The pillaring of Na-montmorillonite with cationic oligomers of hydroxyaluminum (COHA) in the presence of an aqueous solution of polyvinyl alcohol resulted in the formation of a clay having a large surface area and pore volume. The pore-size distribution determined from a N2 adsorption/desorption hysteresis was narrow and centered at about 25 Å. The peak width at half height in the distribution curve was < 5 Å. As a result of delamination, the layer structure of the prepared clay was found from X-ray powder diffraction measurements to be lost. Short-range ordering, however, still existed in this delaminated clay, because exchangeable cations in the montmorillonite completely exchanged with Al3+, a requisite step for pillaring.

The order of adding the starting materials (Na-montmorillonite, polyvinyl alcohol, and COHA) greatly affected the surface area and the pore volume of the delaminated clay. Two orders of addition (Namontmorillonite, then COHA, then polyvinyl alcohol; and COHA, then Na-montmorillonite, then polyvinyl alcohol) gave no measurable surface area and pore volume. Two other orders of addition (polyvinyl alcohol, then COHA, then Na-montmorillonite; and polyvinyl alcohol, then Na-montmorillonite, then COHA) gave surface areas of 107 and 160 m2/g and pore volumes of 0.13 and 0.29 cm3/g, respectively. The amounts of the COHA solution and polyvinyl alcohol added greatly influenced the surface area and pore volume of the delaminated clay. Both properties increased monotonically with increasing amount of added polyvinyl alcohol, and increased to a maximum and then decreased with increasing amount of added COHA solution. The maximum surface area of the prepared delaminated clay was 330 m2/g.

13C and 29Si nuclear magnetic resonance spectroscopy with magic-angle spinning bas been used to study the short-range ordering and bonding in the structures of intercalates of kaolinite with formamide, hydrazine, dimethyl sulfoxide (DMSO), and pyridine-N-oxide (PNO). The 29Si chemical shift indicated decreasing levels of bonding interaction between the silicate layer and the intercalate in the order: kaolinite: formamide (δ) = -91.9, ppm relative to tetramethylsilane), kaolinite: hydrazine (-92.0), kaolinite: DMSO (-93.1). The 29Si signal of the kaolinite:PNO intercalate (-92.1) was unexpectedly deshielded, possibly due to the aromatic nature of PNO. The degree of three-dimensional ordering of the structures was inferred from the 29Si signal width, with the kaolinite: DMSO intercalate displaying the greatest ordering and kaolinite: hydrazine the least. 13C resonances of intercalating organic molecules were shifted downfield by as much as 3 ppm in response to increased hydrogen bonding after intercalation, and in the kaolinite: DMSO intercalate the two methyl-carbon chemical environments were non-equivalent (δ = 43.7 and 42.5).

The stacking order of a 14.30-Å Mg-vermiculite from Santa-Olalla, Spain, has been determined from Weissenberg photographs. The results prove that the Mg-vermiculite structure is not a 2-layer polytype structure, but a semi-ordered one. Because the structure is semi-ordered, its resolution needed a dual approach: (1) a direct approach using an electron density projection along the y axis in conjunction with a one-dimensional electron-density projection onto the z axis; and (2) an indirect approach in which the observed intensities along the (0,k) and (1,k) reciprocal rods were compared to the calculated intensities given by model defect structures. The semi-ordered structure of the Mg-vermiculite results from ±b/3 shifts in the stacking of the silicate layers. The shifts are randomly either along +b or -b.

On the basis of progressive acid dissolution the Thompson-Hower model states that mixed-layer glauconite/smectite and illite/smectite contain potassium in two structural sites: a mica-type K position (site 1) and a position of uncertain structural status more prone to dissolution (site 2). Site 2 was thought not to retain radiogenic argon (40Ar*). Using extensive progressive acid dissolution and K/Ar studies on a sized illite/smectite (I/S), determining the amount of K in site 2 is shown to be somewhat more complicated than previously thought because the dissolution pattern depends on acid normality. More important, site 2 fully retains 40Ar*, and no age correction is thus necessary as is required by the Thompson-Hower model, further affirming the geochronologic value of illite in mixed-layer clay. These data are also relevant to understanding the crystal and particle structure of I/S. Site 2 is probably a partly filled K interlayer that develops as an intermediate kinematic step on the way to being fully filled during the transformation of smectite to illite.

