A fourth-order accurate, Numerov-type, three-point finite-difference discretization of electrochemical reaction-diffusion equations on nonuniform (exponentially expanding) spatial grids in one-dimensional space geometry

The validity for finite-difference electrochemical kinetic simulations, of the extension of the Numerov discretization designed by Chawla and Katti [J Comput Appl Math 1980, 6, 189-196] for the solution of two-point boundary value problems in ordinary differential equations, is examined. The discretization is adapted to systems of time-dependent reaction-diffusion partial differential equations in one-dimensional space geometry, on nonuniform space grids resulting from coordinate transformations. The equations must not involve first spatial derivatives of the unknowns. Relevant discrete formulae are outlined and tested in calculations on two example kinetic models. The models describe potential step chronoamperometry under limiting current conditions, for the catalytic EC, and Reinert-Berg CE reaction mechanisms. Exponentially expanding space grid is used. The discretization considered proves the most accurate and efficient, out of all the three-point finite-difference discretizations on such grids, that have been used thus far in electrochemical kinetics. Therefore, it can be recommended as a method of choice.     

Multidimensional quantum trajectories: applications of the derivative propagation method

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.     

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Patch-adaptive simulation of moving fronts in non-linear diffusion models of the switching of conductive polymers

The finite-difference patch-adaptive strategy (PAS) for electrochemical kinetic simulations, previously described by the author, is extended and applied to two examples of non-linear diffusion in one-dimensional space geometry, characterised by moving fronts, and related to the modelling of redox switching of conductive polymers. The extension consists of an appropriate spatial discretisation of the second derivative diffusion term involving concentration-dependent diffusion coefficient, and of an improved selection of starting approximations for the Newton iterations within the extrapolation time-stepping scheme. Adaptive solutions obtained are reliable, accurate, efficient, and superior to fixed-grid solutions.     

The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry

In this paper, we have applied an accurate and efficient wavelet scheme (due to Legendre polynomial) to find the numerical solutions for a set of coupled reaction–diffusion equations. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of non linear terms appearing in the governing differential equations. The highest derivative in the differential equation is expanded into wavelet series, this approximation is then integrated while the boundary conditions are applied by using integration constants. With the help of operational matrices, the nonlinear reaction–diffusion equations are converted into a system of algebraic equations. Finally, some numerical examples to demonstrate the validity and applicability of the method have been furnished. The use of Legendre wavelets is found to be accurate, efficient, simple, and computationally attractive. This wavelet method can be used for obtaining quick solution in many chemical Engineering problems.     

Numerical simulation of coupled‐stress case II diffusion in one dimension

This article is concerned with the robust and efficient numerical simulation of case II diffusion, which constitutes an important regime of solvent diffusion into glassy polymers. Even in the one‐dimensional case considered here, the numerical simulation of case II diffusion is made difficult by the extreme nonlinearities and coupling in the governing model equations due to a nonlinear flux law needed to produce sharp solvent fronts, a concentration‐dependent relaxation time of the polymer used to model the glass‐rubber transition, and coupling between the diffusion and deformation phenomena. Having an efficient and accurate solution to such equations is central to advancing a clear understanding of the meaning of such models. The difficulties due to coupling and nonlinearities are highlighted by the consideration of a specific, normalized, one‐dimensional case II diffusion model laid out in a general framework of balance laws. Issues such as the stiffness of the spatially discrete differential algebraic equations obtained from the finite element discretization of the governing equations and their bearing on the choice of time‐stepping schemes are discussed. The key requirements of numerical schemes, namely, robustness and efficiency, are addressed by the use of an implicit, adaptive, second‐order backward differentiation formula with error control for time discretization. Error control is used to maximize the step size to satisfy a target error and the radius of convergence requirements while nonlinear algebraic equations are solved at each time step. An example of an initial boundary value problem is solved numerically to show that the chosen model reproduces case II behavior and to validate that the stated objectives for the numerical simulation are met. Finally, the features and numerical implementation of this model are compared with those of a closely related case II diffusion model due to Wu and Peppas. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2091–2108, 2003     

Numerical solution of the two-dimensional time independent Schrödinger equation with Numerov-type methods

The solution of the two-dimensional time-independent Schrödinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and Numerov type methods are used to solve it. Specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe et al. [Int. J. Comp. Math 32 (1990) 233–242], and the minimum phase-lag method of Rao et al. [Int. J. Comp. Math 37 (1990) 63–77] are applied to this problem. All methods are applied for the computation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon–Heils potential. The results are compared with the results produced by full discterization.     

