Adaptive finite element simulation of currents at microelectrodes to a guaranteed accuracy. Theory

In our accompanying paper (K. Harriman et al., Electrochem. Commun. 2 (2000) 150) we demonstrated how, for the finite element method, an automatic mesh-adaptation algorithm can be derived, given an a posteriori error bound on the simulated current. In this paper we give the technical details of the background theory of the finite element method, and the derivation of the a posteriori error bound.     

Adaptive finite element simulation of currents at microelectrodes to a guaranteed accuracy. An E reaction at a channel microband electrode

We extend our earlier work (K. Harriman et al., Electrochem. Commun. 2 (2000) 150) on adaptive finite element methods for disc electrodes to the case of an E reaction mechanism to the increasingly popular channel microband electrode configuration. We use the standard Galerkin finite element method for the diffusion-dominated (low-flow) case, and the streamline diffusion finite element method for the convection-dominated (high-flow) case. We demonstrate excellent agreement with previous approximate analytical results across the range of parameters of interest, on comparatively coarse meshes.     

Adaptive finite element simulation of currents at microelectrodes to a guaranteed accuracy. ECE and EC2E mechanisms at channel microband electrodes

We extend the work in the accompanying paper (K. Harriman et al., Electrochem. Commun. 2 (2000) 567) on the use of adaptive finite element methods to simulate the current for a steady state E reaction mechanism at a channel microband electrode to the more complex ECE mechanism, and the non-linear EC₂E mechanism. We again use the standard Galerkin approach for the diffusion dominated (low-flow) case, and the streamline diffusion finite element method (SDFEM) for convection-dominated (high-flow) case, and compare our results with previous numerical and analytical approximations. We give a general discussion on the implications of our results for numerical simulation at microelectrodes.     

Adaptive finite element simulation of chronoamperometry at microdisc electrodes

In this paper we shall introduce a transient finite element algorithm by considering the simplest problem of numerical simulation of the chronoamperometric current for an E reaction mechanism at a microdisc electrode. Such a numerical simulation is made difficult by the presence of a boundary singularity where the electrode meets the insulator (often known as the ‘edge-effect’) and by a time-singularity caused by the impulsive start of the experiment. We attempt to overcome these problems by using an adaptive finite element algorithm in which we derive an a posteriori bound on the error in the computed current. This is used to guide mesh refinement and adaptive time-stepping, resulting in a fully automated algorithm which is both accurate and efficient.     

Effect of ion pairing on steady-state voltammetric limiting currents at microelectrodes Part I. Theoretical principles

Voltammetric studies in solutions of high resistivity are facilitated by the use of microelectrodes under steady-state conditions. Such solutions are encountered with solvents of low permittivity because of the very sparing solubility of electrolytes. Moreover, in such media the supporting electrolyte, as well as the electroactive ionic species, is usually extensively ion paired. Here we predict the limiting current that will flow in these circumstances, when a monovalent ion undergoes a one-electron transfer at a hemispherical microelectrode to form a neutral product. The ion pairing equilibria are assumed to be fast but all diffusion coefficients are treated as distinct. An analytical solution is elusive in the general case, but a simple numerical procedure allows the limiting current to be predicted for any combination of the system parameters. Several special cases are also discussed, some of which yield explicit formulae for the limiting current. In a companion paper, experimental data are compared with the theoretical predictions.     

Application of the finite element method to molecular quantum mechanics

We present a general formalism of the finite element method and its computational aspects that influence the solution accuracy of spectral quantum mechanical problems. The possibilities of the method are demonstrated using as an example inversion vibrational spectra of structurally non-rigid complexes.High Energy Physics Institute, Protvino. I. V. Kurchatov, Atomic Energy Institute, Moscow. Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 27, No. 4, pp. 442–446. July–August, 1991. Original article submitted April 4, 1991.     

