Molecular dynamics simulation of the structural and dynamic properties of dioctadecyldimethyl ammoniums in organoclays

The structural and dynamic properties of dioctadecyldimethylammoniums (DODDMA) intercalated into 2:1 layered clays are investigated using isothermal-isobaric (NPT) molecular dynamics (MD) simulation. The simulated results are in reasonably good agreement with the available experimental measurements, such as X-ray diffraction (XRD), atom force microscopy (AFM), Fourier transform infrared (FTIR), and nuclear magnetic resonance (NMR) spectroscopies. The nitrogen atoms are found to be located mainly within two layers close to the clay surface whereas methylene groups form a pseudoquadrilayer structure. The results of tilt angle and order parameter show that interior two-bond segments of alkyl chains prefer an arrangement parallel to the clay surface, whereas the segments toward end groups adopt a random orientation. In addition, the alkyl chains within the layer structure lie almost parallel to the clay surface whereas those out of the layer structure are essentially perpendicular to the surface. The trans conformations are predominant in all cases although extensive gauche conformations are observed, which is in agreement with previous simulations on n-butane. Moreover, an odd-even effect in conformation distributions is observed mainly along the chains close to the head and tail groups. The diffusion constants of both nitrogen atoms and methylene groups in these nanoconfined alkyl chains increase with the temperature and methelene position toward the tail groups.

Analytical solutions of linear finite- and small-strain one-dimensional consolidation

Analytical solutions are presented for linear finite-strain one-dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one- and two-way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small-strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small-strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non-linear finite-strain consolidation. They also clarify a formerly perplexing aspect of finite-strain solution charts first noted in numerical solutions. Copyright (C) 2004 John Wiley Sons, Ltd.

Clay mineralogical, geochemical and isotopic tracing of the evolution of the Woodleigh impact structure, Southern Carnarvon Basin, Western Australia

Chaotically structured diamictite from the inner ring syncline surrounding the central uplift of the Woodleigh impact structure contains shocked metamorphic and impact melt-rock fragments, largely derived from Ordovician and Devonian target sandstones. Coarse illite fractions (< 2 mu m) from the sandstones containing no K-feldspar yield K-Ar ages of around 400 Ma, whereas the K-Ar ages of authigenic clays of > 0.2 mu m fractions from the diamictite without smectite and K-feldspar cluster around 360 Ma, consistent with Rb-Sr data. Crystallisation of newly formed illite in the impact melt rock clasts and recrystallisation of earlier formed illite in the sandstone clasts preserved in the diamictite, are attributed to impact-induced hydrothermal processes in the Late Devonian. The illitic clays from the diamictite and from the sandstones have very similar trace element compositions, with significantly enriched incompatible lithophile elements, which increase in concentrations correlatively with those of the compatible ferromagnesian elements. The unusual trace element associations in the clays may be due to the involvement of hot gravity-driven basinal fluids that interacted with rocks of the Precambrian craton to the east of the study area, or with such material transported and reworked in the studied sedimentary succession.

Woodleigh, Southern Carnarvon Basin, Western Australia: history of discovery, Late Devonian age, and geophysical and morphometric evidence for a 120 km-diameter impact structure

The discovery of the Woodleigh impact structure, first identified by R. P. lasky, bears a number of parallels with that of the Chlcxulub impact structure of K-T boundary age, underpinning complications inherent in the study of buried impact structures by geophysical techniques and drilling. Questions raised in connection with the diameter of the Woodleigh impact structure reflect uncertainties in criteria used to define original crater sizes in eroded and buried impact structures as well as limits on the geological controls at Woodleigh. The truncation of the regional Ajona - Wandagee gravity ridges by the outer aureole of the Woodleigh structure, a superposed arcuate magnetic anomaly along the eastern part of the structure, seismic-reflection data indicating a central > 37 km-diameter dome, correlation of fault patterns between Woodleigh and less-deeply eroded impact structures (Ries crater, Chesapeake Bay), and morphometric estimates all indicate a final diameter of 120 km. At Woodleigh, pre-hydrothermal shock-induced melting and diaplectic transformations are heavily masked by pervasive alteration of the shocked gneisses to montmorillonite-dominated clays, accounting for the high MgO and low K2O of cryptocrystalline components. The possible contamination of sub-crater levels of the Woodlelgh impact structure by meteoritic components, suggested by high Ni, Co, Cr, Ni/ Co and Ni/Cr ratios, requires further siderophile element analyses of vein materials. Although stratigraphic age constraints on the impact event are broad (post-Middle Devonian to pre-Early Jurassic) high-temperature (200-250 degrees C) pervasive hydrothermal activity dated by K-Ar isotopes of illite - smectite indicates an age of 359 +/- 4 Ma. To date neither Late Devonian crater fill, nor impact ejecta fallout units have been identified, although metallic meteoritic ablation spherules of a similar age have been found in the Conning Basin.

