The persistence of phase-separation in LiFePO4 with two-dimensional Li+ transport : the Cahn-Hilliard-reaction equation and the role of defects

We examine the solution of the two-dimensional Cahn-Hilliard-reaction (CHR) equation in the xy plane as a model of Li+ intercalation into LiFePO4 material. We validate our numerical solution against the solution of the depth-averaged equation, which has been used to model intercalation in the limit of highly orthotropic diffusivity and gradient penalty tensors. We then examine the phase-change behaviour in the full CHR system as these parameters become more isotropic, and find that as the Li+ diffusivity is increased in the x direction, phase separation persists at high currents, even in small crystals with averaged coherency strain included. The resulting voltage curves decrease monotonically, which has previously been considered a hallmark of crystals that fill homogeneously.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

Two novel numerical methods for solving the two-dimensional fractional Fitzhugh-Nagumo-monodomain model

Fractional mathematical models represent a new approach to modelling complex spatial problems in which there is heterogeneity at many spatial and temporal scales. In this paper, a two-dimensional fractional Fitzhugh-Nagumo-monodomain model with zero Dirichlet boundary conditions is considered. The model consists of a coupled space fractional diffusion equation (SFDE) and an ordinary differential equation. For the SFDE, we first consider the numerical solution of the Riesz fractional nonlinear reaction-diffusion model and compare it to the solution of a fractional in space nonlinear reaction-diffusion model. We present two novel numerical methods for the two-dimensional fractional Fitzhugh-Nagumo-monodomain model using the shifted Grunwald-Letnikov method and the matrix transform method, respectively. Finally, some numerical examples are given to exhibit the consistency of our computational solution methodologies. The numerical results demonstrate the effectiveness of the methods.

Two-dimensional graphene heterojunctions: The tunable mechanical properties

We report the mechanical properties of different two-dimensional carbon heterojunctions (HJs) made from graphene and various stable graphene allotropes, including α-, β-, γ- and 6612-graphyne (GY), and graphdiyne (GDY). It is found that all HJs exhibit a brittle behaviour except the one with α-GY, which however shows a hardening process due to the formation of triple carbon rings. Such hardening process has greatly deferred the failure of the structure. The yielding of the HJs is usually initiated at the interface between graphene and graphene allotropes, and monoatomic carbon rings are normally formed after yielding. By varying the locations of graphene (either in the middle or at the two ends of the HJs), similar mechanical properties have been obtained, suggesting insignificant impacts from location of graphene allotropes. Whereas, changing the types and percentages of the graphene allotropes, the HJs exhibit vastly different mechanical properties. In general, with the increasing graphene percentage, the yield strain decreases and the effective Young’s modulus increases. Meanwhile, the yield stress appears irrelevant with the graphene percentage. This study provides a fundamental understanding of the tensile properties of the heterojunctions that are crucial for the design and engineering of their mechanical properties, in order to facilitate their emerging future applications in nanoscale devices, such as flexible/stretchable electronics.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Diagnostics and two-dimensional simulation of low-frequency inductively coupled plasmas with neutral gas heating and electron heat fluxes

This article presents the results on the diagnostics and numerical modeling of low-frequency (∼460 KHz) inductively coupled plasmas generated in a cylindrical metal chamber by an external flat spiral coil. Experimental data on the electron number densities and temperatures, electron energy distribution functions, and optical emission intensities of the abundant plasma species in low/intermediate pressure argon discharges are included. The spatial profiles of the plasma density, electron temperature, and excited argon species are computed, for different rf powers and working gas pressures, using the two-dimensional fluid approach. The model allows one to achieve a reasonable agreement between the computed and experimental data. The effect of the neutral gas temperature on the plasma parameters is also investigated. It is shown that neutral gas heating (at rf powers≥0.55kW) is one of the key factors that control the electron number density and temperature. The dependence of the average rf power loss, per electron-ion pair created, on the working gas pressure shows that the electron heat flux to the walls appears to be a critical factor in the total power loss in the discharge.

Reactive species in Ar+H2 plasma-aided nanofabrication : two-dimensional discharge modelling

Understanding the generation of reactive species in a plasma is an important step towards creating reliable and robust plasma-aided nanofabrication processes. A two-dimensional fluid simulation of the number densities of surface preparation species in a low-temperature, low-pressure, non-equilibrium Ar+H2 plasma is conducted. The operating pressure and H2 partial pressure have been varied between 70-200 mTorr and 0.1-50%, respectively. An emphasis is placed on the application of these results to nanofabrication. A reasonable balance between operating pressures and H 2 partial pressures that would optimize the number densities of the two working units largely responsible for activation and passivation of surface dangling bonds (Ar+ and H respectively) in order to achieve acceptable rates of surface activation and passivation is obtained. It is found that higher operating pressures (150-200 mTorr) and lower H2 partial pressures (∼5%) are required in order to ensure high number densities of Ar+ and H species. This paper contributes to the improvement of the controllability and predictability of plasma-based nanoassembly processes.

Two-dimensional simulation of nanoassembly precursor species in Ar+H2+C2H2 reactive plasmas

The results of two-dimensional fluid simulation of number densities and fluxes of the main building blocks and surface preparation species involved in nanoassembly of carbon-based nanopatterns in Ar+H2+C2H2 reactive plasmas are reported. It is shown that the process parameters and non-uniformity of surface fluxes of each particular species may affect the targeted nanopattern quality. The results can be used to improve predictability of plasma-aided nanofabrication processes and optimize the parameters of plasma nanotools.KGaA, Weinheim.

Thermal conductivity of configurable two-dimensional carbon nanotube architecture and strain modulation

We reported the thermal conductivity of the two-dimensional carbon nanotube (CNT)-based architecture, which can be constructed through welding of single-wall CNTs by electron beam. Using large-scale nonequilibrium molecular dynamics simulations, the thermal conductivity is found to vary with different junction types due to their different phonon scatterings at the junction. The strong length and strain dependence of the thermal conductivity suggests an effective avenue to tune the thermal transport properties of the CNT-based architecture, benefiting the design of nanoscale thermal rectifiers or phonon engineering.

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.