Aluminal speciation in the crystal nucleus: a mass spectral interpretation

Alan Stewart Hare
3, Marshbarns, Bishop's Stortford, UK. E-mail: alan@alanhare.me.uk; Tel: +44 (0)1279 465527

Received 30th April 2016 , Accepted 22nd August 2016

First published on 24th August 2016


Abstract

Mass spectral line positions m/z are computed from a general formula for aluminium cations, AlnOp(OH)q(OH2)r(3n−2pq)+, 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.


1. General significance

Aluminal species exist naturally in soil, river water, ice melt, plants and food. For many years they have been of interest to soil science,1 especially in regions which have large human populations to feed. Soil acidification, which accompanies the aqueous dissolution of Al3+, may restrict plant growth and limit crop production.2 Such species have no known physiological function, yet because of their ubiquity we ingest them uncritically. Despite their toxicity3 it was thought that only at abnormally high concentrations might the human suffer cellular damage in consequence. Recent findings however suggest that chronic exposure4 may bring an increased risk of the brain disease, Alzheimer's. In soil science hexameric ring species were first postulated over 50 years ago,5 though not confirmed incontrovertibly. Recent experimental advances6 now enable us to re-examine this and other possibilities.

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 − 2pq 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).

Table 1 Calculation of m/z values from model Al3+ structures
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 3nq = 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 − 2pq 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 − 2pLqL 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 − 2pq + 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
DoubleSheeta 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


Table 2 Calculation of Al3+ m/z values in non-prismatic polyhedra
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 + Vχ (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( + )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)    


2. Special cases

2.1 Ring species

2.1.1 General. In crystal growth, the hexagonal shape is found world-wide. The ice crystal is an iconic example … though the general shape is far from unique. In an aluminal structure where the Al3+ cation may be surrounded by six ligands arranged octahedrally, it seems highly plausible that the ultimate crystal system in bulk might be determined as early as the event of the 3rd cation attaching itself to a dimer in solution [whether in a straight line, or at a 60° angle].

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.

2.1.2 Octahedral symmetry. It may be assumed that perfect octahedral symmetry Oh is to be expected only in the special case of the monomer where the Al3+ cation is surrounded by six ligands indistinguishable one from another, e.g. Al(OH2)63+ (s = 6, n = 1, p = q = 0, r = ns = 6).
2.1.3 Hexagonal ring. It may be further assumed that a perfect hexagonal ring is to be expected only in the special case of the hexamer comprising six imperfect but near-perfect octahedra, i.e., Al6(OH)12(OH2)126+ (s = 6, n = 6, p = 0, q = 2n, r = ns − 2q = 12).

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, Rdgeo 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).

2.1.4 Question of existence. Evidence is now sought for existence of the precise hexamer Al6(OH)12(OH2)126+ (as distinct from other hexameric species such as chains).
2.1.5 Spectral value expected. In a mass spectrum containing the hexagonal ring, z = 3n − 2n = 6 and m = 27n + 16(2n + ns − 4n) + 2n + 2(ns − 4n) = n(18s − 11) = (6 × 97) = 582. m/z = 97.
2.1.6 Ambiguity. Unique spectral interpretation may fail: other aluminal species might exist with m/z = 97 (see below). If so, an experimental variation may be required, or even a novel experiment.

2.2 Hexagonal species

2.2.1 Symmetry. Wherever the species is hexagonal, octahedral symmetry (s = 6) is assumed.
2.2.2 A nuclear hexagon?. The simplest aluminal hexagon that might exist is the hexagonal ring described above. If the hexameric species Al6(OH)12(OH2)126+ does exist, it could potentially act as a crystal nucleus. A crystal nucleated by this species might itself belong a hexagonal class (such as a hexagonal prism), although it need not. The crystal could be an aluminal species; though again, it need not be. Given that ice crystals, for example, also exhibit hexagonal shapes, it is not at all inconceivable that octahedral Al3+ species now found in clay9 might have played some earlier role in nucleating ice in a glacier; or that alumina in mud once nucleated ice in a comet.10
2.2.3 Tessellated hexagon. It is possible to describe a serrated hexagon which is itself tiled by regular hexagons. The hexagonal tiles are primitive, congruent with one another and may be arranged in parallel columns, sharing edges. The line drawn through the centre of each of the tiles furthest from the most central tile forms the edge of the largest outer hexagon. In this model, each vertex of each hexagonal tile would provide the site of an Al3+ cation, with cations shared between adjacent columns. See Fig. 1.
image file: c6ra11209a-f1.tif
Fig. 1 2-d tiling by hexagons. The number of “annular rings” x is int(ξmax/2), the integer part of half the number of regular hexagons in the longest chain. Though approximately hexagonal, the tiled area is not itself a regular hexagon, but serrated. Nuclear expansion in 2-d may be represented by the equation image file: c6ra11209a-t19.tif. One aluminal ring structure thought to exist, for example, takes the formula Al54(OH)144(OH2)3618+ (x = 2); for which m/z = 253. Nucleation of the primary hexagonal species Al6(OH)12(OH2)126+ (x = 0) from a dimer such as Al2(OH)4(OH2)62+ would appear first to require a non-linear trimer to form. This could be an essential precursor of, in 3-d, a hexagonal prismatic crystal or honeycomb. Evidence is required of the polymer geometry. The serrated hexagon may also be regarded, again approximately, as a symmetrically bi-truncated rhombus (also serrated). This particular rhombus is not however a golden rhombus.

2.2.3.1 Number of hexagonal tiles. If the number of hexagons in the highest column is an odd finite number, then the total number of hexagonal tiles, or extent of 2-dimensional growth G(x), is:
image file: c6ra11209a-t1.tif
where for regular hexagonal shape x = int(ξ/2), the integer part of half the column-height, and where ξ > 2 (see below, Section 2.2.8). This is one instance of a symmetrically bi-truncated diamond shape or serrated rhombus X.X, where X = 1 + 2x. Because the sum of the first n natural numbers image file: c6ra11209a-t2.tif. A non-trivial example is x = 2, giving G(x) = 19.