Microchemistry and microstructures of hydrothermally altered shock-metamorphosed basement gneiss, Woodleigh impact structure, Southern Carnarvon Basin, Western Australia

Hydrothermally altered shock-metamorphosed gneisses consisting of relic igneous biotite-K-feldspor-Na-rich alkali feldspar - plagioclase - quartz assemblages ( accessory garnet, corundum, titanite, monazite, zircon), and showing extensive replacement by montmorillonite, illite, sericite, and to a lesser extent chlorite, calcite, epidote, zoisite and pyrite, occur in the basement core uplift of the Woodleigh impact structure, Western Australia. The rocks display extensive hydrothermal clay alteration, complicating identification of pre-hydrothermal and pre-impact textures and compositions. Analysis of quartz-hosted planar deformation features (PDFs) indicates a majority of indexed sets parallel to omega{10 (1) over bar3}, a lesser abundance of sets parallel to pi{10 (1) over bar2}, and some sets parallel to the basal plane (0001) and r,z {10 (1) over bar1}, consistent with pressures about or over 20 GPa. Feldspar-hosted FDFs form reticulate vein networks displaying checkerboard-like to irregular and serrated patterns attributable to preferential replacement of shock-damaged PDFs and/or perthitic twin lamella by clay minerals. The gneisses are pervaded by clay-dominated intergranular and intragranular veins of cryptocrystalline material that display marked departures from bulk-rock chemistry and from mineral compositions. XRD analysis identifies the cryptocrystalline components as illite - montmorillonite, illite and chlorite, while laser Raman analysis identifies high-fluorescence sub-micrometre clay assemblage, feldspar, quartz and minor mica. SEM/EDS-probe and laser-ICPMS analysis indicate low-K high-Mg clay mineral compositions consistent with montmorillonite. Quartz PDF-hosted cryptocrystalline laminae display distinct enrichments in Al, Mg, Ca and K. Altered intergranular veins and feldspar-hosted cryptocrystalline components show consistent enrichment in the relatively refractory elements (Al, Cc, Mg, Fe) and depletion in relatively volatile elements (Si, K, Na). The clay alteration retards determination whether clay-dominated vein networks represent altered shock-induced pseudotachylite veins, diaplectic zones and/or shock-damaged twin lamella, and/or result from purely mineralogical and chemical differentiation affected by hydrothermal fluids, Overall enrichment of the shocked gneiss and of the cryptocrystalline components in Mg and trace ferromagnesian elements (Ni, Cc, Cr) may be attributed alternatively to introduction of siderophile element-rich fluid from the projectile, or/and contamination of hydrothermal fluids by MgO from dolomites surrounding the basement uplift. High Ni/Co and Ni/Cr and anomalous DGE (platinum group elements) may support the former model.

Visoelastic studies on thermoplastic polyurethane nanocomposites using a nano-hardness tester

The viscoelastic behaviour of a range of polyurethane thermoplastic elastomer montmorillonite nanocomposites has been studied using a nanohardness tester. For softer Shore hardness 80A materials, the introduction of the organo-clay increased the creep strain obtained while the nano-indentor was held at constant load. The increase in creep strain was greatest for materials containing an organo-clay modified with a more hydrophilic quaternary alkylammonium surfactant and with higher loadings of the hydrophilic organo-clay. This suggested the effect might be due to a plasticising effect of the excess surfactant. For the harder Shore hardness 55D materials, the addition of the organo-clays produced only a small decrease in the creep strain, probably due to the interconnected hard domains in this material.

