A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Two-dimensional hydrogen-bonded polymers in the crystal structures of the ammonium salts of phenoxyacetic acid, (4-fluorophenoxy)acetic acid and (4-chloro-2-methylphenoxy)acetic acid

The structures of the ammonium salts of phenoxyacetic acid, NH4+ C8H6O3- (I), (4-fluorophenoxy)acetic acid NH4+ C8H5FO3- (II) and the herbicidally active (4-chloro-2-methylphenoxy)acetic acid (MCPA), NH4+ C9H8ClO3-. 0.5(H2O) (III) have been determined. All have two-dimensional layered structures based on inter-species ammonium N-H...O hydrogen-bonding associations which give core substructures consisting primarily of conjoined cyclic motifs. Crystals of (I) and (II) are isomorphous with the core comprising R2/1(5), R2/1(4) and centrosymmetric R2/4(8) ring motifs, giving two-dimensional layers lying parallel to (100). In (III), the water molecule of solvation lies on a crystallographic twofold rotation axis and bridges two carboxyl O-atoms in an R4/4(12) hydrogen-bonded motif, creating two R3/4(10) rings which together with a conjoined centrosymmetric R2/4(8) ring incorporating both ammonium cations, generate two-dimensional layers lying parallel to (100). No pi-pi ring associations are present in any of the structures.

Two-dimensional bow and stern flows in a finite-depth fluid with constant vorticity

This thesis is concerned with two-dimensional free surface flows past semi-infinite surface-piercing bodies in a fluid of finite-depth. Throughout the study, it is assumed that the fluid in question is incompressible, and that the effects of viscosity and surface tension are negligible. The problems considered are physically important, since they can be used to model the flow of water near the bow or stern of a wide, blunt ship. Alternatively, the solutions can be interpreted as describing the flow into, or out of, a horizontal slot. In the past, all research conducted on this topic has been dedicated to the situation where the flow is irrotational. The results from such studies are extended here, by allowing the fluid to have constant vorticity throughout the flow domain. In addition, new results for irrotational flow are also presented. When studying the flow of a fluid past a surface-piercing body, it is important to stipulate in advance the nature of the free surface as it intersects the body. Three different possibilities are considered in this thesis. In the first of these possibilities, it is assumed that the free surface rises up and meets the body at a stagnation point. For this configuration, the nonlinear problem is solved numerically with the use of a boundary integral method in the physical plane. Here the semi-infinite body is assumed to be rectangular in shape, with a rounded corner. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterised by a train of waves upstream. In the limit that the height of the body above the horizontal bottom vanishes, the flow approaches that due to a submerged line sink in a $90^\circ$ corner. This limiting problem is also examined as a special case. The second configuration considered in this thesis involves the free surface attaching smoothly to the front face of the rectangular shaped body. For this configuration, nonlinear solutions are computed using a similar numerical scheme to that used in the stagnant attachment case. It is found that these solution exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a corner is also considered. Finally, the flow of a fluid emerging from beneath a semi-infinite flat plate is examined. Here the free surface is assumed to detach from the trailing edge of the plate horizontally. A linear problem is formulated under the assumption that the elevation of the plate is close to the undisturbed free surface level. This problem is solved exactly using the Wiener-Hopf technique, and subcritical solutions are found which are characterised by a train of sinusoidal waves in the far field. The nonlinear problem is also considered. Exact relations between certain parameters for supercritical flow are derived using conservation of mass and momentum arguments, and these are confirmed numerically. Nonlinear subcritical solutions are computed, and the results are compared to those predicted by the linear theory.

Numerical treatment of a two-dimensional variable-order fractional nonlinear reaction-diffusion model

A two-dimensional variable-order fractional nonlinear reaction-diffusion model is considered. A second-order spatial accurate semi-implicit alternating direction method for a two-dimensional variable-order fractional nonlinear reaction-diffusion model is proposed. Stability and convergence of the semi-implicit alternating direct method are established. Finally, some numerical examples are given to support our theoretical analysis. These numerical techniques can be used to simulate a two-dimensional variable order fractional FitzHugh-Nagumo model in a rectangular domain. This type of model can be used to describe how electrical currents flow through the heart, controlling its contractions, and are used to ascertain the effects of certain drugs designed to treat arrhythmia.