2.2.3.2 Number of vertices. In this tessellated hexagon, the number n of Al3+ cations is given by image file: c6ra11209a-t3.tif, or 6(1 + x)2. For x = 0, 1, 2…, n = 6, 24, 54….
2.2.3.3 Number of sides. Each side of each hexagonal tile is assumed to be occupied by two hydroxyl groups, with adjacent hexagons sharing two groups between them. None of the q hydroxyl groups is located elsewhere. So the total number of sides is image file: c6ra11209a-t4.tif. Or, the total number of OH ligands q = 6(2 + 5x + 3x2). Since no O2− ligands need be postulated in the near 2-dimensional wafer (as distinct from in the 3-dimensional prism), p = 0.
2.2.3.4 Number of corners. There are image file: c6ra11209a-t5.tif corners, each occupied by two H2O ligands: r = 12(1 + x).
2.2.3.5 Crystal nucleation and growth?. Given that nucleation and growth mechanisms for hexagonal crystal formation are of interest, it follows that besides seeking evidence for the nuclear hexagon Al6(OH)12(OH2)126+ (x = 0), we should also seek a tessellated hexagon or hexille, such as Al24(OH)60(OH2)2412+ (x = 1) or Al54(OH)144(OH2)3618+ (x = 2); not necessarily ruling out Al96(OH)264(OH2)4824+ (x = 3) or larger species. In this model of near 2-dimensional crystalline growth, or laminar growth, the edge-length of the crystal hexagon (x > 0) is 2Rxsqrt[(1 + cos(π/3)2) + sin(π/3)2], or 2Rxsqrt[2(1 + cos(π/3)]; that is, greater than that of the nuclear hexagon 2R by a factor of xsqrt(3). In a growth pattern such as this where hexagonal species might be expected to be relatively stable, non-hexagonal intermediate species could form by adding tiles from G(x) to G(x + 1), G(x) recurring, but these would be transient.
2.2.3.6 Spectral values. See Table 1. Any of these three values may be interpreted unambiguously (unlike the value of m/z = 97, which may or may not derive from the hexagon, G(0) = 1).
2.2.4 The serrated rhombus. The serrated rhombus is tessellated by image file: c6ra11209a-t6.tif hexagonal tiles. It is also a quadrille, i.e., tessellated by G(x − 1) diamond-shaped tiles or rhombi. For the serrated rhombus, unlike the serrated hexagon (where X odd > 2 is necessary), X > 1 is sufficient and may be odd or even. G(x) = X2. It is natural to ask whether its diagonals exhibit the “golden ratio” τ, which has the property τ = (τ + 1)/τ. For this condition, its half-angle θ would have to equal cot−1[thin space (1/6-em)]τ, or tan−1(2/(1 + sqrt(5))). But in the hexagonal tessellation or hexille, the array of regular hexagons is such that θ = π/6, or tan−1(1/sqrt(3)); which is less than golden. It seems that the rhombus is tarnished!
2.2.5 Lower-symmetry hexagons?. By replacing each sphere by the more general ellipsoid, however, the hexagon could be elongated (as on the face of Fyodorov's elongated dodecahedron,11 perhaps). It might also become narrower; and possibly thinner: ellipsoidal packing may exceed the Kepler maximum (see below). Elongation, by a factor of λ, would increase the half-angle θ above its π/6 value, and could be such as to produce the golden rhombus.

As a 1st approximation, the linear ratio would be λ = tan(cot−1[thin space (1/6-em)]τ)/tan(π/6) = 2sqrt(3)/(1 + sqrt(5)). Despite its lower symmetry, the hexagon may have regained something of its lustre!

2.2.6 The golden ratio. While growth in the direction of cot−1[thin space (1/6-em)]τ may give rise to the “golden” rhombohedron, with all six faces congruent, this is not the sole manifestation of the golden ratio in crystal systems: τ is also one of the Cartesian coordinates defining each of the 12 vertices of an icosahedron (see below, Sections 2.2.7 and 2.6).
2.2.7 The regular icosahedron. With Penrose, it has been recognised that – contrast the hexille – rhombic tilings may be aperiodic: without translational repetition. Also, the classical Bravais lattice is transcended by a hypercubic lattice: in N-dimensional hyperspace, the 4-, 5-, 6-cube et seq are analogous with the cube (or 3-cube). Specifically, a regular icosahedral quasi-crystal12 may be represented visually as the 3-d projection of a crystal from a 6-cubic or hexeractic lattice. Instead of rhombic tiles (in 2-d), in a 3-d tiling the tiles may be rhombohedra.

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.

2.2.8 Chain of hexagons. A chain of hexagons may be regarded as being equivalent to a single row of ξ adjacent hexagons, although its specific formula does not imply a straight line (or indeed any particular conformation): Al2(2ξ+1)(OH)2(5ξ+1)(OH2)4(ξ+2)2(ξ+2)+. Potentially, any small species of this kind could play a role in the nucleation and growth of a crystal. Given that the spectral value m/z = 97 is ambiguous, in that this value is a set containing other aluminal species besides the single hexagon (ξ = 1), complementary evidence may be sought for species with ξ > 1. See Table 1. Again, any of these values could be unambiguously interpreted.

2.3 Chain species

2.3.1 General chain formulation.
2.3.1.1 All cations. In the case of a chain of polyhedra, each of the n Al3+ cations is again assumed to share ligands. Depending on its position in the chain, an Al3+ cation will share either two or four of its s ligands. At either end of the chain, the terminal Al3+ cations will each share only two of their ligands, with the next cation along. In a trimer or chain of greater length, each interior cation shares four. (In a dimer, only the two are shared; in a monomer, none.) This means that the total number of ligands p + q + r = s + (n − 1)(s − 2). In this case, r is not independently variable, but is determined by the chain constrained by the symmetry: r = (s − 2)n + 2 − (p + q).
2.3.1.2 Unipositive cations (z = 1). In this special case, 3n − 2pq = 1. In this case, q is not independently variable, and is determined by the chain limitations: q = 3n − 2p − 1. However, it seems unnecessary to restrict our scope in this way.
2.3.2 Monomer. The monomer is the special case of a general chain where n = 1. This means that the number of ligands p + q + r = s. r = s − (p + q). Given that p = 0, r = sq.

