistinguishable one from another, the six identical spheres may be positioned with each sphere touching two others in a plane so as to form a perfect hexagonal lamina or “wafer” 2R thin and of edge-length 2R with cross-sectional area 6R2sqrt(3).
As a 1st approximation, the linear ratio would be λ = tan(cot−1τ)/tan(π/6) = 2sqrt(3)/(1 + sqrt(5)). Despite its lower symmetry, the hexagon may have regained something of its lustre!
The only instance known to occur naturally is icosahedrite, the aluminium mineral Al63Cu24Fe13, with 5-fold symmetry (rotation through 72°, or 2π/5 or cos−1(τ/2)): not an “alumina” per se. In the regular pentagon, τ is the ratio of chord to edge length. Nucleation and growth mechanisms remain to be elucidated. More generally, quasi-crystalline growth remains possible, involving structural changes to an “amorphous” or chaotic precipitate. When modelling crystal growth, icosahedral symmetry need no longer be debarred.
The leading Al3+ cation of each of the four (trimeric) chains will bond in some way with either an apex, edge or face of the aluminate tetrahedron. This bonding will involve either one, two or three of the cation's six ligands (assuming broadly octahedral symmetry, sL = 6): K = 1, 2 or 3; and leave five, four or three of the six remaining. Of this remainder, two ligands will be shared with the next cation in the chain. As before, the terminal cation (now at the other end of the chain) will share two ligands with its neighbour. (Bonding through the tetrahedral apical oxygen seems intuitively the most likely possibility, retaining a high degree of symmetry such as Td while distorting the tetrahedron the least severely.) This means that the total number of Keggin chain ligands p + q + r = s + (n − 2)(s − 2) + (s − 2 − K). r = n(s − 2) − (K − 2) − (p + q). Given the likely case (K = 1, n = 3, s = 6), r = 4n − (K − 2) − (p + q) = 13 − (p + q). In a mass spectrum containing the tridecameric species, m = 4m(nL, pL, qL) + m(y).
Keggin-like combinations of tetrahedra and octahedra may be significant in crystal growth because, unlike tetrahedra alone, some such combinations have the capacity to fill space contiguously.13 An example is Biliński (see Fig. 2).
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Fig. 2 3-d space-filling by Biliński rhombic dodecahedra. Biliński space-filling may be represented by the equation ![]() |
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Fig. 3 A rhombic dodecahedron. A dodecahedron of the Biliński kind comprises four golden rhombohedra, two prolate and two oblate: |GρD(x)〉 = 2|GV1(x)〉 + 2|GV2(x)〉. Were it to exist, an aluminal species structured with cations at the vertices (one internal) could be formulated as AlO4α(OH)8β[Al3+δ(OH)(10+4δ)ε(OH2)s(3+δ)−(20+8δ)ε−2(α+β)]4 z+, where δi = {[0, 1], twice}i=14, s = sv = 6; and where zi = 3(4δi + 13) − 8[α + β + ε(2δi + 5)], or z = 45 − 8(α + β + 6ε). The Biliński rhombic dodecahedron is a space-filler (see Fig. 2). The formula makes explicit reference to a central tetrahedral aluminate structure. In such a structure the special case of δi = 0 (zero values only), x = 1 may be identified with the typical Keggin tridecamer [AlnOp(OH)q(OH2)rc+]4[AlOy(OH)4−y(y+1)−]z+ (n = 3), in which y = 4α, α + 2β = 1 and z = 3(1 + 4n) − 4(1 + α + 2p + q), where q = 10ε and p = 0. The Biliński dodecahedron |GρD(x)〉 is central to the Fyodorov rhombic icosahedron, |GρI(x)〉, where it appears as a nuclear subtype within “Penrose in 3-d” (see Fig. 4): |GρI(x)〉 = |GρD(x)〉 + 3|GV1(x)〉 + 3|GV2(x)〉; so that the Fyodorov icosahedron has a total of five pairs of golden rhombohedra. It is subject to Penrose enlargement. |
G(x) = jG(x), where j is a finite number of polygonal layers. |
In this model of the prism, v O2− ligands or OH− ligands may be shared between each of the j − 1 pairs of adjacent layers in the prism. Where the layer itself contains no O2− ligands, the formula for the cation is [Aln(x)(OH)q(x,h)−αv(OH2)r(x,h)c+]j[O(j−1)v2v(j−1)−]z+, where z = cj − 2v(j − 1) and where in the cationic layer c = 3n − (q − αv). More generally, if prior to sharing the layer itself were already to contain p O2− ligands, then after sharing, the formula would be [Aln(x)Op(x,h)(OH)q(x, h)−αv(OH2)r(x,h)c+]j[O(j−1)v2v(j−1)−]z+, where now c = 3n − 2p − (q − αv); v ≤ q; α ≤ 1. Throughout the length of the prism, the overall number of O2− ligands shared between the j layers is αvj, which equals (j − 1)v; therefore α = 1 − (1/j), so that the formula becomes Aln(x)Op(x,h)(OH)q(x,h)−v[1−(1/j)](OH2)r(x,h)c+]j[O(j−1)v2v(j−1)−]z+, where c = 3n − 2p − q + v[1 − (1/j)]; j ≥ 2 (j even). In a mass spectrum containing the prismatic species, m = jm(n, p, q, r) + m(j, v).
Given hexagonal close packing of spheres and a unit-cell volume of 8πR3, the unit cell has length 4Rsqrt(2/3) and the Kepler maximum packing-density η = π/3sqrt(2). Cubic close packing gives the same maximum (though this may be surpassed by ellipsoidal packing). The hexagonal prism length is 4R(j − 2)sqrt(2/3). Without spherical close packing, η < ηKepler, e.g. η = π/3sqrt(3), and the volume and length are correspondingly greater.
AlO2(OH)2(OH2)2[AlO2(OH)2]x−3AlO2(OH2)2(3x−5)− |
Also adjacent to this short interior row is either, the opposite edge of the sheet, or, a longer interior row comprising x octahedra; there are h − 1 such longer rows:
Al(OH)2(OH2)2[Al(OH)2]x−2Al(OH2)2(x+2)−. |
Overall then, in the sheet: n = 2x + h(x − 1) + (h − 1)x, or x(2h + 1) − h. p = 2h(x − 1). q = 2 + 2(x − 2) + 2h(x − 2) + 2(h − 1)(x − 1) = 2h(2x − 3).
r = 16 + 4(x − 2) + 4h + 4(h − 1) = 16 + 4x − 8 + 4h + 4h − 4 = 4 + 4x + 8h = 4(1 + 2h + x). |
The model sheet formula becomes Alx(2h+1)−hO2h(x−1)(OH)2h(2x−3)(OH2)4(1+2h+x)(3x−2hx+7h)+.
