Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

Increased charge transfer of Poly (ethylene oxide) based electrolyte by addition of small molecule and its application in dye-sensitized solar cells

A Poly (ethylene oxide) based polymer electrolyte impregnated with 2-Mercapto benzimidazole was comprehensively characterized by XRD, UV–visible spectroscopy, FTIR as well as electrochemical impedance spectroscopy. It was found that the crystallization of PEO was dramatically reduced and the ionic conductivity of the electrolyte was increased 4.5 fold by addition of 2-Mercapto benzimidazole. UV–visible and FTIR spectroscopes indicated the formation of charge transfer complex between 2-Mercapto benzimidazole and iodine of the electrolyte. Dye-sensitized solar cells with the polymer electrolytes were assembled. It was found that both the photocurrent density and photovoltage were enhanced with respect to the DSC without 2-Mercapto benzimidazole, leading to a 60% increase of the performance of the cell.

Influence of electrolyte cations on electron transport and electron transfer in dye-sensitized solar cells

The influence of different electrolyte cations ((Li+, Na+, Mg2+, tetrabutyl ammonium (TBA+)) on the TiO2 conduction band energy (Ec) the effective electron lifetime (τn), and the effective electron diffusion coefficient (Dn) in dye-sensitized solar cells (DSCs) was studied quantitatively. The separation between Ec and the redox Fermi level, EF,redox, was found to decrease as the charge/radius ratio of the cations increased. Ec in the Mg2+ electrolyte was found to be 170 meV lower than that in the Na+ electrolyte and 400 meV lower than that in the TBA+ electrolyte. Comparison of Dn and τn in the different electrolytes was carried out by using the trapped electron concentration as a measure of the energy difference between Ec and the quasi-Fermi level, nEF, under different illumination levels. Plots of Dn as a function of the trapped electron density, nt, were found to be relatively insensitive to the electrolyte cation, indicating that the density and energetic distribution of electron traps in TiO2 are similar in all of the electrolytes studied. By contrast, plots of τn versus nt for the different cations showed that the rate of electron back reaction is more than an order of magnitude faster in the TBA+ electrolyte compared with the Na+ and Li+ electrolytes. The electron diffusion lengths in the different electrolytes followed the sequence of Na+ > Li+ > Mg2+ > TBA+. The trends observed in the AM 1.5 current–voltage characteristics of the DSCs are rationalized on the basis of the conduction band shifts and changes in electron lifetime.

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

A new optimal control model of the interactions between a growing tumour and the host immune system along with an immunotherapy treatment strategy is presented. The model is based on an ordinary differential equation model of interactions between the growing tu- mour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treat- ment to the system. The numerical solution of this model, obtained from a multi species Runge–Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum al- lowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour.

Study of the underlying electrochemistry of polycrystalline gold electrodes in aqueous solution and electrocatalysis by large amplitude Fourier transformed alternating current voltammetry

Polycrystalline gold electrodes of the kind that are routinely used in analysis and catalysis in aqueous media are often regarded as exhibiting relatively simple double-layer charging/discharging and monolayer oxide formation/ removal in the positive potential region. Application of the large amplitude Fourier transformed alternating current (FT-ac) voltammetric technique that allows the faradaic current contribution of fast electron-transfer processes to be emphasized in the higher harmonic components has revealed the presence of well-defined faradaic (premonolayer oxidation) processes at positive potentials in the double-layer region in acidic and basic media which are enhanced by electrochemical activation. These underlying quasi-reversible interfacial electron-transfer processes may mediate the course of electrocatalytic oxidation reactions of hydrazine, ethylene glycol, and glucose on gold electrodes in aqueous media. The observed responses support key assumptions associated with the incipient hydrous oxide adatom mediator (IHOAM) model of electrocatalysis.