2.4 Keggin cage species

In the special case of the Keggin “cage”, it is the tetrahedral aluminate anion Al(OH)4 (again Td) that is believed to be central, rather than an Al3+ cation (except in as much as Al3+ is central to the Al(OH)4 anion itself). It is surrounded by chains. A general Keggin formulation therefore may now be made as the positive species [AlnOp(OH)q(OH2)rc+]4[AlOy(OH)4−y(y+1)−]z+, where z = 4c − (y + 1) and where in the cationic chain cL = 3nL − 2pLqL. In general, each nL value could be different (L = 1, 2…4). Typically, nL = 3 is assumed with octahedral symmetry (sL = 6) throughout. However, the Biliński case (n = 3 + δ, p = 0, q = 10ε) is also of interest (see below, Section 2.6.6 Mass spectra – Kepler–Biliński case.).

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).


image file: c6ra11209a-f2.tif
Fig. 2 3-d space-filling by Biliński rhombic dodecahedra. Biliński space-filling may be represented by the equation image file: c6ra11209a-t20.tif. Its relationship with the 2-d tiling equation cannot go unnoted (see Fig. 1). In m/z for the 3-d case, both {m, z} are linear in G(x); and, as with classical cubic space-filling, cubic in x. Sadly, however, it now seems unlikely that – save in mathematics – this beautiful crystal really exists.

image file: c6ra11209a-f3.tif
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.

2.5 Prismatic species

2.5.1 General prism formulation. The prevalence in recent literature of references to implicit chain structures and the Keggin “cage” does not preclude the existence of other aluminal species. Besides the ring and the chain, the 2-dimensional sheet (where in the cation hn may be formulated, instead of simply n) and doubleSheet (2hn, as in a pseudoBoehmitic structure) are entirely possible. Moreover 3-dimensional systems may also be modelled (jhn), such as those with extended hexagonal rings, sheets and prisms.
2.5.1.1 Polygonal prism. Given an assembly of identical polygons each of size G(x), the extent of 3-dimensional growth G(x, y), is:
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 − (qαv). 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 − (qαv); vq; 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 − 2pq + 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).


2.5.1.2 Hexagonal prism. For the hexagonal layer (above), the number of O2− ligands within any one layer was taken to be zero: p = 0. n, q and r were computed above as functions of the row-length x. For the purpose of this computation h was defined as an explicit function of the row-length, h(x).z = 3njqjαvj − 2vj + 2v = (3nq)j −3v(j − 1).

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.

2.5.2 Sheet of octahedra. Hexagons may be unnecessary. If so, the model sheet comprises instead 2h + 1 rows of either x or x − 1 octahedra with row-length alternating x, x − 1, x… (x ≥ 3), finite integer h ≥ 1. Two edge rows exist, each comprising x octahedra. An edge would form the chain, Al(OH)2(OH2)4[Al(OH)2(OH2)2]x−2Al(OH2)4(x+2)+, except that this now shares x − 1 hydroxyl groups with the adjacent row, giving the reduced formula Al(OH)(OH2)4[Al(OH)(OH2)2]x−2Al(OH2)4(2x+1)+. The shorter, adjacent row comprises x − 1 octahedra (in which the hydroxyl groups shared between two rows have now become O2− ligands); there are h rows of this shorter kind:
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)+.

2.5.3 Double sheet. The doubleSheet is orthorhombic: a special case of the polygonal prism where the polygon is a rectangle and j = 2 (and therefore α = 1/2). Given that in the model one-half of the q hydroxyl groups are shared between the two sheets, then q/2 = αv, q = v and the reduced prismatic formula is [Aln(x)Op(x,h)(OH)q(x,h)/2(OH2)r(x,h)(3n−2p−(q/2))+]2[Oq/2q]z+, q even. In the doubleSheet, q/2 = h(2x − 3), 2p = 4h(x − 1) and so the special-case formula becomes [Al2[x(2h+1)−h]Oh(6x−7)(OH)2h(2x−3)(OH2)8(1+2h+x)]2(3x+7h−2hx)+. See Table 1.
2.5.4 Extended double sheet. Now if αvj hydroxyl groups are shared between adjacent doubleSheets in a prism of length j ≥ 2 (α = 1/2, j even), each group becoming an O2− ligand, the orthorhombic formula becomes [Al2[x(2h+1)−h]Oh(6x−7)(OH)2h(2x−3)−v[1−(1/j)](OH2)8(1+2h+x)c+]j[O(j−1)v2v(j−1)−]z+, where c = 6x + 14h − 4hx + v[1 − (1/j)]. Within the shorter rows of the sheet, 2h(x − 3) hydroxyl groups were available for sharing; but in the doubleSheet, half of these have already been shared. If the remaining half now comprise all the O2− ligands shared with the adjacent doubleSheet, then v = h(x − 3) and the extended DoubleSheet formula becomes [Al2[x(2h+1)−h]Oh(6x−7)(OH)3hx−3h+(h/j)(x−3)(OH2)8(1+2h+x)c+]j[Oh(j−1)(x−3)2h(j−1)(x−3)−]z+, where c = 6x + 11h − 3hx − (h/j)(x − 3). Again, see Table 1.

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.

2.6 Other polyhedral species

2.6.1 Polyhedral formulation. Given that polygonal species may exist, such as the hexagon, it is natural to enquire whether a number of them might come together as faces of a polyhedron. Given that an Al3+ ion could be situated at each vertex, with two or fewer OH ligands on each edge and none elsewhere, the polyhedral formula would become AlV(OH)2εE(OH2)(s−(4εE/V))V(3V−2εE)+ (0 < ε ≤ 1), where for a cationic aluminal species 3V > 2εE (in which V and E are the numbers of vertices and edges, respectively). In this formulation, ε cannot be empirical: 2εE must be a positive integer.
2.6.2 Platonic solids. Were the primitive icosahedron Al12(OH)24(OH2)2412+ to exist (s = 6, V = 12, E = 30, ε = 2/5), it would carry a positive charge: m/z = 97, again (further ambiguity); whereas, by contrast, Al8(OH)24 (s = 6, V = 8, E = 12, ε = 1) would be charge-neutral whatever the crystal system (cubic, golden rhombohedral or otherwise).
2.6.3 Archimedean solids. By Euler, E = F + Vχ (where the polyhedron has F polygonal faces). So, for a convex cation with two uni-negative ligands per edge (χ = 2, ε = 1), V > 2(F − 2). Archimedean solid geometry is a natural candidate; e.g., were Goldberg alumina or “buckminstalumina” to exist, with 60 Al3+ ions (V = 60, F = 32) arranged as a “soccer ball” or truncated icosahedron Al60(OH)180, it would be charge-neutral … although one OH group fewer (one more H2O) would be sufficient to produce a value of m/z = m = 4663 + 18. Yet there is no reason to suppose that such species exist. Similarly, V = 12, 24, 48, 60, 120 could be paired with F = 8, 14, 26, 32, 62, respectively. (Invariably 4E/V = 6.) This would account for seven of the thirteen convex polyhedral arrangements, all neutral. Naturally, the charge-neutrality of a species does not preclude its existence. Neglecting water molecules, each is a multiple of Al(OH)3!