Clay mineralogical and isotopic (K-Ar, delta O-18, delta D) constraints on the evolution of the North Anatolian Fault Zone, Turkey

This study presents the first attempt to constrain the evolution of the North Anatolian Fault Zone (NAFZ) by age dating and isotope tracing of clay minerals formed during near-surface faulting. Extensive illitic clay mineralisation occurred along the NAFZ related to hydrothermal alteration of the fault gouges and pseudotachylytes. Samples representing the pre-fault protoliths outside the fault zone do not contain authigenic illitic clay minerals indicating that hydrothermal processes were confined to the areas within the fault zone. K-Ar age data indicate that the hydrothermal system and the associated illite authigenesis initiated at similar to 57 Ma. This process is interpreted to reflect the onset of significant strike-slip or transtensional faulting immediately after the continental collision related to the closure of the Neotethys Ocean. Following the initiation of the fault movements in the latest Paleocene-Early Eocene, displacements along the NAFZ have continued, with probably intensified fault activities at similar to 26 Ma and later than similar to 8 Ma. Oxygen isotope compositions of the illitic clays from different locations along the NAFZ are similar, with narrow ranges in delta O-18 values indicating clay precipitation from fluids with similar oxygen isotope compositions and crystallisation temperatures. The delta O-18 and delta D values of the calculated fluid isotopic composition (delta O-18=5.9 parts per thousand to 11.2 parts per thousand, delta D=-59 parts per thousand to -73 parts per thousand) are consistent with metamorphic and magmatic origin of fluids mobilised during active tectonism. The interpretation of the fluid flow history of the NAFZ is in agreement with that reported previously for some well-known large-scale high-angle fault zones, which similarly developed along collisional-type orogenic belts and are commonly associated with significant mesothermal ore mineralisation. (c) 2005 Elsevier B.V. All rights reserved.

Flow field analysis of some mixing and conveying screw element regions, within a closely intermeshing, co-rotating twin-screw extruder.

Grafting of antioxidants and other modifiers onto polymers by reactive extrusion, has been performed successfully by the Polymer Processing and Performance Group at Aston University. Traditionally the optimum conditions for the grafting process have been established within a Brabender internal mixer. Transfer of this batch process to a continuous processor, such as an extruder, has, typically, been empirical. To have more confidence in the success of direct transfer of the process requires knowledge of, and comparison between, residence times, mixing intensities, shear rates and flow regimes in the internal mixer and in the continuous processor.The continuous processor chosen for the current work in the closely intermeshing, co-rotating twin-screw extruder (CICo-TSE). CICo-TSEs contain screw elements that convey material with a self-wiping action and are widely used for polymer compounding and blending. Of the different mixing modules contained within the CICo-TSE, the trilobal elements, which impose intensive mixing, and the mixing discs, which impose extensive mixing, are of importance when establishing the intensity of mixing. In this thesis, the flow patterns within the various regions of the single-flighted conveying screw elements and within both the trilobal element and mixing disc zones of a Betol BTS40 CICo-TSE, have been modelled using the computational fluid dynamics package Polyflow. A major obstacle encountered when solving the flow problem within all of these sets of elements, arises from both the complex geometry and the time-dependent flow boundaries as the elements rotate about their fixed axes. Simulation of the time dependent boundaries was overcome by selecting a number of sequential 2D and 3D geometries, used to represent partial mixing cycles. The flow fields were simulated using the ideal rheological properties of polypropylene and characterised in terms of velocity vectors, shear stresses generated and a parameter known as the mixing efficiency. The majority of the large 3D simulations were performed on the Cray J90 supercomputer situated at the Rutherford-Appleton laboratories, with pre- and postprocessing operations achieved via a Silicon Graphics Indy workstation. A mechanical model was constructed consisting of various CICo-TSE elements rotating within a transparent outer barrel. A technique has been developed using coloured viscous clays whereby the flow patterns and mixing characteristics within the CICo-TSE may be visualised. In order to test and verify the simulated predictions, the patterns observed within the mechanical model were compared with the flow patterns predicted by the computational model. The flow patterns within the single-flighted conveying screw elements in particular, showed good agreement between the experimental and simulated results.

