Primary Current Distribution Research Articles

A Semianalytical Method for Predicting Primary and Secondary Current Density Distributions: Linear and Nonlinear Boundary Conditions

A semianalytical solution technique is presented for solving Laplace's equation to obtain primary and secondary potential and current density distributions in electrochemical cells. The potential distribution inside a rectangle with the electrodes facing each other between two insulators is presented to illustrate the method. It is shown that the method yields analytic equations for the potential and the potential gradient along the lines. The unique attribute of the technique developed is that the solution once obtained is valid for nonlinear boundary conditions also. The procedure is applied to some realistic problems encountered in electrochemical engineering to illustrate the utility of the technique developed. © 2000 The Electrochemical Society. All rights reserved.

A primary, secondary and pseudo-tertiary mathematical model of a chlor-alkali membrane cell †

A theoretical study of current density and potential at the anode, membrane and cathode, of a chlor-alkali membrane cell where the electrode blades are placed vertically, is presented. A representative unit cell is modelled in primary, secondary and pseudo-tertiary current distribution models. It is shown that electrolyte and membrane resistance has the greatest effect on current distribution. Furthermore, it is shown that there is a surprisingly small influence of mass transport on current distribution, on the assumption that the diffusion layer is of constant thickness. In converse to this, it is shown that mass transport affects the anode overpotential distribution to the extent that conclusions can be made about the occurrence of side-reactions and where they occur. Finally, it is shown that it is possible to estimate tertiary behaviour with a secondary current distribution model, by using an analytic expression at the anode surface.

Design and Characterization of a Rotating Electrochemical Quartz‐Crystal‐Microbalance Electrode

A novel design for a rotating electrochemical quartz‐crystal microbalance (rEQCM) which includes easily and reproducibly exchangeable quartz‐crystal holders is described. The rEQCM allows for well‐controlled mass‐transport conditions at the working electrode on a 10 MHz quartz crystal. Together with the flexibility of the proposed design, this makes it useful for a wide range of applications in electrochemistry. At a data acquisition rate of 5 Hz, noise level and stability of the resonant frequency were better than 1 Hz at rotation rates of 0–3500 rpm. This corresponds to a mass density on the quartz crystal of . The small recess needed to fix the quartz crystal in the electrode holder was shown not to affect the primary current distribution and the mass‐transport conditions. Electrodeposition of Cu and Ag was used for the in situ characterization of the frequency dependence of the Sauerbrey constant (mass sensitivity) of the quartz as a function of mass load on the crystal. It was found that, in the initial phase of Ag and Cu deposition on Au, the mass sensitivity is affected by internal stresses in the deposit. © 2000 The Electrochemical Society. All rights reserved.

Imaging of Nonuniform Current Density at Microelectrodes by Electrogenerated Chemiluminescence

The chemiluminescence arising from reaction of electrogenerated radical cations of 9,10-diphenylanthracene (DPA) and benzonitrile (solvent) radical anions has been used to image microelectrodes with dimensions in the micrometer range. Experimental conditions including supporting electrolyte, DPA concentration, and excitation frequency were optimized to affect high luminescent intensity. In solutions of high resistance, the light was found to be temporally delayed with respect to the applied potential due to the increased time required to charge the double layer. Spatially nonuniform light at disk- and band-shaped microelectrodes was observed under certain conditions, with the highest intensity occurring at the region of the electrode with highest curvature. The optimum condition for observation of the nonuniform light was with very high electrode currents. Under this condition, the current density approaches that of the primary current distribution, a circumstance where spatially nonuniform potentials occur. This phenomenon was also examined at a conical electrode as a method of reducing the emission area. A submicrometer-size light source was obtained at high frequencies with an electrode that had a significantly larger uninsulated area.

Shape evolution of electrodeposited bumps with additive

Bumps with anisotropic conductive film are promising interconnections for microelectronics. The bumps having humps on their surroundings decreases the interconnection resistivity, since a greater number of conductive particles in the anisotropic conductive film can be trapped. The anodic current density-voltage curves show the increase of i 0 and α c by adding selenic acid. The bump has flat shape without the selenic acid. However, the bump surroundings have humps and the hump height becomes larger with selenic acid of 200 and 1000 g/m 3. The hump heights at the surroundings decreases along with the decrease in current density of 1734, 600 and 350 A/m 2. The bump center height increases and the surrounding hump heights decrease with the decrease of cathode diameter. These changes in bump shapes corresponds well with the numerical analysis of current density in two dimensions. The additive of selenic acid increase i 0 and α c and hence decrease the charge transfer resistance. The ohmic resistance becomes larger than the charge transfer resistance. This converts the secondary to the primary current distributions and causes the bumps with humps in their surroundings.