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.

2.6.4 Nucleation and growth in 3-d. Crystal growth in 3-d, if it is to produce space-filling convex polyhedra, could be taken to involve the sharing of a face between two polyhedra, P0 and P1 (and not merely an edge). Paradoxically then, although the crystal is growing, the overall number of external faces F0 + F1 decreases by 2. Since, following Penrose, one or two species of polyhedron P1 or P1 and P2 may be adding to the crystal, a more general pair of growth equations in x may be written:
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,
where 1 + c1 + c2 is the number of polyhedra sharing faces (in which c1 ≥ 0, c2 ≥ 0; and either c1 [and h1] or c2 [and h2] is zero, though not simultaneously both c1 and c2), and h = h1 + h2 is the increase in the number of shared faces (h ≥ 2). F(0) = 0, G(0) = 0, V(0) = 0. The total number of vertices also diminishes: V(x + h) < V(x) + V1 or V(x) + V2. A more general expression in xi could be written, but may be unnecessary. The paradox is explained if we accept that the system may attain a critical FV(x) state where only 1/(1 + c1 + c2) polyhedra continue to grow, at the expense of both c1/(1 + c1 + c2) and c2/(1 + c1 + c2), each of which is diminishing. If the two additive species of polyhedron are identical (FV1 = FV2 and V1 = V2), then FV(x+h)(x + h) + h = FV(x)(x) + c1FV1(x1 + 1) − 2h. An instance could be the nucleation of a 3-d tiling at x = 0, c1 = h and its growth to x = 4: F14(4) + 4 = 4F8(1) − 8.

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−1[thin space (1/6-em)]τ). 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).

2.6.5 3-d tiling?. It would be interesting to know, (a), whether, and under what conditions, potential tiling species can be identified at the nucleation stage; (b), if so, whether one, two or more distinct tiles can be distinguished from each other in the mass spectrum; and (c), whether 3-d tiling occurs; and, if it does, then whether this is Penrose or non-Penrose.

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.

2.6.6 Mass spectra.
2.6.6.1 General. Given that an Al3+ ion could be situated at each vertex and each closed vertex, with edges internalised there is also the possibility of two or fewer O2− ligands on each internal edge; and, with none elsewhere, the polyhedral formula would become AlC+VOαD(OH)2βD+2εE(OH2)sV−4εE−2(α+β)D[3C+3V−2(α+β)D−2εE]+ (with α + β ≤ 1, 0 < α, β, ε ≤ 1), where C(x) + V(x) is the total number of vertices, in which C(x) are closed points and V(x) completely open points or corners; and where the number of internal edges D(x) depends on the number of internal faces G(x). Again, neither α nor β nor ε may be empirical: any one of them must be either zero or a positive integer, or rational fractions such that αD, 2βD and 2εE are each either zero or a positive integer.
2.6.6.2 Kepler–Biliński case. For either of the two rhombic dodecahedra F14(4) = 12, Kepler or Biliński, the specific formula becomes AlO4α(OH)8βAl14(OH)48ε(OH2)14s−96ε−8(α+β)[45−8(α+β+6ε)]+. See Table 3.

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.


2.6.6.3 Fyodorov case. For the rhombic icosahedron F22(5 + 5) = 20 attributed to Fyodorov,11 were it to exist, the specific formula would in all probability become AlO4α(OH)8βAl10O11α(OH)22β+20ε[[Al(OH)4ε]3]5(OH2)22s−160ε−30(α+β)z+ or Al26O15α(OH)30β+80ε(OH2)22s−160ε−30(α+β)[78−10(3α+3β+8ε)]+. Again, see Table 3. Importantly, the Fyodorov rhombic icosahedron FρI(5 + 5) is also subject to Penrose enlargement; not as an independent species, but as the ρI part of a structural pattern characteristic of aperiodic 3-d tiling (see below, Section 2.6.8 Penrose in 3-d).
2.6.7 Polyhedral growth.
2.6.7.1 Stellar nucleus. For a stellar polyhedral crystal to be nucleated (with any χ) it appears sufficient for the additive species to add to faces of the nuclear species, e.g., FV(28)(28) + 12 = F14(4) + 6F14(4) − 24. Growth might continue in this mode unless perhaps or until it were ever to become disallowed volumetrically (reminiscent of the term, “steric hindrance”!). A general formulation would comprise an indeterminate number of indefinite series, possibly complicated by branching. However, the special case of Ogawa's flower, which maintains a definite shape while still growing, is of particular interest (see below, Section 2.6.8 The growing flower).
2.6.7.2 Convex nucleus. To nucleate a convex polyhedron, however (χ = 2), the original edges must also be closed off, besides the faces. For example, if in addition to the faces all edges of the FV(4) species were to be enclosed, then nucleation could initiate a Haüy growth pattern13 of the form xd + ρFVi2 (where 0 ≤ ρ ≤1, and ρFV is either zero or a positive integer), such as FV(H(x))(H(x)) + 2h(x) = FV(4) + h(x)F14(4) − 4h(x), in which image file: c6ra11209a-t9.tif. In this example, FV=14(4) = 12, ρ = 1/6, ρFV = 2, d = 2 and H(x) = 4(h(x) + 1).

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 image file: c6ra11209a-t10.tif, 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.