Wavelet Derivative: Application in Multicomponent Analysis of Electrochemical Signals

The wavelet derivative was developed for resolving overlapped signals in electrochemical analysis. It has been shown that the wavelet derivative could be used as an effective tool in resolution enhancement of the electrochemical signals. Since signal‐to‐noise ratio (SNR) does not degrade significantly at high order derivatives, the wavelet derivative is much better than the conventional numerical derivative in resolving practical signals with low SNR, particularly in the case when the overlapped degree of complex is high and high order derivatives are required for the further resolution enhancement. In this article, applications of the wavelet derivative in the voltammetric determination of small mounts of substances in the presence of large mounts of potentially interfering species are presented, and the advantages of this technique are discussed as well.     

Electrochemical Conversions of 2-Methyl-3-nitro-4-phenylquinoline, Its Quinolinium Salts, and Hydrogenated Derivatives

Classical polarography, cyclic voltammetry, and EPR spectroscopy was used to study electrochemical reduction and oxidation of 3-nitro derivatives of 2-methyl-4-phenylquinoline, the corresponding quinolinium perchlorates, and 1,2- and 1,4-dihydroquinolines. The nitro derivatives of quinoline and 1,2-dihydroquinoline are reduced in the first step at the nitro group; the quinolinium cations are reduced at the heterocycle followed by reduction of the nitro group; and in 1,4-dihydroquinolines, the nitro group is not reduced. Electrochemical reduction processes associated with electron transfer in the heterocycle mainly display the same behavior as established for pyridine derivatives. But important differences were observed in electrochemical oxidation: the N-methyl derivative of 1,4-dihydroquinoline is oxidized significantly more easily than the corresponding N-unsubstituted derivative of 1,4-dihydroquinoline (in the 1,4-dihydropyridine series, the difference in pot! enti als is fairly small), and even more easily than the corresponding N-methyl derivative of 1,2-dihydroquinoline.     

Electroactive C(2) symmetry receptors based on the biphenyl scaffold and tetrathiafulvalene units

The synthesis of a family of biphenyl-tetrathiafulvalene (TTF) derivatives incorporating a binding site has been carried out in good to moderate yields through functionalization of the biphenyl scaffold. X-ray structure of one derivative (compound 3) of the series is provided and shows a dihedral angle of 74 degrees around the central Ar-Ar bond of the biphenyl unit in a cisoid conformation. (1)H NMR and cyclic-voltammetry studies demonstrate the critical importance of the nature of the substitution on the conformational rigidity and on the electrochemical behavior of the resulting biphenyl-TTF assemblies. This feature is underlined by an original electrochemical recognition process upon binding of Pb(2+), correlated to conformational changes occurring upon metal cation complexation.     

An efficient approach for self-consistent-field energy and energy second derivatives in the atomic-orbital basis

Based on self-consistent-field (SCF) perturbation theory, we recast the SCF and the coupled-perturbed SCF (CPSCF) equations for time-independent molecular properties into the atomic-orbital basis. The density matrix and the perturbed density matrix are obtained iteratively by solving linear equations. Only matrix multiplications and additions are required, and this approach can exploit sparse matrix multiplications and thereby offer the possibility of evaluating second-order properties in computational effort that scales linearly with system size. Convergence properties are similar to conventional molecular-orbital-based CPSCF procedures, in terms of the number of derivative Fock matrices that must be constructed. We also carefully address the issue of the numerical accuracy of the calculated second derivatives of the energy, in order to specify the minimum precision necessary in the CPSCF procedure. It is found that much looser tolerances for the perturbed density matrices are adequate when using an expression for the second derivatives that is correct through second order in the CPSCF error.     