The role of tunnel junction resistances and defects on electron transport mechanism in networks of two-dimensional disordered conductors

The effect of tunnel junction resistances on the electronic property and the magneto-resistance of few-layer graphene sheet networks is investigated. By decreasing the tunnel junction resistances, transition from strong localization to weak localization occurs and magneto-resistance changes from positive to negative. It is shown that the positive magneto-resistance is due to Zeeman splitting of the electronic states at the Fermi level as it changes with the bias voltage. As the tunnel junction resistances decrease, the network resistance is well described by 2D weak localization model. Sensitivity of the magneto-resistance to the bias voltage becomes negligible and diminishes with increasing temperature. It is shown 2D weak localization effect mainly occurs inside of the few-layer graphene sheets and the minimum temperature of 5 K in our experiments is not sufficiently low to allow us to observe 2D weak localization effect of the networks as it occurs in 2D disordered metal films. Furthermore, defects inside the few-layer graphene sheets have negligible effect on the resistance of the networks which have small tunnel junction resistances between few-layer graphene sheets

Two-dimensional coordination polymeric structures in caesium complexes with ring-substituted phenoxyacetic acids

The two-dimensional polymeric structures of the caesium complexes with the phenoxyacetic acid analogues (4-fluorophenoxy)acetic acid, (3-chloro-2-methylphenoxy)acetic acid and the herbicidally active (2,4-dichlorophen­oxy)acetic acid (2,4-D), namely poly[[5-(4-fluorophenoxy)acetato][4-(4-fluorophenoxy)acetato]dicaesium], [Cs2(C8H6FO3)2]n, (I), poly[aqua[5-(3-chloro-2-methylphenoxy)acetato]caesium], [Cs(C9H8ClO3)(H2O)]n, (II), and poly[[7-(2,4-di­chlorophenoxy)acetato][(2,4-dichlorphenoxy)acetic acid]caesium], [Cs(C8H5Cl2O3)(C8H6Cl2O3)]n, (III), are described. In (I), the Cs+ cations of the two individual irregular coordination polyhedra in the asymmetric unit (one CsO7 and the other CsO8) are linked by bridging carboxylate O-atom donors from the two ligand molecules, both of which are involved in bidentate chelate Ocarboxy,Ophenoxy interactions, while only one has a bidentate carboxylate O,O'-chelate inter­action. Polymeric extension is achieved through a number of carboxylate O-atom bridges, with a minimum CsCs separation of 4.3231 (9) Å, giving layers which lie parallel to (001). In hydrated complex (II), the irregular nine-coordination about the Cs+ cation comprises a single monodentate water molecule, a bidentate Ocarboxy,Ophenoxy chelate interaction and six bridging carboxylate O-atom bonding interactions, giving a CsCs separation of 4.2473 (3) Å. The water mol­ecule forms intralayer hydrogen bonds within the two-dimensional layers, which lie parallel to (100). In complex (III), the irregular centrosymmetric CsO6Cl2 coordination environment comprises two O-atom donors and two ring-substituted Cl-atom donors from two hydrogen bis[(2,4-dichlorophenoxy)acetate] ligand species in a bidentate chelate mode, and four O-atom donors from bridging carboxyl groups. The duplex ligand species lie across crystallographic inversion centres, linked through a short O-HO hydrogen bond involving the single acid H atom. Structure extension gives layers which lie parallel to (001). The present set of structures of Cs salts of phenoxyacetic acids show previously demonstrated trends among the alkali metal salts of simple benzoic acids with no stereochemically favourable interactive substituent groups for formation of two-dimensional coordination polymers.