Novel catalytic methods for the upgrading of coal liquids

Reproducible preparation of a number of modified clay and clay~like materials by both conventional and microwave-assisted chemistry, and their subsequent characterisation, has been achieved, These materials are designed as hydrocracking catalysts for the upgrading of liquids obtained by the processing of coal. Contact with both coal derived liquids and heavy petroleum resids has demonstrated that these catalysts are superior to established proprietary catalysts in terms of both initial activity and deactivation resistance, Of particular activity were a chromium-pillared montmorillonite and a tin intercalated laponite, Layered Double Hydroxides (LDH's) have exhibited encouraging thermal stability. Development of novel methods for hydrocracking coal derived liquids, using a commercial microwave oven, modified reaction vessels and coal model compounds has been attempted. Whilst safe and reliable operation of a high pressure microwave "bomb" apparatus employing hydrogen, has been achieved, no hydrotreatment reactions occurred,

Chemical and physical characterization of clay bodies

Suitable methods for the assessment of the effect of freeze-thaw action upon ceramic tiles have been determined. The results obtained have been shown to be reproducible with some work in this area still warranted. The analysis of Whichford Potteries clays via a variety of analytical techniques has shown them to be a complex mix of both clay and non-clay minerals. 57Fe Mössbauer spectroscopy has highlighted the presence of both small and large particleα-Fe203, removable via acid washing. 19F MAS NMR has demonstrated that the raw Whichford Pottery clays examined have negligible fluorine content. This is unlikely to be detrimental to ceramic wares during the heating process. A unique technique was used for the identification of fluorine in solid-state systems. The exchange of various cations into Wyoming Bentonite clay by microwave methodology did not show the appearance of five co-ordinate aluminium when examined by 27Al MAS NMR. The appearance of Qo silicate was linked to an increase in the amount of tetrahedrally bound aluminium in the silicate framework. This is formed as a result of the heating process. The analysis of two Chinese clays and two Chinese clay raw materials has highlighted a possible link between the two. These have also been shown to be a mix of both clay and non-clay minerals. Layered double hydroxides formed by conventional and microwave methods exhibited interesting characteristics. The main differences between the samples examined were not found to be solely attributable to the differences between microwave and conventional methods but more attributable to different experimental conditions used.

Studies in clay chemistry

A study of clay chemistry has been approached with three aims: - to modify the conducting properties by intercalation of tetrathiafulvalene, - to study the electrochemistry of redox-active coordination compounds immobilised on clay coated electrodes, and - to study the role of clays as reagents in inorganic glass forming reactions using mainly solid-state magic-angle-spinning NMR. TTF was intercalated by smectites containing different interlayer and lattice cations. Evidence from ESR and 57Fe Mossbauer indicated charge-transfer from TTF to structural iron in natural montmorillonite, and to interlayer Cu2+ in Cu2+ exchanged laponite. No charge transfer was observed for laponite (Na+ form) itself. Ion exchange of TTF3(BF4)2 with laponite was found to proceed quantitatively. The intercalated species were believed to be (TTF)2+ dimers. Conductivity data showed an order of magnitude increase for the intercalated clays. The mechanism is thought to be ionic rather than CT as Na+ laponite showed a similar enhancement in conductivity. Mechanically robust colloidal clay films were prepared on platinum electrodes. After immersion in solutions containing redox active complexes [Co(bpy)3]3+ and [Cr(bpy)3]3+, the films became electroactive when a potential was applied. Cyclic voltammograms obtained for both complexes were found to be of the diffusion controlled type. For [Co(bpy)3]3+ immobilised on clay coated electrodes, a one-step oxidation and four-step reduction wave was observed corresponding to a one electron stepwise reversible reduction of Co(III), through Co(II), Co(I), Co(O) to Co(I) oxidation state. For [Cr(bpy)3]3+ the electrochemistry was complicated by the presence of additional waves corresponding to the dissociation of [Cr(bpy)3]3+ into the diaquo complex. ESR and diffuse reflectance data supported such a mechanism. 29Si, 27Al and 23Na MAS NMR spectroscopy, supported by powder XRD and FTIR, was used to probe the role of clays as reagents in glass forming reactions. 29Si MAS NMR was found to be a very sensitive technique for identifying the presence and relative abundance of crystalline and non-crystalline phases. In thermal reactions of laponite formation of new mineral phases such as forsterite, akermanite, sillimanite and diopside were detected. The relative abundance of each phase was dependent on thermal history, chemical nature and concentration of the modifier oxide present. In continuing work, the effect of selected oxides on the glass forming reactions of a model feldspar composition was investigated using solid state NMR alone. Addition of network modifying oxides generally produced less negative 29Si chemical shifts and larger linewidths corresponding to a wider distribution of Si-O-Si bond angles and lengths, and a dominant aluminosilicate phase with a less polymerised structure than the starting material. 29Si linewidths and 27Al chemical shifts were respectively correlated with cationic potential and Lewis acidity of the oxide cations. Anomalous Al(4) chemical shifts were thought to be due to precipitation of aluminate phases rather than a breakdown in Lowenstein's aluminium avoidance principle.