Unlike the hexagonal or tetragonal prism and certain rhombohedra (not exclusively golden) where h is a function of x, in a triclinic, monoclinic or orthorhombic system such as the Sheet, doubleSheet or extended DoubleSheet, h and x are independently variable. In neither m(x) nor z(x) are the coefficients of x identical, and in the doubleSheet (or extended DoubleSheet) they are both linear in h. Without close packing, again η < ηKepler; e.g., if cubic, η = π/6.
An attempt to equate m(x)/z(x) for the disparate crystal systems – to test for potential spectral ambiguity – leads to a quartic equation in h and x whose roots are not necessarily integers. In principle, therefore, mass spectrometry may well be capable of distinguishing between (e.g.) sheet growth and hexagonal growth patterns as early as nucleation.
Precise experimental design will depend on the extent of growth G(x, y), and the growth rate, G(x, y, t). In solution, Akihiro Wakisaka's experiment6 will be the preferred approach initially; however, once precipitation has begun,14 then other experimental methods may come into use: SIMS, for example, may become viable.
The remaining six “semi-regular” species would each be anionic. For example, a snub cubic aluminate ion, were this unlikely species to exist, would comprise only squares and equilateral triangles, Al24(OH)12048− (V = 24, F = 38). (Now 4E/V ≤ 10.)
For the species to be cationic, the Euler characteristic χ could be relaxed such that χ > F − (V/2) and the species would be non-convex; although χ might be negative. If, for example, χ = −6, then for a cation, V would have to exceed 2F by at least 12. Alternatively perhaps, given that s ≥ 4, 3V/E > 2ε ≥ 4/5. Or the polyhedral cation might not exist.
FV(x+h)(x + h1 + h2) + h = FV(x)(x) + c1FV1(x1 + 1) + c2FV2(x2 + 1) − 2h, and GV(x+h)(x + h) = GV(x)(x) + h, |
Suppose initially, at x1 = 0, there exist 4 rhombohedra in 2 pairs (V = 8, F8 = 6). If all four come together, each one sharing a face, the product is a rhombic dodecahedron F14(4) with V = 14, F14 = 12. This specific polyhedron could be significant as it can close off vertices and is potentially space-filling. Of the original 6 × 4 = 24 external faces, 12 survive – open to further growth – and 8 have vanished, while the remaining h = 4 have become internalised within the greater polyhedron: G14(4) = 4.
Implicit in pure polyhedral growth – unlike specifically prismatic growth – is the characteristic that, beyond a requirement for mutual face congruence, there appears to be no geometrical constraint to preserve symmetry, provided that the grown structure remains convex. Thermodynamically there will be a tendency for a Wulff construction15 to form, lowering the Gibbs free energy; although the system seems unlikely to attain equilibrium short-term. (See below, Penrose in 3-d.) This may be so whether or not a 3-d tiling is a Penrose analogue, such as might grow from a Biliński F14(2 + 2) species13 (where the 4 rhombohedra comprise 2 non-identical pairs of identical golden twins, each pair having the half-angle θ = cot−1τ). Continued growth, for example from x = 4 to x = 10, could produce a rhombic icosahedron: F22(5 + 5) + 6 = F14(2 + 2) + 6F8(1) − 12; and might go further, to x = 20, say, and the F32 triacontahedron attributed to Kepler: F32(10 + 10) + 10 = F22(5 + 5) + 10F8(1) − 20.
Alternatively the rhombic dodecahedron, whether Kepler or Biliński, could potentially fill a 3-d volume (in a sense comparable with the way in which the hexagon fills a 2-d area).
Unfortunately, however, the distinctive “first” rhombic dodecahedron, or classical Kepler F14(4) species13 (whose four constituent rhombohedra are all identical, though these, having θ = cot−1sqrt(2), are not golden), cannot be expected to show a mass spectrum distinguishable from its Biliński counterpart.
For 3-d tiling, and given the possibility of “Penrose in 3-d” with 2 unlike tiles, the capacity of any one species to fill space may matter less than contiguity of growth FV(x + h) over a long range of x values irrespective of particular species in the growth pattern. FV(x) is, of course, a positive integer; and we might assume that for contiguity, at least one of these external faces must be congruent with at least one face of either a FV1(x1 + 1) or a FV2(x2 + 1) species remaining. (At large x values, neither x1 nor x2 is necessarily still equal to zero.) The number of vertices on the two approaching faces must match.
If, for any given x, we knew the specific set of all polygons remaining, then to count the number of any such vertices could be a trivial task (e.g., for any rhombus, it would be 4); however, for large x, computation of the potential sets may not be straightforward; moreover, to establish facial congruence even in the case of smaller, regular polyhedra, the calculation of growth-angle θ and the edge-length is not always trivial. Again, thermodynamic considerations may help exclude some of the less likely possibilities.
The tetrahedral aluminate structure central to Biliński prompts the observation that the Keggin tridecamer (see above) is a special case with δi = 0 of the Biliński formula AlO4α(OH)8β[Al3+δ(OH)(10+4δ)ε(OH2)s(3+δ)−(20+8δ)ε−2(α+β)]4z+, where δi = {[0, 1], twice}i=14, s = sv = 6. See Fig. 3.
Such tiling by rhombic dodecahedra is in broad terms a 3-d counterpart of the 2-d tiling by hexagons, G(x). (See above, Section 2.2.4.) The first occurrence (x = 1) would be FV(76)(76) + 36 = F14(4) + [(32 − 1) + 2(12 + 22)]F14(4) − 72; the second (x = 2), FV(340)(340) + 168 = F14(4) + [(52 − 1) + 2(12 + 22 + 32 + 42)]F14(4) − 336; and so on.
A forerunner of this example is the classical cubic construction13 with the form , where d = 3. Ultimately, if a crystal were to be nucleated with this geometry, its growth could be expected to be asymptotic in the limitx→∞ h(x) + 1.
In the FV(x3)(x3) species the x3 units would bear a total of 3x3 + 6x2 + 3x edges. Subtracting 12x2, the number of internalised edges is 3x(x − 1)2. The golden rhombohedral formula then becomes Al(x+1)3O6αx(x−1)2(OH)6βx(x−1)2+24εx2(OH2)4[s(3x−1)−12εx2]z+, where z = 3(x + 1)3 − 6(2α + β)x(x − 1)2 − 24εx2. Or expanding, , where:
m0 = 9 − 24s, z0 = 1 |
m1 = 27 + 72s + 32α + 34β − 288ε, z1 = [3 − 2(2α + β)] |
m2 = 27 − 64α − 68β + 136ε, z2 = [3 + 4(2α + β) − 8ε] |
m3 = 9 + 32α + 34β, z3 = [1 − 2(2α + β)]. |
See Table 3.