Two-dimensional self-assembly of a symmetry-reduced tricarboxylic acid

Investigations of the self-assembly of simple molecules at the solution/solid interface can provide useful insight into the general principles governing supramolecular chemistry in two dimensions. Here, we report on the assembly of 3,4′,5-biphenyl tricarboxylic acid (H3BHTC), a small hydrogen bonding unit related to the much-studied 1,3,5-benzenetricarboxylic acid (trimesic acid, TMA), which we investigate using scanning tunneling microscopy (STM) and density functional theory (DFT) calculations. STM images show that H3BHTC assembles by itself into an offset zigzag chain structure that maximizes the surface molecular density in favor of maximizing the number density of strong cyclic hydrogen bonds between the carboxylic groups. The offset geometry creates “sticky” pores that promote solvent coadsorption. Adding coronene to the molecular solution produces a transformation to a high-symmetry host–guest lattice stabilized by a dimeric/trimeric hydrogen bonding motif similar to the TMA flower structure. Finally, we show that the H3BHTC lattice firmly immobilizes the guest coronene molecules, allowing for high-resolution imaging of the coronene structure.

Predicting a new phase (T′′) of two-dimensional transition metal di-chalcogenides and strain-controlled topological phase transition

Single layered transition metal dichalcogenides have attracted tremendous research interest due to their structural phase diversities. By using a global optimization approach, we have discovered a new phase of transition metal dichalcogenides (labelled as T′′), which is confirmed to be energetically, dynamically and kinetically stable by our first-principles calculations. The new T′′ MoS2 phase exhibits an intrinsic quantum spin Hall (QSH) effect with a nontrivial gap as large as 0.42 eV, suggesting that a two-dimensional (2D) topological insulator can be achieved at room temperature. Most interestingly, there is a topological phase transition simply driven by a small tensile strain of up to 2%. Furthermore, all the known MX2 (M = Mo or W; X = S, Se or Te) monolayers in the new T′′ phase unambiguously display similar band topologies and strain controlled topological phase transitions. Our findings greatly enrich the 2D families of transition metal dichalcogenides and offer a feasible way to control the electronic states of 2D topological insulators for the fabrication of high-speed spintronics devices.

Two-dimensional tin disulfide nanosheets for enhanced sodium storage

Sodium-ion batteries (SIBs) are considered as complementary alternatives to lithium-ion batteries for grid energy storage due to the abundance of sodium. However, low capacity, poor rate capability, and cycling stability of existing anodes significantly hinder the practical applications of SIBs. Herein, ultrathin two-dimensional SnS2 nanosheets (3-4 nm in thickness) are synthesized via a facile refluxing process toward enhanced sodium storage. The SnS2 nanosheets exhibit a high apparent diffusion coefficient of Na+ and fast sodiation/desodiation reaction kinetics. In half-cells, the nanosheets deliver a high reversible capacity of 733 mAh g-1 at 0.1 A g-1, which still remains up to 435 mAh g-1 at 2 A g-1. The cell has a high capacity retention of 647 mA h g-1 during the 50th cycle at 0.1 A g-1, which is by far the best for SnS2, suggesting that nanosheet morphology is beneficial to improve cycling stability in addition to rate capability. The SnS2 nanosheets also show encouraging performance in a full cell with a Na3V2(PO4)3 cathode. In addition, the sodium storage mechanism is investigated by ex situ XRD coupled with high-resolution TEM. The high specific capacity, good rate capability, and cycling durability suggest that SnS2 nanosheets have great potential working as anodes for high-performance SIBs. © 2015 American Chemical Society.

Generalized self-assembly of scalable two-dimensional transition metal oxide nanosheets

Two-dimensional (2D) transition metal oxide systems present exotic electronic properties and high specific surface areas, and also demonstrate promising applications ranging from electronics to energy storage. Yet, in contrast to other types of nanostructures, the question as to whether we could assemble 2D nanomaterials with an atomic thickness from molecules in a general way, which may give them some interesting properties such as those of graphene, still remains unresolved. Herein, we report a generalized and fundamental approach to molecular self-assembly synthesis of ultrathin 2D nanosheets of transition metal oxides by rationally employing lamellar reverse micelles. It is worth emphasizing that the synthesized crystallized ultrathin transition metal oxide nanosheets possess confined thickness, high specific surface area and chemically reactive facets, so that they could have promising applications in nanostructured electronics, photonics, sensors, and energy conversion and storage devices.