2.6.7.3 Classical cubic model. If a species F8(1) were to nucleate a rhombohedral species FV(x3)=8(x3) = 6 with no facial growth (ρ = 0) and all six faces congruent – such as a cube, or golden rhombohedron – this would have (x + 1)3 cations, one at each of the 23 vertices and the (x − 1)3 closed vertices. Of the remaining 6(x2 − 1) cations, 12(x − 1) would be situated on the edges of the “cube” and 6(x − 1)2 on its faces. Each corner unit of the V(x3) corners would present 3 open edges; and every other edge unit, 1: a total of 3V + (F + Vχ)(x − 2) or 12x open edges. On each face, the x2 facial units would each present one open face: 2FVx(x − 1) additional “facial edges” overall. In total, then, the FV(x3)(x3) species would present 24 + 12(x − 2) + 12x(x − 1) or 12x2 open edges. All other edges would have been internalised.

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, image file: c6ra11209a-t11.tif, 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.

Table 3 Calculation of Al3+ m/z values for 3-d space-fillers
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: image file: c6ra11209a-t7.tif
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 image file: c6ra11209a-t8.tif 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)      



2.6.7.4 The golden rhombohedron. The two distinct golden rhombohedra F8(1), prolate and oblate (sometimes called “acute” and “obtuse”, as if they were 2-dimensional), form one important special case (x = 1): Al8(OH)24ε(OH2)8(s−6ε)24(1−ε)+. Charge-neutrality would arise if ε = 1: [Al(OH)3]8. Again, see Table 3.
2.6.7.5 Biliński tiling model. If in this model of nucleation, the rhombic dodecahedron F14(4) is now substituted for the primitive cube or rhombohedron F8(1), the model may support 3-d tiling of a Penrose or non-Penrose kind. The F14(4) species is not itself primitive; it has G14(4) or 4 internal edges already: Dinternal = 4. If the Biliński species F14(2 + 2) were to nucleate a crystal of the kind FV(H(x))(H(x)), then the latter would again have cations from corner, edge, face and interior units. Additionally, because x = 4 initially, each unit would have already come with its own single internalised cation: Cinternal = 1. (See above, Fig. 2.)

The number of Biliński components image file: c6ra11209a-t12.tif, 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,

image file: c6ra11209a-t13.tif

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+,
where z = 4[3 − 2(2α + β + 2ε)]H(x − 1) + 24(4x 2 + 3x + 2) − 16(2α + β)(8x2 + 1) − 16ε(6x2 + 9x + 2). Or expanding again, in this instance:
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 + 2βε)

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 image file: c6ra11209a-t14.tif.


2.6.7.6 Non-Penrose nucleation. Despite the possible existence of a Biliński dodecahedral species comprising two unlike pairs of rhombohedra, a 3-d tiling of Haüy construction – such as the above examples – might always be regarded as non-Penrose, by dint of the degree of repetition under translation (albeit limited repetition). Either the cubic or Biliński nucleus would sit well on a classical lattice.
2.6.8 Penrose nucleation.
2.6.8.1 Penrose in 3-d. If evidence were to be found either of a rhombohedron (of either kind) or of a rhombic icosahedron, but none for a rhombic dodecahedron (after Haüy, Biliński above), then we might reasonably seek a distinct Penrose figure. Ogawa defines a 3-d Penrose transformation as a matrix operation16 on a set of space-filling polyhedra such that the enlarged Penrose figure comprises only polyhedra similar to those in the unenlarged set. The enlargement operation relies on a “self-similarity” factor. In formulating this definition Ogawa discovers a non-convex structure, again pentagon-based, with icosahedral symmetry: his “5-petalous floral dodecahedron”. (Temporarily I continue with what seems, in the absence of a sine curve, a misnomer; the structure is probably neither quinquepetalous nor dodecahedral but rather a stellated rhombic triacontahedron, a Great Stella stellation; or in our simple notation, F62(20 + 0) = 5 × 12, or 60. Mathematically the triacontahedron itself, F32 above, was discovered by Kepler in 1611. Its hexecontahedral stellation F62 = 60 seems to have lain undiscovered until after 1940.)
2.6.8.2 Fibonacci growth. The space-filling polyhedra are taken to be either or both of the two golden rhombohedra F8(1). In the F62(20 + 0) example, all 20 are of the prolate kind; none is oblate. In the Penrose enlargement, both kinds of rhombohedron contribute: the prolate and the oblate. Ogawa discovered an enlarged oblate having two F62(20 + 0) “flowers”, one at each of the two vertices of its principal diagonal, the two flowers sharing an enlarged prolate; each of the two larger rhombohedra comprising multiple smaller ones. The Penrose-nucleated crystal would grow in accordance with a pair of simultaneous equations
|GV1(x + 1)〉 = Φn+1|GV1(x)〉 + Φn|GV2(x)〉
and
|GV2(x + 1)〉 = Φn|GV1(x)〉 + Φn−1|GV2(x)〉,
where {Φk}k=1 is the Fibonacci number sequence, Φk+2 = Φk + Φk+1 (Φ1 = Φ2 = 1; here denoted Φ, as distinct from F in Euler's equation). Under the transform, n is invariant. The matrix is a 2 × 2 matrix. The two eigenstates |GV1(1)〉 and |GV2(1)〉 are both F8(1): the prolate and oblate case, respectively (x = 1). In this instance, Ogawa invokes the special case Φn−1 = 21 and Φn = 34 (n = 9). The self-similarity is the golden rhombohedral edge length. See Fig. 4(a). More generally, whilst in Φn the value of n itself appears to depend on Fibonacci numbers Φr, in these Φr the value of r is not always determined by Φr and recently the series of atom cluster sizes {Φ(Φr)}r=36 has been recognised in Coelfen's silver nanoparticle precursors.17 (The requirement for the solution to comprise integer numbers obviates alternative formulations such as those based on exp((r − 2)/2) or its ceiling.)