Special case | n | p | q | r | s | m | z | m/z | Text reference |
---|---|---|---|---|---|---|---|---|---|
Polyhedron (3-d space-filler, non-prismatic) | 2.6.4–7 | ||||||||
including | |||||||||
Non-Penrose: classical cubic (or rhombohedral) | 2.6.7 | ||||||||
(x + 1)3 | 6αx(x − 1)2 | 6βx(x − 1)2 + 24εx2 | 4[s(3x − 1) − 12εx2] | 4 or 6 | 27n + 16p + 17q + 18r | 3(x + 1)3 − 6(2α + β)x(x − 1)2 − 24εx2 | Both m and z are cubic in x: ![]() |
||
of which the single rhombohedron (including the cube) is the special case, x = 1 | 2.6.7 | ||||||||
8 | 0 | 24ε | 8(s − 6ε) | 4 or 6 | 24(1 − ε): neutral if ε = 1 | m/z = (9 + 6s − 19ε)/(1 − ε) e.g. s = 6, ε = 2/3 | |||
Rhombic dodecahedron – Biliński, ρD | 2.6.6 | ||||||||
C + V = CρD(2 + 2) + VρD(2 + 2) = 1 + 14 | αD = αDρD(2 + 2) = 4α | 2βD + 2εE = 2βDρD(2 + 2) + 2εEρD(2 + 2) = 8(β + 6ε) | sV − 4εE − 2(α + β)D = 14s − 8(α + β) − 96ε | 4 or 6 | 27n + 16p + 17q + 18r | 45 − 8(α + β + 6ε) | If s = 6, then m/z = (1917 − 80α − 8β − 912ε)/[45 − 8(α + β + 6ε)] | ||
Non-Penrose: non-classical – Biliński rhombic dodecahedra | 2.6.7 | ||||||||
C + V = 4H(x − 1) + 8(4x2 + 3x + 2), where ![]() |
D = 4(H(x − 1) + 2(8x2 + 1)); 2αD = 8α(H(x − 1) + 2(8x2 + 1)) | E = 8H(x − 1) + 8(6x2 + 9x + 2); 2βD + 2εE = 8(β + 2ε)(H(x − 1)) + 16(8β + 6ε)x2 + 144εx + 16(β + 2ε) | s(3H(x − 1) + 2(8x2 + 12x + 7)) − 4ε(8H(x − 1) + 8(6x2 + 9x + 2)) | 4 or 6 | 27n + 16p + 17q + 18r | Again both m and z are cubic in x; z = 4[3 − 2(2α + β + 2ε)]H(x − 1) + 24(4x2 + 3x + 2) − 16(2α + β)(8x2 + 1) − 16ε(6x2 + 9x + 2) | Again, cubic: Biliński coefficients {(m, z)}i=03 differ from the classical cubic. If (e.g.) α = 0, β = 1, ε = 2/3, then m/z = 1301/5 or 260.2 | ||
Penrose in 3-d – the assembly | Table 4 | ||||||||
(C + V){[φ+, φ−] + ρ[φ+, φ−] + 3ρI} | αD {[φ+, φ−] + 3ρI} | 2βD{[φ+, φ−] + 3ρI} + 2εE{[φ+, φ−] + ρ[φ+, φ−] + 3ρII} | sV − 4(βD + εE) | 4 or 6 | 27n + 16p + 17q + 18r | Potentially comprises an aluminal ϕ+ species paired with a corresponding ϕ− species | [m1(2) + τ−1m2(2)]/[z1(2) + τ−1z2(2)], in the limit as n (in Φn) tends towards infinity | ||
including | |||||||||
Ogawa's flower, {ϕ+, ϕ−} | Table 5 | ||||||||
(C + V){ϕ+, ϕ−} | αD{ϕ+, ϕ−} | 2βD{ϕ+, ϕ−} + 2εE{ϕ+, ϕ−} | |||||||
Fyodorov: the ρI component (cf. Biliński, ρD above) | 2.6.6 | ||||||||
C + V = CρI(5 + 5) + VρI(5 + 5) = 4 + 22 | αD = αDρI(5 + 5) = 15α | 2βD + 2εE = 2βDρI(5 + 5) + 2εEρI(5 + 5) = 30β + 80ε | sV − 4εE − 2(α + β)D = 22s − 30(α + β) − 160ε | 4 or 6 | 27n + 16p + 17q + 18r | 78 − 10(3α + 3β + 8ε) | If s = 6, then m/z = (3078 − 300α − 30β − 1520ε)/[78 − 10(3α + 3β + 8ε)] | ||
Gap-filling rhombohedra ρ{ϕ+, ϕ−}, with contributions to {m, z} reduced by vertex-, edge- or face-sharing | 2.6.8 | ||||||||
Reduced from 8 | 0 | Reduced from 24ε | Reduced from 8(s − 6ε) |
The number of Biliński components , or 2[12 + 22 + 32 +… + (2x)2] + (2x + 1)2. Of these, H(x − 1) are interior units. Subtracting, the number of facial units, including corner, edge and facial dodecahedra, is not 8x(x + 1), but 2(8x2 + 1) [because 2x is even]. H(0) = 1. We may refer to the subset (2x + 1)2 as the one “principalSquare”, and to the other squares as “minor”. Of the 2(8x2 + 1) units, 2 are polar, 4 are principalSquare corners, 4(2x − 1) are principalSquare edge units and each of the 2(2x − 1) minorSquares has 4 corners, leaving 8(2x2 − 3x + 1) minorSquare edge units. Now, from the geometry:
Vinterior = 3; Einterior = 8; Vpolar = 5; Epolar = 12; VprincipalSquareCorner = 5; EprincipalSquareCorner = 10; VprincipalSquareEdge = 2; EprincipalSquareEdge = 6; VminorSquareCorner = 2; EminorSquareCorner = 6; VminorSquareEdge = 1; EminorSquareEdge = 3. |
Multiplying out, and adding the products:
C = CinternalH(x) = H(x) = H(x − 1) + 2(8x2 + 1); D = DinternalH(x) = 4H(x); V = VinteriorH(x − 1) + 2Vpolar + 4VprincipalSquareCorner + 4(2x − 1)VprincipalSquareEdge + 8(2x − 1)VminorSquareCorner + 8(2x2 − 3x + 1)VminorSquareEdge; |
V = 3H(x − 1) + 2(8x2 + 12x + 7); C + V = 4H(x − 1) + 8(4x2 + 3x + 2); similarly, E = 8H(x − 1) + 8(6x2 + 9x + 2). Summing squares, |
Unsurprisingly then, the Biliński equation for either m or z is also a cubic in x. See Table 3. The specific formula is
Al4H(x−1)+8(4x2+3x+2)O8α(H(x−1)+2(8x2+1))(OH)8(β+2ε)(H(x−1))+16(8β+6ε)x2+144εx+16(β+2ε)(OH2)s(3H(x−1)+2(8x2+12x+7))−4ε(8H(x−1)+8(6x2+9x+2))z+, |
m0 = 4(81 + 67s + 32α + 34β − 76ε), z0 = 4[9 − 2(2α + β + 2ε)] |
m1 = (4/3)[864 + 513s + 448α + 476β + 3116ε], z1 = (16/3)[24 − 14α − 7β − 41ε] |
m2 = −16(9s − 64α − 68β − 38ε), z2 = −32(4α + 2β − ε) |
m3 = (32/3)[57 + 27s + 64α + 68β − 152ε], z3 = (64/3)[3 − 2(2α + β + 2ε)]. |
In this Biliński formulation, each & every coefficient differs from its classical cubic counterpart. Just as classical construction continues by adding a cube, here growth may continue through addition of a Biliński rhombic dodecahedron. Their space-filling capacity may not differ hugely, but the two non-Penrose tilings are distinct from each other. Compare 2-d area tiling by hexagons (see above, Section 2.2.3), in 3-d Biliński space-filling may be represented by the equation .