Auto-inhibition of hydrogen gas evolution on gold in aqueous acid solution

It was demonstrated recently that dramatic changes in the redox behaviour of gold/aqueous solution interfaces may be observed following either cathodic or thermal electrode pretreatment. Further work on the cathodic pretreatment of gold in acid solution revealed that as the activity of the gold surface was increased, its performance as a substrate for hydrogen gas evolution under constant potential conditions deteriorated. The change in activity of the gold atoms at the interface, which was attributed to a hydrogen embrittlement process (the occurrence of the latter was subsequently checked by surface microscopy), was confirmed, as in earlier work, by the appearance of a substantial anodic peak at ca. 0.5 V (RHE) in a post-activation positive sweep. Changes in the catalytic activity of a metal surface reflect the fact that the structure (or topography), thermodynamic activity and electronic properties of a surface are dependent not only on pretreatment but also, in the case of the hydrogen evolution reaction, vary with time during the course of reaction. As will be reported shortly, similar (and often more dramatic) time-dependent behaviour was observed for hydrogen gas evolution on other metal electrodes.

The role of defects, or active states, in surface electrochemistry with particular reference to gold in neutral solution

Metastable, active, or nonequilibrium states due to the presence of abnormal structures and various types of defects are well known in metallurgy. The role of such states at gold surfaces in neutral aqueous media (an important electrode system in the microsensor area) was explored using cyclic voltammetry. It was demonstrated that, as postulated in earlier work from this laboratory, there is a close relationship between premonolayer oxidation, multilayer hydrous oxide reduction and electrocatalytic behaviour in the case of this and other metal electrode systems. Some of the most active, and therefore most important, entities at surfaces (e.g., metal adatoms) are not readily imageable or detectable by high resolution surface microscopy techniques. Cyclic voltammetry, however, provides significant, though not highly specific, information about such species. The main conclusion is that further practical and theoretical work on active states of metal surfaces is highly desirable as their behaviour is not simple and is of major importance in many electrocatalytic processes.

The formation of gold nanoparticles using hydroquinone as a reducing agent through a localized pH change upon addition of NaOH to a solution of HAuCl4

We demonstrate a rapid synthesis of gold nanoparticles using hydroquinone as a reducing agent under acidic conditions without the need for precursor seed particles. The nanoparticle formation process is facilitated by the addition of NaOH to a solution containing HAuCl4 and hydroquinone to locally change the pH; this enhances the reducing capability of hydroquinone to form gold nucleation centres, after which further growth of gold can take place through an autocatalytic mechanism. The stability of the nanoparticles is highly dependent on the initial solution pH, and both the concentration of added NaOH and hydroquinone present in solution. The gold nanoparticles were characterized by UV–visible spectroscopy, transmission electron microscopy, Fourier transform infrared spectroscopy, atomic force microscopy, dynamic light scattering, and zeta potential measurements. It was found that under optimal conditions that stable aqueous suspensions of 20 nm diameter nanoparticles can be achieved where benzoquinone, the oxidized product of hydroquinone, acts as a capping agent preventing nanoparticles aggregation.

Voltammetric monitoring of gold nanoparticle formation facilitated by glycyl-L-tyrosine : relation to electronic spectra and transmission electron microscopy images

Voltammetric techniques have been introduced to monitor the formation of gold nanoparticles produced via the reaction of the amino acid glycyl-L-tyrosine with Au(III) (bromoaurate) in 0.05 M KOH conditions. The alkaline conditions facilitate amino acid binding to Au(III), inhibit the rate of reduction to Au(0), and provide an excellent supporting electrolyte for voltammetric studies. Data obtained revealed that a range of time-dependent gold solution species are involved in gold nanoparticle formation and that the order in which reagents are mixed is critical to the outcome. Concomitantly with voltammetric measurements, the properties of gold nanoparticles formed are probed by examination of electronic spectra in order to understand how the solution environment present during nanoparticle growth affects the final distribution of the nanoparticles. Images obtained by the ex situ transmission electron microscopy (TEM) technique enable the physical properties of the nanoparticles isolated in the solid state to be assessed. Use of this combination of in situ and ex situ techniques provides a versatile framework for elucidating the details of nanoparticle formation.