If anilinium ions are intercalated into Llano vermiculite, the stacking order of adjacent silicate layers is increased, resulting in a relatively sharp single crystal X-ray diffraction (XRD) pattern. The packing of intercalated organic members forms a superstructure and produces bonding from layer to layer which favors the stacking order. Superlattice reflections occur which, although sharp in the a*b* plane, are streaked along c*. Apparently there is little coherence between adjacent layers of ordered organic units.

A three-dimensional set of XRD reflections for a triclinic sub-cell having the following lattice parameters was measured: a = 5.326(3), b = 9.264(4), c = 14.82(5) Å, α = 90.31(7), β = 96.70(6), and γ = 89.55(5)°. In this unit cell (symmetry Cl), ditrigonal cavities in adjacent silicate layers are approximately opposite. Differential Fourier analyses and least-squares refinements showed that the principal axes of the anilinium ions, i.e., N-C(1)-C(4), are nearly perpendicular to the silicate layers. The planes of the aromatic rings, however, are about ±30° to X, neither parallel nor perpendicular to that direction, as indicated by earlier studies.

Inorganic cations and water molecules are also present in the interlayer; the former and some of the latter occupy sites near the middle of the layer. Anilinium-rich and anilinium-poor domains coexist. In the latter, the cation-water system predominates and apparently conforms to the superstructure. Although the cation-water structure could not be uniquely established from the reflections produced by the sub-cell, possible positional coordinates were obtained. From structural data for the silicate layers, no evidence was found for long-range Si/Al ordering in the tetrahedral sites.

Stable, three-dimensionally ordered complexes were formed from synthetically hydrated, highly ordered kaolinite (d(001) = 8.4 Å) and several organic compounds. Removal of the intercalated organic compound by drying or by water washing recovered the 8.4-Å hydrate with its ordered layer stacking essentially unchanged. Some of the complexes were stable for less than a day, whereas others appeared to be stable indefinitely. The compounds that formed ordered complexes were dimethylsulfoxide, formamide, hydrazine-hydrate, 1,1-dimethylhydrazine, ethylene glycol, glycerol, and pyridine. Clay-organic complexes that were prepared from methanol, ethanol, 1- and 2-propanol, acetone, acetic acid, propionic acid, acetaldehyde, N-methylformamide, methylethyl ketone, tetrahydrofuran, and K-acetate were stable only if they were immersed in the intercalating medium and had little or no stacking order.

Many of the organic compounds intercalated by the 8.4-Å hydrate are not known to be intercalated by non-hydrated kaolinite either directly or indirectly. Isolated water molecules appear to be keyed into the ditrigonal holes formed by the basal oxygen of the silicate tetrahedra of the 8.4-Å hydrate. These water molecules, referred to as “hole water,” and fluorine ions that had replaced ~20% of the inner-surface hydroxyls of the 8.4-Å phase sufficiently altered the interlayer bonding to allow an expansion of the inner-layer spaces by a variety of guest molecules. The presence of these guest molecules between the clay layers not only changed the basal spacing and perturbed the infrared (IR) bands arising from the inner-surface hydroxyls, it also shifted the position of the IR band arising from the inner hydroxyl.

Hydrous phyllosilicate minerals exhibit many variations in unit-cell shape and symmetry, distortion due to cation ordering, polytypic repeats along Z, and lateral modulations that can affect the ease of indexing of X-ray powder diffraction patterns. In indexing the patterns of monoclinic and triclinic hydrous phyllosilicates it is important that the indices of all overlapping but nonequivalent lines be listed and used in the indexing program, with weights assigned according to their multiplicities. Deviations from orthogonal unit-cell geometry show up by a broadening or a splitting of nearly superimposed lines, as occurs also if the observed ß angle differs from that produced by a layer shift of ideal magnitude (e.g., a/3). Observed powder intensities should be correlated with those expected from the structure, and an index should not be assigned solely on the basis of the calculated d-value that is closest to that observed. Assigned indices should not violate space group extinctions. Multilayer structures show up in reflections of type k ≠ 3n for C-centered unit cells, not in 00/reflections.