Influence of Counter Electrode Positioning on Solution Resistance in Impedance Measurements of Reinforced Concrete

Abstract Erratic impedance measurements of steel in concrete were traced to variability in the contact configuration between the counter electrode (CE) and the surface of the concrete. A stable contact over a small portion of the concrete surface improved reproducibility. The relative placement of the CE, reference electrode (RE), and working electrode (WE) could be manipulated to select effective solution resistance over a wide range (4.66:1) of values. Results agreed with predictions from a numerical model of the primary current distribution of the system. Except for the offset in the real component, the electrochemical impedance response of the specimens essentially remained unchanged by controlled variations of the CE configuration. However, this may not be the case when the interfacial impedance is not distributed evenly over the surface of a WE.

Analysis of terminal effects in rectangular electrochemical cells

The current distribution in electrochemical cells consisting of resistive parallel rectangular plates is determined by evaluating the appropriate analytical solution of Laplace's equation within the electrodes and electrolyte. Boundary conditions corresponding to potential continuity (primary current distribution) or linear electrode kinetics (secondary current distribution) at the electrode-electrolyte interface are considered and the analysis does not make the usual assumption that current flow in the resistive electrode is one-dimensional. Fourier series for cells in which resistive effects are significant for one or both electrodes are derived. These series express the potential in terms of the electrolyte:electrode conductivity ratio, the ratio of electrolyte resistance to polarization resistance (the Wagner number) and the ratios of relevant geometrical parameters. Calculations that illustrate the perturbation of the potential distribution within the electrolyte, the effect of electrolyte:electrolyte conductivity ratio, the Wagner number and the effect of the width and relative disposition of the external electrical contacts are presented. The predictions of one- and two-dimensional models for the resistive electrodes are compared.

Optimization of bath composition for hard Fe–C alloy plating

Fairly uniform Fe–C alloy deposits with bright appearance and characteristic black colour were obtained in Hull cell tests at 1.0A from baths with the following composition: FeSO4 0.5m or higher, citric acid 0.01m or lower, and pH around 2.0. The carbon content of the deposits from these baths ranged between 1.0 and 1.2wt% and the Vickers hardness was HV700 or above over a wide area of the Hull cell cathode. The thickness distribution of the deposits on the Hull cell cathode from these baths suggested that the current efficiency was 100% and the local current density obeyed the primary current distribution, except for the region near the low-current-density (LCD) end where the thickness decreased unexpectedly. The addition of chloride ion as NaCl, at concentrations of 1.0m or above, improved the thickness in the LCD region, and the thickness distribution on the Hull cell cathode suggested that current efficiency was 100% even at the LCD end. Chloride ion did no t affect the carbon content or the hardness of the deposits. Polarization measurements showed that the hydrogen ion reduction occurred at more noble potentials than Fe deposition. The unexpected thickness decrease near the LCD end, observed in Hull cell tests, and the improvement of this by adding chloride ion were both accounted for by the change in the preceding hydrogen ion reduction rate. The increase in citric acid concentration enhanced hydrogen ion reduction and therefore caused a lowering of current efficiency at low current densities. On the other hand, the addition of chloride ion inhibited hydrogen ion reduction and caused an increase in current efficiency at low current density.

Estimation of Front Velocity in Electrodeposition onto Highly Resistive Substrates

The primary current distribution for electrodeposition onto a surface composed of a perfectly conducting section and a highly resistive section is examined, both by direct numerical calculation and by singular‐perturbation analysis. Results of the study show how the current density and the region over which electrodeposition reactions are appreciable scale with the conductivity and thickness of the resistive section of the electrode. The perturbation analysis leads to an approximate expression (Eq. 34 in text) for the front velocity of a growing metallic film, a result which is of practical interest for design estimates of the speed of coverage in metallization processes on resistive substrates.

Electric Potential Field Calculations for the FCF Mark-IV Electrorefiner

The electric potential field and resulting current densities are studied for the FCF Mark-IV electrorefiner during the direct transport mode of operation. The results are used in estimating primary current distributions and ohmic resistances between components of the electrorefiner for various cell configurations. Each configuration is analyzed for operation with a single power supply, dual anodes and serial cathodes, and dual power supplies. The effect of cathode deposit growth from a bare mandrel to a 10'' diameter cylinder is also studied. A comparison of the calculated results with data from initial electrorefiner experiments in direct transport mode indicates a good agreement for cell resistance and reference electrode readings at the beginning of an experiment. The difference between the calculated results and experimental data provides information regarding overpotentials that form on the surfaces of the electrorefiner components.