The manufacture and utilisation of architectural terracotta and faience.

The revival of terracotta and faience in British architecture was widespread, dramatic in its results and, for two decades, the subject of intense debate. However the materials have been frequently denigrated and more generally disregarded by both architects and historians. This study sets out to record and explain the rise and fall of interest in terracotta and faience, the extent and nature of the industry and the range of architectural usage in the Victorian, Edwardian and inter-war periods. The first two chapters record the faltering use of terracotta as an 'artificial stone', until the material gained its own identity, largely through the appreciation of Italian architecture. In the mid-Victorian period, terracotta will be seen to have become symbolic of the philosophy of the Victoria and Albert Museum and its Art School in attempting to reform both architecture and the decorative arts. The adoption of terracotta was furthered as much by industrial as aesthetic factors; three chapters examine how the exploitation of coal-measure clays, developments in the processes of manufacture, the changing motivation of industrialists and differing economics of production served to promote and then to hinder expansion and adaptation. The practical values of economy, durability and fire-resistance and the aesthetic potential, seen in terms of colour and decorative and sculptural modelling, became inter-related in the work of the architects who made extensive use of architectural ceramics. A correlation emerges between the free Gothic style, exemplified by the designs of Alfred Waterhouse and the use of red terracotta supplied from Ruabon, and between the eclectic Renaissance style and a buff material produced by different manufacturers.These patterns were modified as a result of the adoption of faience for facing external walls as well as interiors, and because of the new architectural requirements and tastes of the twentieth century. The general timidity in exploiting the scope for polychromatic decoration and the increasing opposition to architectural ceramics is contrasted with the most successful schemes produced for cinemas, chain-stores and factories. In the last chapter, those undertaken by the Hathern Station Brick and Terracotta Company between 1896 and 1939 are used as a case study; they confirm that manufacturers, architects and clients were all committed to creating a modern and yet decorative architecture, appropriate for new building types and that would appeal to and be comprehensible to the public.

The sedimentology and diagenesis of Lower Permian (Rotliegend) sediments: (onshore U.K. and southern North Sea). Available in 2 volumes.

The thesis provides a comparative study of both sedimentology and diagenesis of Lower Permian (Rotliegend) strata, onshore and offshore U.K. (Southern North Sea). Onshore formations studied include the Bridgnorth, Penrith and Hopeman Sandstone, and are dominated by aeolian facies, with lesser amounts of interbedded fluvial sediments. Aeolian and fluvial strata in onshore basins typically grade laterally into alluvial fan breccias at basin margins. Onshore basins represent proximal examples of Rotliegend desert sediments. The Leman Sandstone Formation of the Ravenspurn area in the Southern North Sea displays a variety of facies indicative of a distal sedimentological setting; Aeolian, fluvial, sabkha, and playa lake sediments all being present. "Sheet-like" geometry of stratigraphical units within the Leman Sandstone, and alternation of fluvial and aeolian deposition was climatically controlled. Major first order bounding surfaces are laterally extensive and were produced by lacustrine transgression and regression from the north-west. Diagenesis within Permian strata was studied using standard petrographic microscopy, scanning electron microscopy, cold cathodo-Iuminescence, X-ray diffraction clay analysis, X-ray fluorescence spectroscopy, fluid inclusion microthermometry, and K-Ar dating of illites. The diagenesis of Permian sediments within onshore basins is remarkably similar, and a paragenetic sequence of early haematite, illitic clays, feldspar, kaolinite, quartz and late calcite is observed. In the Leman Sandstone formation, authigenic mineralogy is complex and includes early quartz, sulphates and dolomite, chlorite, kaolinite, late quartz, illite and siderite. Primary lithological variation, facies type, and the interdigitation and location of facies within a basin are important initial controls upon diagenesis. Subsequently, burial history, structure, the timing of gas emplacement, and the nature of sediments within underlying formations may also exersize significant controls upon diagenesis within Rotliegend strata.