image file: c6ra11209a-f4.tif
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[thin space (1/6-em)]ln[thin space (1/6-em)]τ/ln[thin space (1/6-em)]2. In fractal geometry this suggests asymmetric Cantor dust in 3-d, of capacity dimension κcapacity = 3 (d is approximately 6.25). Because Φn+1Φn−1Φn2 = (−1)n, it follows that |GVj(x + 3)〉 = (Φn+1 + Φn−1)|GVj(x + 2)〉 + |GVj(x + 1)〉 (n odd). Golden rhombohedra cement two floral Ogawa species ϕ+, ϕ and three rhombic icosahedra ρI into a regular pentagonal shape. In the primitive state x = 1, this would potentially comprise an aluminal ϕ+ species Al187O12+45α(OH)30(4+3β)+482ε(OH2)30+100s−9[(α+β)+8ε]z+ where z = 417 − [90(α + β) + 482ε], paired with the ϕ species Al149O12+45α(OH)30(4+3β)+268ε(OH2)30+70s−18[5(α+β)+28ε]z+ in which z = 303 − [90(α + β) + 268ε]. (b) Penrose in 3-d: aperiodicity. Ogawa's flower sits at either end of the principal diagonal of a major golden rhombohedron. The “floral” structure is here represented by a non-convex, stellar polyhedron, comprising twenty minor golden rhombohedra. (Fig. 5 appends a more precise description.) Along this diagonal, the two stellar polyhedra share one minor rhombohedron. Others – r0, unshared – are positioned between the two. The stellations |G62,ϕ+(x)〉 centre on the two major principal vertices: the diagonal joining these two centres is now taken to be the chord of a regular pentagon. On each of the pentagon's three remaining vertices sits a Fyodorov rhombic icosahedron, with the centre of its constituent Biliński dodecahedron positioned at the local vertex. This structure allows for five nuclei, each one equidistant from its two nearest neighbours. (Both these nuclear subtypes derive from Kepler's rhombic triacontahedron.) As described in Fig. 3, each Fyodorov icosahedron |GρI(x)〉 comprises the four Biliński and six extra rhombohedra of its own, all golden. Each Fyodorov is now orientated so as to allow a further r1 + r2 rhombohedra to be added (including three beyond the local vertex); with the remainder cementing the five gaps between nearest neighbour nuclei. Consistency with Ogawa requires |GV1(x + 1)〉 equal to |G62,ϕ+(x)〉 + |GV1(x)〉 + r0|GV2(x)〉 + 3[|GρI(x)〉 + r1|GV1(x)〉 + r2|GV2(x)〉] + 5[((19 − 3r1)/5)|GV1(x)〉 + ((19 − r0 − 3r2)/5)|GV2(x)〉]; with, for example, r0 = 5 and r1 = r2 = 3: a special case of the equation |GVi(x + 1)〉 = Φn+1|GV1(x)〉 + Φn|GV2(x)〉.

image file: c6ra11209a-f5.tif
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

image file: c6ra11209a-t15.tif
or as the polynomial
image file: c6ra11209a-t16.tif
where each coefficient Cijn = cijΦn−1i (C = {A, B}). For example, in Ogawa's special case, Φn = 34 and
image file: c6ra11209a-t17.tif


2.6.8.3 Mass spectrum. The formula of each of the smallest rhombohedra would be M8P0Q24εR8(s−6ε)z+, z = 8(c − 3), where for instance Mc+ = Al3+, Pa = O2−, Qb = OH, R = OH2: Al8(OH)24ε(OH2)8(s−6ε)24(1−ε)+. Note that α = 0, initially; however, see below. In 3-d Penrose relations as written above there would be nothing to imply face- or edge-sharing were it not for the supposed quinquepetalousness (see above, Penrose in 3-d; and again below, The computation). Given its essential position in the structure, Ogawa's special-case equations with Φn = 34 are recast as
|GV1(x + 1)〉 = |G62,ϕ+(x)〉 + 20|GV1(x)〉 + 19|GV2(x)〉 + 3|GρI(x)〉
and
|GV2(x + 1)〉 = |G62,ϕ−(x)〉 + 6|GV2(x)〉 + 3|GρI(x)〉,
where the rhombic icosahedron FρI(5 + 5) above is explicitly accommodated in the ρI enlargement (see below).

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.

Table 4 Calculation of m/z for “Penrose in 3-d”
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) + νnB1*(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) + νqB1*(x), where: νq+ = 30(4 + 3β) + 482ε; νq = νq+ − 214ε sV − 4(βD + εE) = νr+A*(x) + νrB1*(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) + νzB1*(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) + νnC*(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) + νqC*(x) sV − 4(βD + εE) = νr+B2*(x) + νrC*(x) 4 or 6 27n + 16p + 17q + 18r νz+B2*(x) + νzC*(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.


2.6.8.4 The growing flower. The basic formula of Ogawa's flower F62,ϕ+(20 + 0)(x = 1) is MP2[M2P2αQ4βR]5[MQ5]2[[M2Q4R]5]5z+. If this were to exist as an aluminal species, for example, its basic formula would be AlO2[Al2O2α(OH)4βOH2]5[Al(OH)5]2[[Al2(OH)4OH2]5]5z+ or Al63O2+10α(OH)10(11+2β)(OH2)30[75−20(α+β)]+. See Table 5. Correspondingly, F62,ϕ−(19 + 0) might be formulated by subtracting from F62,ϕ+(20 + 0) a single (prolate) rhombohedron, F8(1), or M8Q24εR8(s−6ε) or Al8(OH)24ε(OH2)8(s−6ε). However, if as expected there is vertex-, edge- or face-sharing by the rhombohedron, the subtrahend will be correspondingly smaller. For example, if three faces that in F62,ϕ+(20 + 0) were shared are no longer shared in F62,ϕ−(19 + 0), then Al7(OH)18ε(OH2)7(s−6ε) vanishes from the subtrahend, leaving the latter representing a single Al3+ cation with its 6ε hydroxyl (or for arithmetic purposes “hydroxide”) and s − 6ε water components. It will be assumed that, in the flower at least, this face-sharing does occur. Similarly, E62,ϕ− and V62,ϕ− may be computed.
Table 5 Calculation of m/z for a Great Stella stellation
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−      


Table 6 A reinterpretation of m/z values observed7,8 by ESI mass spectrometry in aqueous Al3+ solutions
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.