X-ray powder diffraction (XRD) studies of the clay fraction of Upper Cretaceous mudstones from a shallow (2.5 km) drill hole in the Niger delta indicate a high geothermal gradient (about 100°C/km) during diagenesis. The mineralogy of the clays is similar to that observed elsewhere and consists of interstratified illite/smectite (I/S), kaolinite, and chlorite. Detrital mica and K-feldspar are also present throughout the section. The composition-depth relationship of the I/S is different from that observed in deeply buried Gulf Coast shales. The Niger delta rocks show a linear change in composition of the I/S as a function of depth in the drill hole from 60 to 10% smectite layers. The I/S ordering is R=0 in shallower samples and progresses to R=1 in deeper samples. The I/S in the deepest samples has R=3 ordering. No R=1 I/S showing a first-order 27-Å XRD reflection was found. The Niger delta sequence differs from the Gulf Coast sequence b. (1) a lack of R=1 I/S showing a 27-Å XRD reflection (which are common in Gulf Coast samples and contain 20% smectite layers), an. (2) the existence of a simple, continuous linear relation between the composition of the R=1 and R=0 I/S and depth.

A comparison of composition-temperature curves for I/S formed under different diagenetic regimes shows different types of I/S ordering in which the presence of R=1 I/S showing a 27-Å reflection and R=3 I/S types changes the composition-depth (and thus, composition-temperature) relations. These changes suggest a difference in the energy necessary to form the various ordering types. Also, geothermal gradient during burial appears to be responsible for different composition-temperature gradients found for the same type of I/S ordering.

Junction probability diagrams show variation in both composition and layer arrangement in mixed-layer clay minerals. These diagrams can represent short-range and long-range ordered, random, and segregated interstratifications. Mineralogical analyses of illite/smectite from shale cuttings, bentonites, and hydrothermally altered tuffs define characteristic reaction pathways through these diagrams. Shale and bentonite analyses fall along pathways joining smectite and illite on diagrams showing nearest-neighbor (R1) layer arrangements. Transition from random to R1-ordered interstratifications occurs in shale samples containing 60–70% illite layers, and in bentonites containing 55–67% illite layers. Analyses of alteration products, however, fall near a line connecting rectorite and illite, which represents the maximum degree of R1 layer ordering. No mineralogical evidence is available to suggest that these alteration samples formed from a smectite precursor. All samples develop next-nearest (R2) and thrice-removed (R3) neighbor ordering along similar pathways. Transition to R2 ordering occurs gradually in samples composed of 65–80% illite layers, and samples containing more than 85% illite layers may show strong R3 ordering.

The stacking order of a bi-ionic K/Mg vermiculite from Malawi (Nyasaland), has been determined from Weissenberg data. The sample corresponds to a K/Mg interstratified vermiculite containing 50% K layers (dK = 1.01 nm) and 50% Mg layers with two layers of water (dMg = 1.44 nm). The observed intensities along (0, 0), (h, 0), (0, k) and (1, k) reciprocal rods were compared to the calculated intensities given by model defect structures. It was found that: 1) The (h, 0), (0, k) and (1, k) rods reveal the interstratification phenomenon which was previously observed on the (0, 0) rod; 2) The exchange of Mg by K does not occur randomly in a single interlayer, but occurs in interlayer domains sufficiently extensive to allow the reorganisation of the layer stacking and development of the ordered K-vermiculite structure from the semi-ordered structure characteristic of magnesium vermiculites; 3) For this sample, the interlayer water of the Mg phase occurs in two coexisting configurations; one configuration has water molecules forming octahedral coordination around the Mg2+, the other has water which is not linked to the Mg2+ cation forming two planes.