|GV1(x + 1)〉 = Φn+1|GV1(x)〉 + Φn|GV2(x)〉 |
|GV2(x + 1)〉 = Φn|GV1(x)〉 + Φn−1|GV2(x)〉, |
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Fig. 4 (a) “Penrose in 3-d”, showing self-similar enlargement. Ogawa's flower, at either end of the principal diagonal, is perhaps the most striking feature of Penrose in 3-d. The overall Penrose structure appears to be a special case n = Φr + 2 − i, r = Φr − 2 (i = 1, 2) of the Fibonacci enlargement |GVi(x + 1)〉 = Φn+1|GV1(x)〉 + Φn|GV2(x)〉. Each polyhedron is self-similar and enlarged by the linear factor, τ3 = 2 + sqrt(5); or volumetrically, τ9. In this diagram |GV1(x)〉, |GV1(x − 1)〉 and even |GV1(x − 2)〉 may be discerned visually; though not the smaller moiety |GV1(x − 3)〉. There is no semblance of a “unit cell” in the classical Bravais sense. The self-similar enlargement has a fractal dimension d = 3κ![]() ![]() ![]() |
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Fig. 5 A stellation of Kepler's rhombic triacontahedron. A rhombic hexecontahedral stellation of the Great Stella class, this non-convex polyhedron has 62 vertices and an internal vertex, with icosahedral symmetry. Comprising 20 golden rhombohedra (all prolate), it gives a precise 3-d representation of Ogawa's flower: |G62,ϕ+(x)〉 = 20|GV1(x)〉; |G62,ϕ−(x)〉 = 19|GV1(x)〉. The stellation is shown enlarged, with self-similarity, by Penrose in 3-d: |GV1(x + 1)〉 = |G62,ϕ+(x)〉 + 20|GV1(x)〉 + 19|GV2(x)〉 + 3|GρI(x)〉. Given that with Shechtman icosahedral symmetry is now regarded as crystalline and that earlier an Al alloy19 was ascribed the stellar structure, it is reasonable to seek near-icosahedral aluminal species such as AlO2[Al(OH)5]2[[Al2OH2(OH)4]5]655+, with m/z = 81.5; or, as is perhaps more likely, AlO2[Al2O2(OH)2OH2]5[Al(OH)5]2[[Al2(OH)4OH2]5]545+, where m/z = 99.4. In contrast to classical crystals generally and to Guyot's alloy, presumed dense, the fractal nature of the 3-d Penrose enlargement and its peculiar characterisation as Cantor dust suggests the possibility of a quasi-crystal which may be “nowhere dense” (my emphasis). My conjecture is that neither polyhedron of a pair {|GV1(x)〉, |GV2(x)〉} in which both are nowhere dense may be space-filled contiguously to every edge by similar polyhedra |GV1(x − 1)〉 or |GV2(x − 1)〉 in any combination (x ≥ 2). Can we then, using mass spectrometry, test for the physical existence of Cantor dust? |
Ogawa constructed an aperiodic 3-d tiling with linear self-similarity factor τ3 (or volumetrically, τ9); τ = (1 + sqrt(5))/2. τ3 = 2 + sqrt(5). τ9 = 38 + 17sqrt(5).
The above equations suggest that Penrose enlargement may recur endlessly, and may be written in the form
|GV1(x + 1)〉 = |G62,ϕ+(x)〉 + 20|GV1(x)〉 + 19|GV2(x)〉 + 3|GρI(x)〉 |
|GV2(x + 1)〉 = |G62,ϕ−(x)〉 + 6|GV2(x)〉 + 3|GρI(x)〉, |
Now allowing for internal face-sharing when x = 1 (α > 0, but in the floral case we let αϕ = 2βϕ = 1), the generic formula becomes the pair of the sums of the floral (ϕ+, ϕ−), rhombohedral ρ and ρI contributors, MP2[M2P2Q2R]5[MQ5]2[[M2Q4R]5]5M8Q28εR4(s−6ε)[MQ18εRs−8ε]4[MP4αQ8βM10P11αQ22β+20ε[[MQ4ε]3]5M2Q10εM8Q28εR8s−56εR22s−160ε−30(α+β)]3M4Q28εR2(s−8ε)z+ or potentially the aluminal ϕ+ species Al187O12+45α(OH)30(4+3β)+482ε(OH2)10[3+10s−9(α+β+8ε)]z+, together with the ϕ− species Al149O12+45α(OH)30(4+3β)+268ε(OH2)30+70s−18[5(α+β)+28ε)]z+. See Table 4.