2.6.8.5 Fractal geometry. If Ogawa's flower |G62,ϕ+(x)〉 is really the enlarged rhombicHexecontahedron ρH, such that |GρH(x)〉 = 20|GV1(x)〉, then for edge length a1 its most primitive volume will be V1 = 4a13sqrt[2(5 + sqrt(5))]. Its enlargement by the linear self-similarity factor τ3 gives rise to a volume τ9 times greater: V2 = 4τ9a13sqrt[2(5 + sqrt(5))]. The fractal dimension d = [ln(V2) − ln(V1)]/[ln(2) − ln(1)] = 9[thin space (1/6-em)]ln[thin space (1/6-em)]τ/ln[thin space (1/6-em)]2. Again, see Fig. 4(a).

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.


2.6.8.6 The ρI enlargement. Again, under Penrose the enlarged rhombicIcosahedron ρI would now comprise not 10 rhombohedra (5 + 5), but 5 enlarged prolate + 5 enlarged oblate rhombohedra: |GρI(x)〉 = 5|GV1(x)〉 + 5|GV2(x)〉.
2.6.8.7 The computation. Bringing together the above pair of equations |GVj(x + i)〉, where i is a small number (j = 1, 2), we have for Penrose in 3-d
Φn−1|GV1(x + 2)〉 − Φn+1Φn−1|GV1(x + 1)〉 = Φn|GV2(x + 2)〉 − Φn2|GV1(x + 1)〉,
or
(Φn2Φn+1Φn−1)|GV1(x + 1)〉 = Φn|GV2(x + 2)〉 − Φn−1|GV1(x + 2)〉.

Now Φn+1Φn−1Φn2 = (−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

image file: c6ra11209a-t18.tif
so that |GV1(2x)〉 = (A + B)x−2[A|GV1(2)〉 + B|GV2(2)〉], where A = Φn+12 + Φn2 and B = Φn(Φn+1 + Φn−1). Similarly |GV1(2x + 1)〉 = (A + B)x−2 [A|GV1(3)〉 + B|GV2(3)〉]. |GVj(2x)〉 and |GVj(2x − 1)〉 are linear combinations of |GV1(2)〉 and |GV2(2)〉. For the |GVj(2x)〉 pair therefore (x > 1), if the major |GVj(2)〉 rhombohedra share a Mc+ cation at a vertex only, a straight multiplication may be written (with no subtraction for edge- or face-sharing). See Table 4.

In the aluminal case for example, where again the Al3+ cation is shared only at a vertex, the ϕ+ species would be

image file: c6ra11209a-t21.tif
paired with the ϕ species
image file: c6ra11209a-t22.tif

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.


2.6.8.8 A subtrahend. The above approximation for [m(x)/z(x)]i is predicated on the absence of all face- and edge-sharing between one primitive major |GV1(2)〉 or |GV2(2)〉 and another. Any sharing beyond that of a single vertex necessitates subtraction of numbers from m1(2) and m2(2), and the re-computation of z1(2) and z2(2). Again however either [m(x)/z(x)]i value would be x-independent, and would still approximate to [m*1(2) + τ−1m*2(2)]/[z*1(2) + τ−1z*2(2)], in which m*1(2) now equals m1(2) − σ1(2); also m*2(2), correspondingly.

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.


2.6.8.9 A reduced subtrahend. It may be asserted that an exterior major rhombohedron |GVj(2x + 1)〉 is unlike an interior major, in that there is a limit to the extent of possible edge- and face-sharing (cf. Biliński tiling, above). The exterior major has no adjoining major |GVj(2x + 1)〉 in which its constituent rhombohedra |GVj(2x)〉 could share. The potential subtrahends σ1(2x) and σ2(2x) are diminished.

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.


2.6.8.10 Exact reduction. Since |GV1(2x + 1)〉 = (A + B)x−2[A|GV1(3)〉 + B|GV2(3)〉], a precise subtrahend could be obtained by regarding the interior as a single |GV1(2x − 1)〉, for then the interior fraction would be (A + B)x−3/(A + B)x−2, or 1/(A + B). Now σ*1(2) = σ1(2)/(A + B). But A + B = 6765, and B + C = 4181: huge reduction factors compared with Φn+1 + Φn−1, which equals 76. (Φn+1 + Φn−1)−1 = 0.0131579. A |GV1(2x)〉 interior must be preferred. This comprises (A + B)x−2[A|GV1(2)〉 + B|GV2(2)〉]. The fraction is x-independent, and equals [A|GV1(2)〉 + B|GV2(2)〉]/[A|GV1(3)〉 + B|GV2(3)〉]. Comparing like |GVj(2)〉 components, the two interior fractions are (Φn+12 + Φn2)/[Φn+1(Φn+12 + Φn2) + Φn2(Φn+1 + Φn−1)] and Φn(Φn+1 + Φn−1)/[Φn(Φn+12 + Φn2) + Φn−1 Φn(Φn+1 + Φn−1)]. Each fraction is equal to 0.0131556, which does differ slightly from (Φn+1 + Φn−1)−1, though only by <200 ppm and not in the first 5 significant figures.
2.6.8.11 The emperor's new crystal?. Should crystalline evidence be found for Penrose nucleation and growth, any such structure would most certainly have been regarded as “amorphous” in the pre-Penrose era (we could say “X-ray amorphous”). No crystal lattice is implied. Perhaps the more disordered aperiodic tiling may be favoured – over Biliński, say – by the entropic part of its Gibbs free energy. As crystal growth tends to the infinite, however, then in the limitn→∞(Φn/Φn−1) = τ. (Given the icosahedral symmetry,16 this limiting case appears fully consistent with non-repetition under translation, without any semblance of “unit cell”.)

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

2.6.9 Stochastic nucleation. If rhombohedral species Al8(OH)24ε(OH2)8(s−6ε)24(1−ε)+ were to exist, but without immediately nucleating a well-ordered 3-d growth pattern such as a space-filler or Penrose moiety, two or more species could perhaps connect to one another through a shared vertex, edge or face. This could produce species seemingly at random such as Al[Al7(OH)24ε(OH2)(s−6ε)7]23(15−16ε)+, Al2O2α(OH)2β(OH2)(s−(2β+8ε))2[Al6(OH)22ε(OH2)(s−6ε)6]22[21−(2α+β+22ε)]+, or Al4O8α(OH)8β(OH2)(s−(4β+4ε))4[Al4(OH)16ε(OH2)(s−6ε)4]24[9−2(2α+β+4ε)]+, and so on: including, perhaps, Biliński (from Keggin) and the stellation of Kepler's triacontahedron; any or all of which could develop into well-ordered crystal nuclei. Ultimately, however, we should not be surprised to discover some degree of structuring, even in an X-ray amorphous precipitate.