基于多阶导数拉曼光谱组合技术的矿物油模式分类

为了实现对法庭科学领域重质矿物油物证的快速、准确、无损的鉴定,该文基于光谱分析技术提出了一种多阶导数光谱数据组合分析的方法。收集了80种不同型号、不同厂家的重质矿物油样本,利用傅里叶变换拉曼光谱分析法采集样本的原始光谱数据和导数光谱数据,并通过结合化学计量学构建分类模型。在构建的主成分分析(PCA)结合径向基函数神经网络(RBF)分类模型中,对单独的原始光谱、一阶导数谱和二阶导数谱数据的训练集准确率分别为80.0%、86.7%和86.2%,测试集准确率分别为73.3%、80.0%和72.7%;对组合后的原始光谱+一阶导数谱、原始光谱+二阶导数谱和一阶导数谱+二阶导数谱数据的分类中,训练集准确率分别为97.0%、96.7%和100%,测试集准确率分别为85.7%、90.0%和100%。结果表明,对组合后的导数光谱与原始光谱构建分类模型,准确率更高。其中,基于一阶导数谱+二阶导数谱数据构建的PCA结合RBF分类模型的结果最为理想,准确率达100%。而K最近邻算法模型由于受到样本不均匀的影响,整体分类准确率均较低。利用组合的导数光谱与原始光谱数据构建分类模型能够实现对重质矿物油样本的快速、准确、无损鉴别,可为光谱组合技术在法庭科学及其他分析测试领域的应用提供一定的借鉴和参考。     

Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation

The generalized master equation or the equivalent continuous time random walk equations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from short-term microscopic simulation data. The computation of the mean first passage time and additional low-order FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steady-state relaxation, an extension of steady-state flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a gamma distribution or rational function approximation to its Laplace transform.     

Pyrene-fluorene hybrids containing acetylene linkage as color-tunable emitting materials for organic light-emitting diodes

New blue- to yellow-emitting materials have been developed by incorporating fluorene-based chromophores on pyrene core with acetylene linkage and using multifold palladium-catalyzed cross-coupling reactions. Both mono- and tetrasubstituted derivatives have been synthesized and characterized. The tetrasubstituted derivatives displayed red-shifted emission when compared to the monosubstituted derivative indicative of an extended conjugation in the former. End-capping with a diphenylamine unit further red-shifted the absorption and emission profiles and imparted a weak dipolar character to the molecules. Amine-containing derivatives displayed positive solvatochromism in the fluorescence spectra indicating a more polar excited state due to an efficient charge migration from the diphenylamine donor to the pyrene π-acceptor. All of the derivatives were tested as emitting dopants with host material 4,4'-bis(9H-carbazol-9-yl)biphenyl (CBP) in a multilayered OLED and found to exhibit bright blue or yellow electroluminescence. The device utilizing 1,3,6,8-tetrasubstituted pyrene derivative as a dopant emitter displayed highest maximum luminescence 4630 cd/m(2) with power efficiency 3.8 lm/W and current efficiency 7.1 cd/A at 100 cd/m(2) attributable to the proper alignment of energy levels that led to the efficient harvesting of excitons. All of the devices exhibited color purity over a wide range of operating voltages.     

Evolution of the Size Spectrum of Aggregates in the Disperse System with Reversible Coagulation

Kinetics of coagulation–fragmentation in a disperse system represented by different-scale fractions was studied using two-fraction model (Dolgonosov, B.M., Kolloidn. Zh., 2001, vol. 63, no. 1, p. 27). A model accounting originally only for the interaction between different fractions was extended, by introducing interactions between the particles of coarse fraction, to describe considerable deviations from the equilibrium. The threshold dependence of the coagulation efficiency on the sizes of colliding aggregates was introduced. According to this dependence, the efficiency vanishes when the aggregate sizes exceed a certain threshold value. The scheme of discretization of integro-differential equations of a model was proposed to numerically solve these equations. The results of calculations demonstrated the evolution of dispersed phase to the equilibrium, which is accompanied by (1) the transformation of initial bimodal spectrum into polymodal, (2) complex evolution of the latter with the change in the position and height of peaks and their merging, and, finally, (3) a return to bimodal spectrum with changed characteristics.     