Special case | n | p | q | r | s | m | z | m/z | Text reference |
---|---|---|---|---|---|---|---|---|---|
Penrose in 3-d – the assembly | 2.6.8 | ||||||||
(C + V){[φ+, φ−] + ρ[φ+, φ−] + 3ρI} | 2αD{[φ+, φ−] + 3ρI} | 2βD{[φ+, φ−] + 3ρI} + 2εE{[φ+, φ−] + ρ[φ+, φ−] + 3ρI} | sV − 4(βD + εE) | ||||||
pairing | |||||||||
A species, denoted φ+ [deriving from Ogawa's flower, φ+] | 2.6.8 | ||||||||
C + V = A*(x)[C(ϕ+,2) + V(ϕ+,2)] + B1*(x)[C(ϕ−,2) + V(ϕ−,2)] = νn+A*(x) + νn−B1*(x), where: A*(x) = A(A + B)x−2 and B1*(x) = B(A + B)x−2, in which A = Φn+12 + Φn2 and B = Φn(Φn+1 + Φn−1); and where: νn+ = 187; νn− = 149. Note C = Φn−12 + Φn2, also | 2αD = 2α[A*(x)D(ϕ+,2) + B1*(x) D(ϕ−,2)] = νp+[A*(x)+B1*(x)], where: νp+ = 3[4 + 15α] | 2βD + 2εE = 2A*(x)[βD(ϕ+,2) + εE(ϕ+,2)] + B1*(x)[βD(ϕ−,2) + εE(ϕ−,2)] = νq+A*(x) + νq−B1*(x), where: νq+ = 30(4 + 3β) + 482ε; νq− = νq+ − 214ε | sV − 4(βD + εE) = νr+A*(x) + νr−B1*(x), where: νr+ = 10[3 + 10s − 9(α + β + 8ε)]; νr− = 30 + 70s − 18[5(α + β) + 28ε] | 4 or 6 | 27n + 16p + 17q + 18r | νz+A*(x) + νz−B1*(x), where: νz+ = 417 − [90(α + β) + 482ε]; νz− = 303 − [90(α + β) + 268ε] | m/z = [m(x)/z(x)]1 = [m1(2) + τ−1m2(2)]/[z1(2) + τ−1z2(2)], in the infinite limit (independent of x > 2); prior to correction for subtrahend. If α = β = ε and ε = 1/2, then m/z = 168.9. | ||
with | |||||||||
A corresponding species, denoted φ− (an approximation derived from vertex-sharing only) | 2.6.8 | ||||||||
in which approximation (without subtrahend *) | |||||||||
*An exact reduction may be obtained by formulating a precise subtrahend based on the preferred |G(2x)〉 interior (see text) | |||||||||
C + V = B2*(x)[C(ϕ+,2) + V(ϕ+,2)] + C*(x)[C(ϕ−,2) + V(ϕ−,2)] = νn+B2*(x) + νn−C*(x), where: B2*(x) = B(B + C)x−2 and C*(x) = C(B + C)x−2 | 2αD = 2α[B2*(x)D(ϕ+,2) + C*(x)D(ϕ−,2)] = νp+[B2*(x) + C*(x)] | 2βD + 2εE = 2B2*(x)[βD(ϕ+,2) + εE(ϕ+,2)] + 2C*(x)[βD(ϕ−,2) + εE(ϕ−,2)] = νq+B2*(x) + νq−C*(x) | sV − 4(βD + εE) = νr+B2*(x) + νr−C*(x) | 4 or 6 | 27n + 16p + 17q + 18r | νz+B2*(x) + νz−C*(x) | m/z = [m(x)/z(x)]2, which in the limit again equals [m1(2) + τ−1m2(2)]/[z1(2) + τ−1z2(2)]. By the time Ogawa has been reached (n = 9), we may expect to discern a single line only. Mass spectrum independent of G(2x), the extent of 3-d tiling |
In this formulation the first M8Q28ε component derives from five rhombohedra (r0) positioned between the two Ogawa flowers, while the M2Q10ε and M8Q28ε together derive from six (r1 + r2) added to each ρI; with M4Q28ε from three in the gap between the two ρI neighbours (see text below Fig. 4(b)). The [MQ18ε]4 component allows for three rhombohedra to cement each of the other four gaps between floral and ρI nuclei, where face-sharing necessitates only one additional vertex but nine additional edges. See Fig. 4(b).
For the Ogawa case Φn = 34 then, the result is a pair of expected spectral lines {(m/z)j}j=12, where the Al3+ values in this first instance (x = 1) are (m/z)1 = [7821 + 1800s − (900α + 90β + 4766ε)]/[417 − [90(α + β) + 482ε]], and (m/z)2 = [6795 + 1260s − (900α + 90β + 4992ε)]/[303 − [90(α + β) + 268ε]]; again, see Table 4. This pair derives from the primitive state: |GV1(2)〉 and |GV2(2)〉 (x = 1).
Alternatively, and without invoking Ogawa's flower, (m/z)1 and (m/z)2 could each be computed as the rational quotient mj(Φn)/zj(Φn) of paired polynomials {mj(Φn), zj(Φn)} of degree x in Φn; though to do so might imply the property α = 0, suggesting (dubiously for alumina) an absence of O2− ligands. For this reason, it is physically significant to retain the floral and ρI characteristics explicitly.