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(+)D−2eεE]+, with a = 0, b = 0, c = 1, e = 1. A Poisson distribution may be expected!

2.6.10 N-Dimensional growth?. The n-dimensional hypercube, or n-cube, has 2n vertices and n2n−1 edges. A general formulation is MC+VPαDQ2β(D+E)RsV−4β(12+V)z+. If an aluminal species were to exist with such a structure, then approximating, z = 2n3 − [α(n − 1) + 2βn]2n−1. See Table 2.

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.

2.7 Other species

Many kinds of 3-dimensional growth could potentially be modelled, with or without hexagonal rings; but including prismatic systems such as the monoclinic (e.g., incipient bayerite or gibbsite crystals) or orthorhombic. Any or all of these special cases could be the subject of further symmetry-based computations. Close packing may not be assumed. However, where it does exist, spherical expansion may give rise to space-filling polyhedra such as the hexagonal prism or rhombic dodecahedron. Spectral evidence could perhaps be sought for tessellated space or hyperspace. (In alumina, the 3-cube itself seems intuitively unlikely. However, given the icosandritic 6-cube projection it could be interesting to ask which symmetries besides the icosahedral are 3-d projections of an N-dimensional polytopic crystal, and what if any others might be expected.)

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.

3. Results and discussion

Table 6 shows a fresh interpretation of the observations made by Urabe and his group.7,8 In their experiments, the intensity of the strong mass spectral line at m/z = 97 appears to provide good evidence that in the chloride solutions, AlCl3 (described in the group's 2007 paper7), the monomer exists as a cation in a tetrahedral symmetry: Al(OH)2(OH2)2+ (m = 97). (However see below, m/z = 133.) It is possible that, with no loss of symmetry, this species could readily give rise to an aluminate anion (observed by the same Group in 2011 (ref. 8)):
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
or possibly (to m/z = 97, m = 582):
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.

4. Conclusions

The appearance in mass spectra of the line having a value of m/z = 97 is not sufficient to confirm the existence of the hexameric ring species with octahedral symmetry, Al6(OH)12(OH2)126+ (m = 582). It seems far more likely that this spectral line originates primarily from the tetrahedral monomer, Al(OH)2(OH2)2+ (m = 97). Yet the existence of the ring is not disproven. Further experimentation is needed; and, quite possibly, novel experimental design beforehand.

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−1[thin space (1/6-em)]τ. 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.

Note added after first publication

This article replaces the version published on 16th September 2016, which contained an error in the binomial expansion in Section 2.6.8.7.

Acknowledgements

This review stems from a 40-year fascination with the question of how the 3rd cation attaches itself to a dimer in solution [whether in a straight line or at a 60° angle, and why], and whether the answer might explain the nucleation of hexagonal or hexagonal prismatic crystals. This particular enquiry has been stimulated and revitalised by Akihiro Wakisaka's 2007 presentation6 to the Faraday Division of the Royal Society of Chemistry. The paper is written in belated celebration of John C. Vickerman's 70th birthday in 2013, and in recognition of his life's work in the field of mass spectrometry. It is dedicated to the memory of C. Stewart Hare.

Notes and references

  1. P. H. Hsu and C. I. Rich, Soil Sci. Soc. Am. Proc., 1960, 24, 21 CrossRef CAS .
  2. J. Siecińska, D. Wiącek and A. Nosalewicz, Acta Agrophys., 2016, 23(1), 97 Search PubMed .
  3. C. Exley, Cellular and Tissular effects of Aluminium, Morphologie, 2016, 100(329), 51 CrossRef CAS PubMed  , special issue.
  4. Z. Wang, X. Wei, J. Yang, J. Suo, J. Chen, X. Liu and X. Zhao, Neurosci. Lett., 2016, 610, 200 CrossRef CAS PubMed .
  5. P. H. Hsu and T. F. Bates, Mineral. Mag., 1964, 33, 749 CAS .
  6. A. Wakisaka, Faraday Discuss., 2007, 136, 299 RSC .
  7. T. Urabe, M. Tanaka, K. Kumakura and T. Tsugoshi, J. Mass Spectrom., 2007, 42, 591 CrossRef CAS PubMed .
  8. T. Urabe and M. Tanaka, Rapid Commun. Mass Spectrom., 2011, 25, 2933 CrossRef CAS PubMed .
  9. B. J. Mason, Q. J. R. Meteorol. Soc., 1960, 86, 552 CrossRef CAS .
  10. J.-F. Lambert, BIO Web of Conferences, 2015, vol. 4, p. 12 Search PubMed .
  11. E. S. Fyodorov, Convex Space Filling Polyhedra, cited by, I. Sato and R. Takaki, Forma, 2009, vol. 24, p. 79, 1891 Search PubMed .
  12. D. Shechtman, The Discovery of Quasicrystals, cited by The Royal Swedish Academy of Sciences, for the Nobel Prize in Chemistry, 2011 Search PubMed .
  13. E. W. Weisstein, Rhombic Dodecahedron, from Mathworld-A Wolfram Web Resource, http://mathworld.wolfram.com/RhombicDodecahedron.html Search PubMed .
  14. B. Shuping, W. Chenyi, C. Qing and Z. Caihua, Coord. Chem. Rev., 2004, 248, 441 CrossRef .
  15. D. Kondrashov, Geometric Theory of Crystal Growth, http://math.arizona.edu/%7Eflaschka/Topmatter/527files/termpapers/kondrashov.pdf Search PubMed .
  16. T. Ogawa, Mater. Sci. Forum, 1987, 22, 187 CrossRef .
  17. H. Coelfen, D. Gebauer and C. Voelkle, Faraday Discuss., 2015, 179, 59 RSC  and 171.
  18. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., 1990, vol. xxv. ISBN 0-470-84862-6 Search PubMed.
  19. P. Guyot, Nature, 1987, 326, 640 CrossRef .

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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