On the Electrochemical Behaviour of Some 2,3-Dihydrothiazolo[3,2-a]Benzimidazol-3-One and 1-Ethylmercapto-3(4H)-Isoquinolones Derivatives

The electrochemical behaviour of several derivatives containing the sequence S-C-CO-N in their structure presented as such or in masked form in both aqueous and non-aqueous media are reported. The results of electrooxidation indicated that when the S atom is present as hetero atom in the structure of the molecule the main product will be the sulphone derivative, on the other hand when present in the form C=S the preferential product is the dimer. The overall redox study indicated that the azo group at C-2 is very active in both aqueous and non aqueous media. On the other hand derivatives substituted at C-2 the preferential electrochemical step is the attack of the hetero ring itself.     

STM study on 2D molecular assemblies of luminescent quinacridone derivatives: structure fine-tuned by introducing bulky substitutes and co-adsorption with monofunctional/bifunctional acid

We describe the 2D assemblies of a series of N,N'-dialkyl-substituted quinacridone derivatives on highly oriented pyrolytic graphite observed by scanning tunneling microscopy. Our experiments have demonstrated that pure quinacridone derivatives take contractive conformations, but quinacridone derivatives take extended conformations when co-adsorbed with dicarboxylic acid. Interestingly, by co-adsorption with monofunctional acid stearic acid, quinacridone derivative bearing two smaller substituted groups of trifluoromethyl takes an extended conformation, while quinacridone derivative bearing two larger substituted groups of tert-butyl still takes a contractive conformation. Therefore, the 2D structure of the quinacridone derivatives can be fine-tuned by co-adsorbing with monofunctional/bifunctional acid through hydrogen bonds.     

Molecular properties from perturbation theory: A Unified treatment of energy derivatives

The definition of a molecular property as a derivative of the electronic energy with respect to one or more applied perturbations is reviewed. The explicit enumeration of terms entering the derivative formulas is performed by considering in turn the various parameter spaces on which the energy and wave function depend. After deriving general expressions for first, second, and third derivatives for different types of perturbation, the parameter spaces involved in MCSCF and CI cases are identified and used to obtain expressions for the first and second derivatives. An example of an MCSCF third derivative is also given. In addition, the various equation systems defining the perturbed wave functions in each order are derived. Some attention is given to the efficient computer implementation of derivative calculations, and the present work is compared with that of other authors.     

Mathematical Modelling of a Non‐enzymatic Amperometric Electrochemical Biosensor for Cholesterol

Thickness of the electro‐polymerized layer grown on a substrate and used as the recognition element for the analyte is critical to measuring the response of a biosensor, with high sensitivity and accuracy. However, it is difficult to control the thickness during synthesis. A mathematical model is developed in this study that considers thickness of the electro‐polymerized layer in simulating the electrochemical response of a non‐enzymatic biosensor for cholesterol in blood. The model includes transient kinetics and one‐dimensional diffusion of the analyte in the poly‐methyl orange (PMO) recognition layer electrochemically grown on the electrode. The governing partial differential equations resulting from the species conservation balances in the PMO layer are numerically solved. Time and spatial concentration profiles of the analyte in the PMO layer are determined. Model predictions are calibrated with the experimental data for different PMO thicknesses. Interestingly, model predictions show a linear response over the calibrated concentration range of cholesterol for all PMO layer thicknesses. Based on the chronoamperometry measurements, the model predictions for the cholesterol concentrations measured in the laboratory samples were also found to be remarkably accurate. This is the first mathematical model developed to understand the transport and kinetics of an analyte in the electro‐polymerized layer used as the recognition element of a non‐enzymatic biosensor.