Decoupling of the nernst-planck and poisson equations. Application to a membrane system at overlimiting currents

This paper deals with one-dimensional stationary Nernst-Planck and Poisson (NPP) equations describing ion electrodiffusion in multicomponent solution/electrode or ion-conductive membrane systems. A general method for resolving ordinary and singularly perturbed problems with these equations is developed. This method is based on the decoupling of NPP equations that results in deduction of an equation containing only the terms with different powers of the electrical field and its derivatives. Then, the solution of this equation, analytical in several cases or numerical, is substituted into the Nernst-Planck equations for calculating the concentration profile for each ion present in the system. Different ionic species are grouped in valency classes that allows one to reduce the dimension of the original set of equations and leads to a relatively easy treatment of multi-ion systems. When applying the method developed, the main attention is paid to ion transfer at limiting and overlimiting currents, where a significant deviation from local electroneutrality occurs. The boundary conditions and different approximations are examined: the local electroneutrality (LEN) assumption and the original assumption of quasi-uniform distribution of the space charge density (QCD). The relations between the ion fluxes at limiting and overlimiting currents are discussed. In particular, attention is paid to the "exaltation" of counterion transfer toward an ion-exchange membrane by co-ion flux leaking through the membrane or generated at the membrane/solution interface. The structure of the multi-ion concentration field in a depleted diffusion boundary layer (DBL) near an ion-exchange membrane at overlimiting currents is analyzed. The presence of salt ions and hydrogen and hydroxyl ions generated in the course of the water "splitting" reaction is considered. The thickness of the DBL and its different zones, as functions of applied current density, are found by fitting experimental current-voltage curves.     

Computational electrochemistry: finite element simulation of a disk electrode with ultrasonic acoustic streaming

Computer simulations using the finite element method (FEM) are used to predict the correlation between the transport limiting current (I(lim)) and parameters such as diffusion coefficient, source to electrode separation, source power, and medium viscosity for a sonicated disk electrode in "face on" mode. The fluid dynamics and diffusion layer are modeled directly using FEM and predict that the electrode is uniformly accessible, I(lim) is proportional to the diffusion coefficient to the 2/3 power and I(lim) is proportional to the square root of the source power. Curves are also calculated relating I(lim) to the source to electrode separation and liquid viscosity.     

Application of the finite element method to time-dependent quantum mechanics: II. H 2 + in a laser field

A finite element method in Cartesian coordinates in three dimensions is described to solve the time-dependent Schrödinger equation for H ₂ ⁺ in the presence of time-dependent electromagnetic fields. The ionization rates, nonlinear optical polarizabilities and harmonic generation spectrum of H ₂ ⁺ have been calculated for field directions parallel or perpendicular to the hydrogen molecule ion axis. Comparisons of the present numerical results with previously published calculations show that the finite element method reproduces perturbative results and can treat nonperturbativity arbitrary intense short pulses as it includes automatically both bound and continuum electronic states.     

Application of the finite element method to time-dependent quantum mechanics: I. H and He in a laser field

A finite element approach is described to solve the time- dependent Hartree-Fock equation of atoms in the presence of time-dependent electromagnetic fields. Time-dependent energy changes, ionization rates and high order nonlinear optical polarizabilities, 2n+1 (n >, 0) for the atoms H and He have been calculated. The finite element method is shown to be easily adaptable to treat intense short pulses and includes automatically both bound and continuum electronic states.     

Steady state simulation of electrode processes with a new error bounded adaptive finite element algorithm

In a series of papers, Harriman et al. [1], [2], [3], [4], [5] have presented a reliable means of simulating steady state currents with adaptive finite element. They have demonstrated that multi-step, even non-linear, mechanisms, alongside convection, can be incorporated. However, there is considerable complication in the mathematical approach taken, and it would seem to be limited as it stands to certain types of electrode reaction models – notably those without heterogeneous kinetics or transient effects. In this paper we discuss alternative approaches to error estimation and adaptivity, and present a simpler formulation, capable of simulating systems with heterogeneous kinetics; transient simulations also appear more attainable. We introduce, apparently for the first time in electrochemistry, the use of gradient recovery methods [6] to both error estimation and accurate current calculations. The result is an algorithm with considerably more potential for generalisation, closer to the ideal of an entirely flexible automatic simulation program, capable of dealing with any mechanism or electrode geometry. In tests we find our method to perform more efficiently than that cited above, producing accurate results with simpler meshes in less time.     