Refractory high-entropy alloys (RHEAs) are promising candidates for next-generation high-temperature materials. RHEAs containing Al, often exhibit a checkered pattern microstructure comprising a combination of disordered BCC and ordered B2 phases. Since the ordered B2 phase is based on the BCC parent matrix, distinguishing these two phases can be rather challenging. Advanced characterization techniques are necessary for a reliable qualitative and quantitative analysis of BCC and B2 phases in RHEAs. Additionally, there is a tendency for transformation of the ordered B2 phase into more complex ordered-omega type phases that are usually deleterious to mechanical properties. The current study focuses on the phase stability of a candidate RHEA, Al0.5Mo0.5NbTa0.5TiZr. Correlative transmission electron microscopy (TEM) and atom probe tomography (APT) have been employed to investigate the phase stability and transformation pathway of this RHEA when isothermally annealed at 800°C. The results show that a metastable two-phase BCC + B2 microstructure formed at the early stages of decomposition, eventually transforming into a three-phase BCC + B2 + hP18 microstructure. The hP18 phase is an ordered omega derivative of the ordered B2 phase. The correlative microscopy techniques (TEM and APT) reveal a very interesting interplay of compositional partitioning between the different phases and their respective stability.

Ordering is a well-established concept in mathematics and also plays an important role in many areas of computer science, where quasi-orderings, most notably well-founded quasi-orderings and well-quasi-orderings, are of particular interest. This paper deals with quasi-orderings on first-order terms and introduces a new notion of unification based on a special quasi-order, known as homeomorphic tree embedding. Historically, the development of unification theory began with the central notion of a most general unifier based on the subsumption order. A unifier $\sigma$ is most general, if it subsumes any other unifier $\tau$ , that is, if there is a substitution $\lambda$ with $\tau=_{E}\sigma\lambda$ , where E is an equational theory and $=_{E}$ denotes equality under E. Since there is in general more than one most general unifier for unification problems under equational theories E, called E-Unification, we have the notion of a complete and minimal set of unifiers under E for a unification problem $\varGamma$ , denoted as $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ . This set is still the basic notion in unification theory today. But, unfortunately, the subsumption quasi-order is not a well-founded quasi-order, which is the reason why for certain equational theories there are solvable E-unification problems, but the set $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ does not exist. They are called type nullary in the unification hierarchy. In order to overcome this problem and also to substantially reduce the number of most general unifiers, we extended the well-known encompassment order on terms to an encompassment order on substitutions (modulo E). Unification under the encompassment order is called essential unification and if $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ exists, then the complete set of essential unifiers $e\mathcal{U}\Sigma_{E}(\Gamma)$ is a subset of $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ . An interesting effect is that many E-unification problems with an infinite set of most general unifiers (under the subsumption order) reduce to a problem with only finitely many essential unifiers. Moreover, there are cases of an equational theory E, for which the complete set of most general unifiers does not exist, the minimal and complete set of essential unifiers however does exist. Unfortunately again, the encompassment order is not a well-founded quasi-ordering either, that is, there are still theories with a solvable unification problem, for which a minimal and complete set of essential unifiers does not exist. This paper deals with a third approach, namely the extension of the well-known homeomorphic embedding of terms to a homeomorphic embedding of substitutions (modulo E). We examine the set of most general, minimal, and complete E-unifiers under the quasi-order of homeomorphic embedment modulo an equational theory E, called $\varphi U\Sigma_{E}(\Gamma)$ , and propose an appropriate definitional framework based on the standard notions of unification theory extended by notions for the tree embedding theorem or Kruskal’s theorem as it is called. The main results are that for regular theories the minimal and complete set $\varphi\mathcal{U}\Sigma_{E}(\Gamma)$ always exists. If we restrict the E-embedding order to pure E-embedding, a well-known technique in logic programming and term rewriting where the difference between variables is ignored, the set $\varphi_{\pi}\mathcal{U}\Sigma_{E}(\Gamma)$ always exists and it is even finite for any theory E.