Special case | n | p | q | r | s | m | z | m/z | Text reference |
---|---|---|---|---|---|---|---|---|---|
Non-convex, stellar polyhedron (Penrose in 3-d, stellation of Kepler's rhombic triacontahedron) | 2.6.8 | ||||||||
(C + V){φ+, φ−} | αD{φ+, φ−} | 2βD{φ+, φ−} + 2εE{φ+, φ−} | |||||||
comprising | |||||||||
Ogawa's flower, φ+ | 2.6.8 | ||||||||
n62,ϕ+ = C + V = C62,ϕ+(20 + 0) + V62,ϕ+(20 + 0) = 1 + 62 = 63 | D = D62,ϕ+(20 + 0) = 50; however, p62,ϕ+ = p62,ϕ = 2(1 + 5αϕ) – see basic formula | E = E62,ϕ+(20 + 0) = 120; however, q62,ϕ+ = 5(2 + 4(5 + βϕ)), or 10(11 + 2βϕ) | r62,ϕ+ = 5(1 + 5) = 30 | 6 | 27n62,ϕ+ + 16p62,ϕ+ + 17q62,ϕ+ + 18r62,ϕ+ = 4143 + 160αϕ − 340βϕ | 75 − 20(αϕ + βϕ) | m/z = (4143 + 160αϕ − 340βϕ)/[75 − 20(αϕ + βϕ)] or if αϕ = 2βϕ then (1381 + 220βϕ)/5(5 − 4βϕ); e.g., if βϕ = 1/2, then m/z = 2982/30 or 99.4. Or if αϕ were 0 and βϕ were 1, then m/z would be 4483/55 or 81.5 | ||
paired with | |||||||||
a corresponding species, φ− (with subtrahend derived from assumed triple face-sharing) | 2.6.8 | ||||||||
F = F62,ϕ−(19 + 0) = F62,ϕ+(20 + 0) − 3 = 60 − 3 = 57 | |||||||||
from which subtrahend may be derived | |||||||||
n62,ϕ− = C + V = C62,ϕ−(19 + 0) + V62,ϕ−(19 + 0) = C62,ϕ+(20 + 0) + V62,ϕ+(20 + 0) − 1 = n62,ϕ+ − 1 = 63 − 1 = 62 | p62,ϕ− = p62,ϕ (which remains unchanged) | q62,ϕ− = q62,ϕ+ − 6εϕ = 10(11 + 2βϕ) − 6εϕ | r62,ϕ− = r62,ϕ+ − (s − 6εϕ) = 30 − (s − 6εϕ) = 6(4 + εϕ) | 6 | 27n62,ϕ− + 16p62,ϕ− + 17q62,ϕ− + 18r62,ϕ− |
m/z (observed) | Interpretation (most likely) | Possible ambiguity? | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Symmetry | n | p | q | r | z | Species | ||||
79 | Td | 2 | 4 | 2 | 2 | Al2(OH)4(OH2)22+ | ||||
97 | Td | 1 | 2 | 2 | 1 | Al(OH)2(OH2)2+ | Y | Aln(OH)2n(OH2)2nn+ | ||
including | ||||||||||
n | ||||||||||
2 | Dimer | |||||||||
Al2(OH)4(OH2)42+ | ||||||||||
4 | Tetramer | |||||||||
Al4(OH)8(OH2)84+ | ||||||||||
6 | Hexameric ring | |||||||||
Al6(OH)12(OH2)126+ | ||||||||||
8 | Rhombohedron | |||||||||
Al8(OH)16(OH2)168+ | ||||||||||
(or similar formulation) | ||||||||||
OR | ||||||||||
AlO4α(OH)8β[Al3(OH)2(5+2δ)ε(OH2)6(3+δ)−4(5+2δ)ε−(α+2β)]4z+, where α = 2β = 1/2, δ = 0, ε = 1/2 | ||||||||||
which equals | ||||||||||
n | ||||||||||
13 | Tridecamer | |||||||||
AlO2(OH)2[Al3(OH)5(OH2)7]413+ | ||||||||||
115 | Oh | 2 | 4 | 6 | 2 | Al2(OH)4(OH2)62+ | ||||
133 | Oh | 1 | 2 | 4 | 1 | Al(OH)2(OH2)4+ | Y | [An unlikely hexamer] | ||
157 | Td | 2 | 5 | 1 | 1 | Al2(OH)5(OH2)+ | ||||
217 | Td | 3 | 8 | 1 | Al3(OH)8+ | |||||
277 | Td | 4 | 4 | 3 | 3 | 1 | Al4O4(OH)3(OH2)3+ | |||
337 | Td | 5 | 4 | 6 | 2 | 1 | Al5O4(OH)6(OH2)2+ |
A similar flower growing under the Penrose transformation, however, would now comprise not simply 20 prolate rhombohedra, or 19; but 20 enlarged prolate; or, pairwise, 19 enlarged prolate rhombohedra:
|G62,ϕ+(x)〉 = 20|GV1(x)〉: |G62,ϕ−(x)〉 = 19|GV1(x)〉. |
See Fig. 5.
This fractal characteristic of Penrose in 3-d closely resembles asymmetric Cantor dust18 in 3-d. As this is nowhere dense, it may suggest space filling that is less than contiguous. This is my conjecture.
Φn−1|GV1(x + 2)〉 − Φn+1Φn−1|GV1(x + 1)〉 = Φn|GV2(x + 2)〉 − Φn2|GV1(x + 1)〉, |
(Φn2 − Φn+1Φn−1)|GV1(x + 1)〉 = Φn|GV2(x + 2)〉 − Φn−1|GV1(x + 2)〉. |
Now Φn+1Φn−1 − Φn2 = (−1)n; from this, it follows that |GVj(x + 3)〉 = (Φn+1 + Φn−1)|GVj(x + 2)〉 + |GVj(x + 1)〉 (n odd). A special case of this relation is |GVj(4)〉 = (Φn+1 + Φn−1)|GVj(3)〉 + |GVj(2)〉.
Expanding |GV1(3)〉, |GV1(4)〉 = [(Φn+12 + (Φn+1Φn−1 + 1)]|GV1(2)〉 + Φn(Φn+1 + Φn−1)|GV2(2)〉. More generally for x ≥ 2, |GV1(2x)〉 is the binomial expansion
In the aluminal case for example, where again the Al3+ cation is shared only at a vertex, the ϕ+ species would be
Since (A + B)x−2 = A*(x)/A, B1*(x) = (B/A)A*(x) and therefore [m(x)/z(x)]1 = [m1(2) + (Φ2n/Φ2n+1)m2(2)]/[z1(2) + (Φ2n/Φ2n+1)z2(2)] = 168.9, independent of x. While Φ2n/Φ2n−1 is not equal to Φ2n+1/Φ2n below infinite n (the values oscillate), in the limit the ratio tends towards τ.
Either value [m(x)/z(x)]i = [m1(2) + τ−1m2(2)]/[z1(2) + τ−1z2(2)], approximately (independent of x > 2). Again, [m(x)/z(x)]2 = 168.9. By the time n = 9 (Ogawa) has been reached, we may expect to discern a single line only.
Should Penrose in 3-d exist as a pair of physical species, its mass spectrum would be independent of G(2x), the extent of the 3-d tiling development. This asymmetric form of Cantor dust, the Ogawa solution, is unlike any other crystal nucleus so far examined.
If, for each Fyodorov icosahedron |GρI(1)〉 in the |GV1(2)〉 primitive, three golden rhombohedra – those beyond the local vertex – share faces with those in an adjoining |GV1(2)〉 major, then the subtrahend σ1(2) will derive from three M4Q28ε moieties, or three units of Al4(OH)28ε(OH2)2(s−8ε)4(3−7ε)+. If σ2(2) derives from one rhombohedron for each ρI, then this will represent MQ6ε or Al(OH)6ε(OH2)s−6ε3(1−2ε)+.
If ε = 1/2 then [m*1(2) + τ−1m*2(2)]/[z*1(2) + τ−1z*2(2)] would equal 151, approximately (in contrast to the value from vertex-sharing alone). The question of face-, edge- and vertex-sharing between major rhombohedra assumes a critical importance.