Graphene-like two-dimensional ionic boron with double dirac cones at ambient condition

Recently, partially ionic boron (γ-B28) has been predicted and observed in pure boron, in bulk phase and controlled by pressure [Nature, 457 (2009) 863]. By using ab initio evolutionary structure search, we report the prediction of ionic boron at a reduced dimension and ambient pressure, namely, the two-dimensional (2D) ionic boron. This 2D boron structure consists of graphene-like plane and B2 atom pairs, with the P6/mmm space group and 6 atoms in the unit cell, and has lower energy than the previously reported α-sheet structure and its analogues. Its dynamical and thermal stability are confirmed by the phonon-spectrum and ab initio molecular dynamics simulation. In addition, this phase exhibits double Dirac cones with massless Dirac fermions due to the significant charge transfer between the graphene-like plane and B2 pair that enhances the energetic stability of the P6/mmm boron. A Fermi velocity (vf) as high as 2.3 x 106 m/s, which is even higher than that of graphene (0.82 x 106 m/s), is predicted for the P6/mmm boron. The present work is the first report of the 2D ionic boron at atmospheric pressure. The unique electronic structure renders the 2D ionic boron a promising 2D material for applications in nanoelectronics.

Versatile single-layer sodium phosphidostannate(II): Strain-tunable electronic structure, excellent mechanical flexibility, and an ideal gap for photovoltaics

Density functional theory (DFT) calculations were performed to study the structural, mechanical, electrical, optical properties, and strain effects in single-layer sodium phosphidostannate(II) (NaSnP). We find the exfoliation of single-layer NaSnP from bulk form is highly feasible because the cleavage energy is comparable to graphite and MoS2. In addition, the breaking strain of the NaSnP monolayer is comparable to other widely studied 2D materials, indicating excellent mechanical flexibility of 2D NaSnP. Using the hybrid functional method, the calculated band gap of single-layer NaSnP is close to the ideal band gap of solar cell materials (1.5 eV), demonstrating great potential in future photovoltaic application. Furthermore, strain effect study shows that a moderate compression (2%) can trigger indirect-to-direct gap transition, which would enhance the ability of light absorption for the NaSnP monolayer. With sufficient compression (8%), the single-layer NaSnP can be tuned from semiconductor to metal, suggesting great applications in nanoelectronic devices based on strain engineering techniques.

Ink-jet printing of graphene for flexible electronics: An environmentally-friendly approach

Mechanical flexibility is considered an asset in consumer electronics and next-generation electronic systems. Printed and flexible electronic devices could be embedded into clothing or other surfaces at home or office or in many products such as low-cost sensors integrated in transparent and flexible surfaces. In this context inks based on graphene and related two-dimensional materials (2DMs) are gaining increasing attention owing to their exceptional (opto)electronic, electrochemical and mechanical properties. The current limitation relies on the use of solvents, providing stable dispersions of graphene and 2DMs and fitting the proper fluidic requirements for printing, which are in general not environmentally benign, and with high boiling point. Non-toxic and low boiling point solvents do not possess the required rheological properties (i.e., surface tension, viscosity and density) for the solution processing of graphene and 2DMs. Such solvents (e.g., water, alcohols) require the addition of stabilizing agents such as polymers or surfactants for the dispersion of graphene and 2DMs, which however unavoidably corrupt their properties, thus preventing their use for the target application. Here, we demonstrate a viable strategy to tune the fluidic properties of water/ethanol mixtures (low-boiling point solvents) to first effectively exfoliate graphite and then disperse graphene flakes to formulate graphene-based inks. We demonstrate that such inks can be used to print conductive stripes (sheet resistance of ~13 kΩ/□) on flexible substrates (polyethylene terephthalate), moving a step forward towards the realization of graphene-based printed electronic devices.