Electrochemically-induced TCNQ/Mn TCNQ 2 (H2O) 2 (TCNQ= 7, 7, 8, 8-Tetracyanoquinodimethane) solid-solid interconversion : two voltammetrically distinct processes that allow selective generation of nanofiber or nanorod network morphologies

Unlike the case with other divalent transition metal M\[TCNQ](2)(H(2)O)(2) (M = Fe, Co, Ni) analogues, the electrochemically induced solid-solid phase interconversion of TCNQ microcrystals (TCNQ = 7,7,8,8-tetracyanoquinodimethane) to Mn\[TCNQ](2)(H(2)O)(2) occurs via two voltammetrically distinct, time dependent processes that generate the coordination polymer in nanofiber or rod-like morphologies. Careful manipulation of the voltammetric scan rate, electrolysis time, Mn(2+)((aq)) concentration, and the method of electrode modification with solid TCNQ allows selective generation of either morphology. Detailed ex situ spectroscopic (IR, Raman), scanning electron microscopy (SEM), and X-ray powder diffraction (XRD) characterization clearly establish that differences in the electrochemically synthesized Mn-TCNQ material are confined to morphology. Generation of the nanofiber form is proposed to take place rapidly via formation and reduction of a Mn-stabilized anionic dimer intermediate, \[(Mn(2+))(TCNQ-TCNQ)(2)(*-)], formed as a result of radical-substrate coupling between TCNQ(*-) and neutral TCNQ, accompanied by ingress of Mn(2+) ions from the aqueous solution at the triple phase TCNQ/electrode/electrolyte boundary. In contrast, formation of the nanorod form is much slower and is postulated to arise from disproportionation of the \[(Mn(2+))(TCNQ-TCNQ)(*-)(2)] intermediate. Thus, identification of the time dependent pathways via the solid-solid state electrochemical approach allows the crystal size of the Mn\[TCNQ](2)(H(2)O)(2) material to be tuned and provides new mechanistic insights into the formation of different morphologies.

The facile formation of silver dendritic structures in the absence of surfactants and their electrochemical and SERS properties

In this work a simple approach to the creation of highly dispersed electrocatalytically active silver microstructured dendrites on indium tin oxide in the absence of any surface modification or surfactant is presented. It is found that the addition of low concentrations of supporting electrolyte to the AgNO3 solution dramatically influences the morphology of electrodeposited silver which is independent of both the anion and the cation employed. The silver dendrites are characterized by SEM, XRD, XPS as well as by cyclic voltammetry under alkaline conditions. It is found that the surface oxide formation and removal processes are significantly influenced by the microstructured morphology of the silver electrodeposits compared to a smooth macrosized silver electrode. The facile formation of dendritic silver microstructures is also shown to be beneficial for the electrocatalytic oxidation of both formaldehyde and hydrazine and oxygen reduction. The formation of a continuous film of dendritic silver is also investigated for its SERS activity where the connectivity between the individual dendrites is found to be particularly important.

Study of the initial stage of solid electrolyte interphase formation upon chemical reaction of lithium metal and N-Methyl-N-Propyl-Pyrrolidinium-Bis (Fluorosulfonyl) Imide

Chemical reaction studies of N-methyl-N-propyl-pyrrolidinium-bis(fluorosulfonyl)imide-based ionic liquid with the lithium metal surface were performed using ab initio molecular dynamics (aMD) simulations and X-ray Photoelectron Spectroscopy (XPS). The molecular dynamics simulations showed rapid and spontaneous decomposition of the ionic liquid anion, with subsequent formation of long-lived species such as lithium fluoride. The simulations also revealed the cation to retain its structure by generally moving away from the lithium surface. The XPS experiments showed evidence of decomposition of the anion, consistent with the aMD simulations and also of cation decomposition and it is envisaged that this is due to the longer time scale for the XPS experiment compared to the time scale of the aMD simulation. Overall experimental results confirm the majority of species suggested by the simulation. The rapid chemical decomposition of the ionic liquid was shown to form a solid electrolyte interphase composed of the breakdown products of the ionic liquid components in the absence of an applied voltage.