Effect of extrusion and photo-oxidation on polyethylene/clay nanocomposites

Polyethylene (a 1:1 blend of m-LLDPE and z-LLDPE) double layer silicate clay nanocomposites were prepared by melt extrusion using a twin screw extruder. Maleic anhydride grafted polyethylene (PEgMA) was used as a compatibiliser to enhance the dispersion of two organically modified monmorilonite clays (OMMT): Closite 15A (CL15) and nanofill SE 3000 (NF), and natural montmorillonite (NaMMT). The clay dispersion and morphology obtained in the extruded nanocomposite samples were fully characterised both after processing and during photo-oxidation by a number of complementary analytical techniques. The effects of the compatibiliser, the organoclay modifier (quartenary alkyl ammonium surfactant) and the clays on the behaviour of the nanocomposites during processing and under accelerated weathering conditions were investigated. X-ray diffraction, transmission electron microscopy (TEM), scanning electron microscopy (SEM), rheometry and attenuated reflectance spectroscopy (ATR-FTIR) showed that the nanocomposite structure obtained is dependent on the type of clay used, the presence or absence of a compatibiliser and the environment the samples are exposed to. The results revealed that during processing PE/clay nanocomposites are formed in the presence of the compatibiliser PEgMA giving a hybrid exfoliated and intercalated structures, while microcomposites were obtained in the absence of PEgMA; the unmodified NaMMT-containing samples showed encapsulated clay structures with limited extent of dispersion in the polymer matrix. The effect of processing on the thermal stability of the OMMT-containing polymer samples was determined by measuring the additional amount of vinyl-type unsaturation formed due to a Hoffman elimination reaction that takes place in the alkyl ammonium surfactant of the modified clay at elevated temperatures. The results indicate that OMMT is responsible for the higher levels of unsaturation found in OMMT-PE samples when compared to both the polymer control and the NaMMT-PE samples and confirms the instability of the alkyl ammonium surfactant during melt processing and its deleterious effects on the durability aspects of nanocomposite products. The photostability of the PE/clay nanocomposites under accelerated weathering conditions was monitored by following changes in their infrared signatures and mechanical properties. The rate of photo-oxidation of the compatibilised PE/PEgMA/OMMT nanocomposites was much higher than that of the PE/OMMT (in absence of PEgMA) counterparts, the polymer controls and the PE–NaMMT sample. Several factors have been observed that can explain the difference in the photo-oxidative stability of the PE/clay nanocomposites including the adverse role played by the thermal decomposition products of the alkyl ammonium surfactant, the photo-instability of PEgMA, unfavourable interactions between PEgMA and products formed in the polymer as a consequence of the degradation of the surfactant on the clay, as well as a contribution from a much higher extent of exfoliated structures, determined by TEM, formed with increasing UV-exposure times.

The role of pinning and instability in a class of non-equilibrium growth models

We study the dynamics of a growing crystalline facet where the growth mechanism is controlled by the geometry of the local curvature. A continuum model, in (2+1) dimensions, is developed in analogy with the Kardar-Parisi-Zhang (KPZ) model is considered for the purpose. Following standard coarse graining procedures, it is shown that in the large time, long distance limit, the continuum model predicts a curvature independent KPZ phase, thereby suppressing all explicit effects of curvature and local pinning in the system, in the "perturbative" limit. A direct numerical integration of this growth equation, in 1+1 dimensions, supports this observation below a critical parametric range, above which generic instabilities, in the form of isolated pillared structures lead to deviations from standard scaling behaviour. Possibilities of controlling this instability by introducing statistically "irrelevant" (in the sense of renormalisation groups) higher ordered nonlinearities have also been discussed.