Because |GV1(2x + 1)〉 comprises (Φn+1 + Φn−1) |GV1(2x)〉 + |GV1(2x − 1)〉, of which the interior comprises a single |GV1(2x)〉, the interior fraction of rhombohedra is approximately (Φn+1 + Φn−1)−1; consequently we may now expect m1(2) to be decreased by the reduced subtrahend σ*1(2) = (Φn+1 + Φn−1)−1σ1(2), a reduction to the sharing fraction only. Similarly, σ*2(2) = (Φn+1 + Φn−1)−1σ2(2). Now [m*1(2) + τ−1m*2(2)]/[z*1(2) + τ−1z*2(2)] = 168.6, approximately.
Introducing the complication of non-sharing exterior rhombohedra appears to have restored the spectral line pair position to one more closely resembling the “vertex-only” result.
Also, because at infinity the alternating series of inverse products 1/ΦiΦi+1 converges to a value of τ − 1, an interesting question may arise: whether some Ogawa flowers grow less than others, and can wither as well as grow; whether there exists a Penrose transform under which a series or even an infinity of successive Fibonacci pairs {Φi−1, Φi}i=1∞ may be found [besides Ogawa's two successive pairs with n fixed in the crystal nucleus]; and, if so, whether τ – the infinite series – might represent a seemingly more jumbled-up or possibly truly chaotic precipitate;14 or now perhaps, Wakisaka's solution.6
Adopting the general formula MVQ2εER8(s−6ε)(3V−2εE)+, the rhombohedron (having E = 12, V = 8) may be seen as merely one special case of a more general polyhedral formulation McVPaαDQ2bβD+2eεERsV−4εE[3cV−2(aα+bβ)D−2eεE]+, with a = 0, b = 0, c = 1, e = 1. A Poisson distribution may be expected!
Charge-neutrality would arise when β = 3/n (if α = 0); or, more generally, when n=(6 + α)/(α + 2β), e.g. in the hexeractic [Al(OH)3]64(OH2)8 (α = 0, β = 1/2, s = 6). The icosahedritic icosahedron (see above, The regular icosahedron) has been represented as the 3-d projection of a 6-cubic structure.
The 3-cube may be regarded a special case of the rhombohedron with half-angle θ = π/4, or cot−1(1). If an aluminal species were to exist with such an unlikely structure then again, if the sole difference were to be the direction of its crystal growth, this species too would remain indistinguishable by mass spectrometry. The 6-cube, however, might readily be distinguished as a special case of the n-cube described above.
Formulation of the number of ligands p + q + r would again be determined by the specific polyhedron or polytope (again, s would equal 6 or 4). In the solid or 3-d geometry, vertex, edge, facial and internal polyhedra would differ from one another, as would their ligand numbers p, q, r.
To reduce gaps in the overall range of computed m/z values, and so improve interpretation of the mass spectra, there would be merit in performing a fuller range of such computations at an early stage. For the present, however, only the specific 3-dimensional arrays above have been selected (and so there may still be gaps). Despite the gaps, then, accepting the Group's interpretation7 of the interval of 18 between certain (m/z) values, few of the Group's observations7,8 remain unexplained. As is sometimes the case, however, there are minor spectral lines, less intense, still requiring explanation.
Al(OH)2(OH2)2+ + 2OH− ⇔ Al(OH)4− + 2H2O |
Dimeric cations appear to exist both in tetrahedral symmetry as Al2(OH)4(OH2)22+ and Al2(OH)5(OH2)+ (with lines at m/z = 79 and 157, respectively) and in octahedral symmetry as Al2(OH)4(OH2)62+, with the mass spectral line of the latter at m/z = 115 by far the most intense of the three.
Trimeric cations appear to be tetrahedral only, e.g. Al3(OH)8+ (m/z = 217). It may be that the octahedral trimer has been converted into a more stable tetrahedral tetramer or pentamer (m/z = 277 and 337, respectively):
Al3(OH)8(OH2)6+ + Al(OH)2(OH2)2+ + OH− ⇔ Al4O4(OH)3(OH2)3+ + 9H2O |
Al4O4(OH)3(OH2)3+ + Al(OH)2(OH2)2+ + OH− ⇔ Al5O4(OH)6(OH2)2+ + 3H2O |
Or dimer could react with dimer to form a tetramer directly (so bypassing the trimeric stage):
2Al2(OH)4(OH2)62+ + 3OH− ⇔ Al4O4(OH)3(OH2)3+ + 9H2O |
Structural rearrangements between tetrahedral and octahedral symmetry are also not impossible, especially under the moderately acidic conditions pertaining in (e.g.) an Al2(SO4)3 solution. In the pentamer, both cases (s = 4, m/z = 337 and s = 6, m/z = 173) are just visible, although the octahedral line is not at all intense:
Al5O4(OH)6(OH2)2+ + 8H2O + 2H3O+ ⇔ Al5(OH)12(OH2)103+. |
Yet there remains the further possibility, especially in the nitrate solutions, Al(NO3)3 (described in the Group's 2011 paper8), that the trimer could have vanished under nucleophilic (SNx) attack by successive hydroxyl groups of the Al(OH)4− anion:
Al3(OH)8(OH2)6+ + Al(OH)4− ⇔ [Al3(OH)8(OH2)5][Al(OH)4] + H2O |
[Al3(OH)8(OH2)5][Al(OH)4] + Al3(OH)8(OH2)6+ ⇔ [Al3(OH)8(OH2)5]2[Al(OH)4]+ + H2O… |
[Al3(OH)8(OH2)5]3[Al(OH)4]2+ + Al3(OH)8(OH2)6+ ⇔ [Al3(OH)8(OH2)5]4[Al(OH)4]3+ + H2O. |
The tetrahedral structure of the tridecamer could stabilise the octahedral trimer chain particularly well, offering plenty of space for chloride or nitrate ions to surround it. This would explain its long-term persistence reported in quasi-amorphous alumina.8 Longer chains than the trimer could be similarly held (and n could be variable); but if formed from the trimer by accretion of the monomer, the tetramer (and therefore pentamer) might be relatively slow to appear. A hydroxyl group in the cationic monomer would be expected to carry a lesser negative charge density than the corresponding group in the aluminate anion. Assuming p = 0, q ≥ 2(n − 1) for simplicity, no evidence exists for a simple chain of length n = 13; nor indeed for any linear instance of n > 6 (save three unlikely exceptions with n > 10).
Computation of m/z for the hexameric ring Al6(OH)12(OH2)126+ postulated on the basis of octahedral symmetry produces the value m/z = 97 (m = 582, z = 6). This value is indistinguishable from the value ascribed to the unipositive tetrahedral monomer, Al(OH)2(OH2)2+. However, the spectral lines at m/z = 154 and 199, ascribable to linear hexamers, are exceedingly weak. The ring could be more stable than the chain! But because of the ambiguous interpretation of m/z = 97, its existence is neither supported nor disproven. Indeed, the observed m/z values of 154 and 199 distinguish a hexameric chain from the corresponding ring.
