Electrodes Of Arbitrary Shape Research Articles

Multiphysics Modeling of the Tertiary Current Distribution in Pulse/Pulse-Reverse Electrochemical Processing of Microstructured Workpieces

The physical phenomena governing the current distribution on an electrode of arbitrary shape are typically categorized as falling into primary, secondary, and/or tertiary effects. Primary current distribution effects are defined by the geometry of the system and the electrical properties of the relevant materials, whereas secondary and tertiary effects incorporate additional position-dependent polarization that respectively arises from electrochemical-kinetic and mass-transfer/concentration physics. In industrial electrochemical processes, the uniformity of the current distribution across a workpiece is of vital concern. In electrodeposition processes, for example, it is usually desirable for the deposited metal to be as uniformly distributed as possible, regardless of the form of the workpiece. Conversely, in electropolishing processes, it is critical to focus the current density onto the tops of asperities on the workpiece surface, in a highly non-uniform fashion, in order to minimize material removal irrelevant to the goal of decreased surface roughness. In general, the primary current distribution leads to the most non-uniform current distribution possible for a given geometry, from which the secondary and tertiary effects tend to have varying degrees of a “leveling” effect, leading to a comparative increase in processing uniformity. In electrodissolution processes, saturation of the dissolved metal at the workpiece surface is an important mechanism by which the tertiary current distribution effects influence practical electrochemical processes. This saturation phenomenon leads both to an increase in the local overpotential, via concentration polarization, and also has the potential to occlude locally a fraction of the workpiece exposed area due to the formation of insoluble precipitates. As noted, both of these effects tend to increase the uniformity of the resulting overall current distribution, and thus it is important to be able to predict, even if approximately, when a given process will be operating in this regime and to what extent the uniformity of the current distribution might be affected. This talk will summarize results from multiphysics simulations developed to represent this occluded-surface aspect of the tertiary current distribution, in addition to primary and secondary current distribution effects. These simulations incorporate pulse/pulse-reverse waveforms applied to workpieces with structured surfaces, in an attempt to approximate a surface finishing application of industrial relevance. In particular, focus was placed on simulating a “microprofile,” the scenario where surface structures have characteristic dimensions much smaller than the hydrodynamic boundary layer for mass transfer (indicated by δ in Figure 1)—this choice simplifies the modeling by obviating consideration of the macroscopic fluid dynamics of the system. The effect of pulse waveform parameters on the uniformity of the overall current distribution will be discussed, and simulation results will be shown illustrating the tendency of suitably-chosen waveform parameters to “collapse” toward the workpiece surface (δp in Figure 1) the subdomain of the boundary layer in which the local concentration of dissolved material oscillates significantly in response to the applied electric field. Figure 1. Schematic of the hydrodynamic boundary layer above a structured workpiece (δ), and a “collapsed” pulsating boundary layer (δp) arising from the use of a pulsed electric waveform. Figure 1

Boundary Galerkin Method of a Skew-Derivative Problem in the Exterior of an Open Arc Based on Chebyshev Polynomials

A problem modeling Hall effect in a semiconductor film from an electrode of arbitrary shape is considered, which is a skew-derivative problem. Boundary Galerkin method for solving the problem in Sobolev spaces is developed firstly. The solution is represented in the form of the combined angular potential and single-layer potential. The final integral equations do not contain hypersingular integrals. Uniqueness and existence of the solution to the equations are proved. The weakly singular and Cauchy singular integral arising in these equations can be computed directly by truncated series of Chebyshev polynomials with their weighting function without approximation. The numerical simulation showing the high accuracy of the scheme is presented.

Numerical treatment of a skew-derivative problem for the Laplace equation in the exterior of an open arc

The skew-derivative problem for harmonic functions in the exterior of an open arc in a plane is considered. This problem models the electric current in a semiconductor film from an electrode of arbitrary shape in the presence of a magnetic field. A numerical method for solving the problem is proposed. The method is based on a boundary-integral-equation approach. The proposed numerical method is tested. The numerical simulation is presented for different values of the parameters and different shapes of the electrode. Physical effects found in numerical experiments are discussed.

Acceleration sensitivity of crystal resonators affected by the mass and location of electrodes

Predominant thickness-shear frequencies and modes of a crystal plate with electrodes of arbitrary shape and mass distribution are obtained by a finite-element method, based on Mindlin's first-order equations with platings. These frequencies and modes are used in a perturbation method for computing the acceleration sensitivity of crystal resonators with electrodes. Computations are made for a square AT-cut quartz plate that is supported by a four-point mount and coated with identical square and uniform electrodes on the upper and lower faces of the plate. To study the effect of uneven distribution of electrode mass, acceleration sensitivities are calculated when a small mass is added at various locations near the edges of the square electrodes. It is found that the percent increase of the acceleration sensitivity of the resonator with a small added mass to that of the resonator without added mass ranges from 3.8% to 541.7%, depending on the location of the small mass placed at the edges of the electrodes.

Dimensional analysis of corona discharges: the small current regime for rod-plane geometry in air

The technique of dimensional analysis is applied to a general corona discharge between two electrodes of arbitrary shape. This theory is then simplified to apply to rod-plane geometry for currents near inception. A series of experiments with hemispherically tipped rods has been carried out in ambient air. The variations with geometry of the inception voltage and of the slope of the current-voltage curve at inception have been recorded with considerable care. Analysis of the results confirmed known relationships, but also revealed previously unsuspected geometric dependencies. These are discussed in terms of the known behaviour of glow discharges. The dimensional methods have led to the formulation of a new current-voltage law appropriate to the inception regime for small points.

A theory of vacuum tunneling microscopy

A theory of scanning electron microscope, recently developed by Binning et al., is presented. The theory follows Tersoff and Hamann in applying the transfer hamiltonian formalism to vacuum tunneling between a sample consisting of an infinitely extended single crystal plane and a field emission tip. A general expression for the tunneling current for any two electrodes of arbitrary shape is derived and discussed. Reasoning by analogy with field ion microscopy it is argued that the operation of the device depends on the strong enhancement of local electric fields due to microscopic imperfections of the topmost atomic layers of the central facet, such as small atomic clusters and kink sites. These local fields and the equally important multiple image potential lead to a strong dependence of the tunneling barrier on the polar angle. Accounting for this effect, which was not considered by Tersoff and Hamann, complicates the analysis significantly. The tunneling current is expressed as a convolution type integral involving the spectral densities of the samples and of the tip. It is argued that this integral can be interpreted in terms of an instrument function depending only on the tip or probe parameters, which is convoluted with a function depending only on characteristics of the sample. Simple limiting cases of the theory are considered. A discussion of the resolution predicted by the theory is presented.

PERTURBATION METHOD FOR VARIABLE BOUNDARY PROBLEMS AND APPLICATION TO THE EVALUATION OF ERRORS IN PRECISE CAPACITORS

In evaluation of errors of precise capacitors due to geometric, imperfection, one has to solve Laplace equation subject to irregular boundary condition. To solve such problems, a special method of calculation "Perturbation Method for Variable Boundary Problems" has been put forward, including the corresponding condition of convergence. On the basis of this method, a general formula is derived for the calculation of the Variation of charges on electrodes of arbitrary shape. Applying these results to cross-capacitor based on Thompson-Lampard theorem, some general characteristics of this kind of capacitor can be predicted.