Brownian dynamics simulation of a model simple electrolyte in solvents of low dielectric constant

Brownian dynamics simulations are performed to investigate the ionic transport of model simple electrolytes, in which ions are interacting with each other through the repulsive core and Coulombic interactions. The equivalent conductivity and self-diffusion coefficient show minima as the function of the number density of ions when the dielectric constant of the solvent is low. Although the minimum of the former is in harmony with various experiments, no experiment has ever been reported on that of the latter. The analysis of time-dependent transport coefficients reveals that the presence of the minima is ascribed to the slow dynamics, rather than to static association models. The inclusion of a model function that resembles the short-range part of the potential of mean force induced by solvent affects the transport coefficients qualitatively, which suggests the importance of solvent-induced potential of mean force in the conduction mechanism of electrolytes in solvents of low dielectric constant.     

Numerical simulation of the BZ reaction of oxalic acid with a simple four-variable model

A simple model of the bromate-cerium-oxalic acid batch oscillatory system has been constructed and examined by calculations. The model predicts a parameter range and a frequency which agree rather well with the observed values, but it does not describe the peculiar saddlenode infinite period bifurcations found experimentally in the system.

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Heat transfer simulation in cavity of twin screw compressor under coupling of clearance leakage-heat by utilizing fuzzy beamlet finite element model

Journal of Thermal Analysis and Calorimetry - In order to optimize structure of twin screw compressor, and improve the working efficiency of twin screw refrigeration compressor, the heat transfer...     

Application of a simple preconcentration scheme to the determination of isopropylmethylphosphonofluoridate at trace levels in water

The procedure for the preconcentration and determination of isopropyl-methylphosphonofluoridate (Sarin) involves extraction of the sample with chloroform, preconcentration of the vaporized extract on a Porapak Q plug, and direct thermal desorption of the adsorbed material into a gas chromatograph. Precise determinations at concentrations down to approximately 67 pg ml^-1 are possible.     

Application of a large assembly of microelectrodes to the study of cytochrome-c and other redox proteins

A novel microelectrode assembly containing more than 1000 carbon microdisks has been used to study the interfacial electron transfer kinetics of cytochrome c and several other redox proteins. It is demonstrated that near-steady-state current–voltage curves can be obtained without solution stirring, and without significant mains interference or capacitive charging currents. In addition, it is demonstrated that concentrations of redox proteins as low as 1 μM can be detected in solution. Finally, it is shown that the surfaces of carbon microdisk assemblies can be successfully coated with thin films of poly(ester sulfonic acid), which allows reactants to be studied in a membrane-like environment.     

Mathematical simulation of atomic hydrogen diffusion transfer through a multilayer metal membrane at finite pressures

Diffusion transfer of atomic hydrogen through multilayer metal membranes has been studied within the lattice model of an ideal gas, with the transfer being described by a set of nonlinear algebraic equations. It has been shown that, for multilayer membranes composed of less than four layers, an analytical expression describing a diffusion flux can be derived. Atomic hydrogen transfer through a membrane consisting of a vanadium layer, the surfaces of which are coated with palladium films, has been analyzed in detail. It has been found that the value of the flux may depend on the transfer direction. The effect of diffusion asymmetry arises at finite pressures of hydrogen on the outer membrane surfaces, when its dissolution in metals is described by nonlinear sorption isotherms. The degree of the diffusion asymmetry increases with a rise in hydrogen pressure and depends on the arrangement of the layers composing a membrane.