To disprove the existence of the ring, it would become important to know whether only unipositive cations may be found experimentally (as the group reported in 2007 (ref. 7)); or whether other cations may also be found, besides (as reported in 2011 (ref. 8)).
Amongst the observed spectral values there is, besides m/z = 97, one other instance of ambiguous symmetry: m/z = 133. This could be ascribed to a hexamer comprising six tetrahedra; however, it seems far more likely that m/z = 133 should be ascribed to the octahedral monomer, Al(OH)2(OH2)4+, albeit dwarfed by the tetrahedral monomer (see above, m/z = 97).
To bring evidence for the special case (m = 582, z = 6), experimental variation might be required. For example, if an experiment could be devised where both m/z and m/zu were measurable in the same system and u was determined or pre-determined (u > 0, say), then it would be possible to evaluate m = (m/z)u/(u−1)(m/zu)−1/(u−1). For example, if u could be set equal to 2, then m would become calculable as (m/z)2/(m/z2).
Once such an experiment had become feasible it would then be worthwhile to study, besides AlCl3 solutions, Al2(SO4)3 solutions, where it seems that the tridecameric species do not form. Given that the SO42− ion may block the path to the tridecamer it is possible that, especially in the sulphate case, formation of a hexamer may occur preferentially (a line at m/z = 307 is just visible):
Al5(OH)12(OH2)103+ + 4H3O+ ⇔ Al5(OH)8(OH2)147+ + 4H2O, followed by |
Al5(OH)8(OH2)147+ + Al(OH)2(OH2)2+ + 6OH− ⇔ Al6(OH)16(OH2)102+ + 6H2O |
Al5(OH)8(OH2)147+ + Al(OH)2(OH2)2+ + 2 OH− ⇔ Al6(OH)12(OH2)126+ + 4H2O. |
If the latter process were to occur, relative stability of the hexamer might explain the non-appearance of larger polymers (with n > 6) in solution.
If existence of the cyclic hexamer were to be confirmed, a follow-up question would be whether its concentration could be determined; and if so, whether the ratio of this to that of the hexameric chain would be any greater than would be expected statistically.
Clearly the hydroxide ion plays an important role in the crystal nucleation process. Moreover the results presented above are not exhaustive in terms of symmetrically similar permutations of the ligands, O2−, OH− and OH2. In some instances, multiple combinations are possible. At m/z = 217, for example, the trimer could exist variously as any one or more of the tetrahedral species Al3(OH)8+, Al3O2(OH)4(OH2)2+ or Al3O4(OH2)4+. Usually the results presentation has been such as to show the species bearing the lowest charge z+ (where there is a choice) or the smallest variety of ligands, e.g. Al3(OH)8+ in this instance (having all hydroxyl groups); or to give a simple example of the possible reaction mechanisms.
Few of the group's observations7,8 remain to be explained. The value of m/z = 415 reported in their 2011 paper8 equates to m/z = 397 + 18, where m/z = 397 could be ascribed to a partially developed chain of tetrahedra, such as Al6O8(OH)(OH2)5+, that is still reactive.
Curiously, the regular interval of 60 in the series m/z = 97 + 60(n − 1), where n = 1, 2… 6 and z is constant throughout (z = 1), equates numerically to the historical stoichiometry Al(OH)3 minus a single water molecule! 27 + (17 × 3) − 18 = 60. Coincidentally, this matches the formula for the pseudoBoehmitic moiety, AlO(OH). (This reaffirms the merit that lies in computing the values of m/z that might arise from octahedra in a doubleSheet.)
By removing the cube as a classical “building block” and substituting instead either or both of two golden rhombohedra, any of a variety of possible structures might conceivably be built, including:
* a triclinic system,
* the Biliński rhombic dodecahedron, which is a 3-d space-filler (and of which the Keggin tridecamer is a special case),
* 3-d tiling by Biliński, which would be crystalline (and may never have been seen physicochemically),
* a non-convex stellar quasi-crystal, or
* “Penrose in 3-d”.
So far, however, it is believed that, beyond the tetrahedral and octahedral species, no simple polyhedral aluminal species has been observed in solution; neither classical (no “football”), nor Penrose tile nor Ogawa flower.
The next question is whether, given a suitable choice of salt solution, it would also be feasible to study for a given concentration the effect of a chemical reagent, such as an alkali, and possibly as a function of time. If in Akihiro Wakisaka's experiment6 we were to choose AlCl3 instead of KCl, an issue might be whether the mass spectrum would be sensitive enough to identify, and distinguish between, clusters such as the ring Al6(OH)12(OH2)126+ or chains of the kind Aln(OH)2(n−1)(OH2)2(n+2)(n+2)+. The effect of introducing the smallest aliquots of KOH into the solution might then be studied (and the K+(KCl)x contribution perhaps subtracted out in some way). Over time, it might be anticipated that small n values would be superseded by large ones, for example; or possibly even results uniquely ascribable to unambiguous 2hn values.
Crystal nucleation begins even in the acidic solution, prior to introduction of any alkali, through accretion of the monomeric species (though not beyond the hexamer). In the dimer, ESI quadrupole mass spectrometry distinguishes tetrahedral from octahedral symmetry. But the aluminate anion Al(OH)4− can, where present, arrest the aggregation process by removing chains of octahedra as small as the trimer and stabilising them with the tetrahedral form of the tridecamer, [Al3(OH)8(OH2)5+]4[Al(OH)4−]3+.
Of its nature, mass spectrometry cannot of course elucidate the geometry of the trimer or tetramer (irrespective of whether these comprise tetrahedra or octahedra); m/z is independent of any directional angle, θ, π/6, cot−1sqrt(2) or cot−1τ. Yet the onset of this geometry may remain crucial in determining the ultimate crystal shape. Again, an additional experiment is required, possibly involving 27Al NMR in solution; or, given the prospect of isotropy in the closed ring (the non-linear hexamer) but anisotropy in its trimeric (or tetrameric) precursor, perhaps some form of Raman spectroscopy.
Footnotes |
† Electronic supplementary information (ESI) available: Computational results may be found at www.alanhare.me.uk/RSC/Aluminal_Speciation-Mass_Spectral_Interpretation(symmetrical,non-polyhedral).ods or .xlsx (copy-paste full url into browser, if necessary). See DOI: 10.1039/c6ra11209a |
‡ MRSC, without affiliation (retired) |
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