Mathematical modelling of aluminium cells with prebaked anodes: Part II: Current distribution and influence of sideledge

The current distribution at the electrodes in an industrial aluminium cell with prebaked anodes was calculated. The difference between the primary and secondary current distribution was determined for three different gaps between anodes or between anode and sidewall (2.5, 10 and 30 cm). The calculated current densities at the vertical sides of the anode were higher for the secondary current distribution than for the primary (almost double at the uppermost part of the anode), while the differences were much smaller at the cathode (6–20%). If the conducting carbon sidelining is exposed to the electrolyte it will draw an average current density of 0.045 A cm−2 for a 30 cm distance to the cathode. If all this current leads to the formation of aluminium carbide with subsequent dissolution into the electrolyte, the sidelining will corrode at a rate of 0.08 cm d−1 of exposure. The influence of the shape and position of the ledge on the anode current distribution was studied. When the distance between the ledge and anode was greater than 15 cm the shape of the sideledge does not affect the anode current densities significantly, while the current density at the upper part of the anode increases with increasing distance.

Workpiece, Pattern, and Feature Scale Current Distributions

Results of primary and secondary current distributions in geometries containing different length scales are presented. Both two‐ and three‐dimensional problems are considered. Recommendations for treating multiple‐length scale problems are made, and one methodology for solving primary and secondary current distributions on workpieces containing numerous features is presented. It is assumed that the workpiece is composed of patterns, each of which are composed of features. It is shown how the mathematical model gives good estimates of current distribution on the workpiece, pattern, and feature scales.

Calculation of the Primary Current Distribution in Cells with Curved Electrodes Using the Finite Difference, Conservative Scheme, and Finite Element Methods

The primary current distribution was calculated in cells with a curvilinear shape of the electrodes by the finite difference (FDM), the conservative scheme (CS), and the finite element methods (FEM). These methods were used for the solutions of the Laplace equation (LE) for a 2D cross-section of a cell consisting of two concentric cylinders (tubes) as electrodes and the inter-electrode space filled with electrolyte. For this cell the analytical solution of LE is known. The local current density on the approximated shape of the electrodes was calculated. The error in the normalized local current density relative to the mean was 5.2%, 52% or 0.2% with FDM using a 64 o 64 mesh, CS using 64 o 64 mesh or FEM using 969 nodes, respectively. Also the boundary element method (BEM) has been used. With 199 elements at the electrode the error in the normalized current density was 0.2%. Taking into account the simplicity of programming and the possibility of using previously developed modules in other calculations, FEM and BEM showed the best performance.

Numerical methods for electrochemical modelling—I

This is a case study for the treatment of the partial differential equations (PDEs) of cell modelling. This part deals with the calculation of the primary and secondary current distribution in a cell using finite difference (FD) methods for the discretization. For nonuniform resistivity and shaped boundaries, conservative FD methods are discussed.

Steady‐State Characterization of the Uniform‐Injection Cell: I . Theoretical Analysis

Presented is the theoretical characterization of an electrochemical cell that is uniformly accessible to mass transfer and exhibits a nearly uniform primary current distribution over most of the electrode. Conceptually, the cell consists of two parallel, coaxial disks of radius with a gap between the disks of length . One disk forms the working electrode, and the other disk is a porous surface through which electrolyte is assumed to be injected with a uniform axial velocity . The aspect ratio of the gap between the disks is small compared to unity. Analytic expressions analogous to the Levich equation are derived for the mass‐transfer‐limited current density as a function of the physical properties of the electrolyte, the injection velocity , and the gap length . The approximate analytic results are compared to numerical computations and are found to agree within 2% for typical operating conditions. The effect of gap aspect ratio on the primary current distribution is also explored. Numerical solutions show that the primary current distribution is flat over most of the electrode when the gap aspect ratio is much less than unity, but, because of the cell geometry, the local current density always becomes infinite at the electrode/insulator boundary.

Primary current distribution in a thin-film battery. Application to power-density calculations for lithium batteries

The mathematical formulation required to determine the power density of a thin-film battery that is governed by ohmic drop in the electrodes and electrolyte is presented along with representative results for a lithium polymer battery. The problem statement is two dimensional, reflecting the ohmic drop along the length of the electrodes and across the electrolyte separating the electrodes. Although the analysis corresponds to steady-state charge transport and current passage, the form of the governing equations is common to electrostatic-field problems in the absence of separated charge, with the electric potential satisfying Laplace's equation. The power density calculations are a function of seven dimensionless groups: four geometric ratios, two conductivity ratios, and a group representing the cell's end-plate volume contribution.

A Probabilistic Method for Evaluating Primary Current Distribution in Electrodeposition Cells

SummaryA probabilistic method is presented for determination of primary current density profiles in electrodeposition cells. It is shown that the current distribution at the cathode can be determined from the simple relationship: where ri, is the proportion of the total trials that result in particles terminating at the anode equipotential boundary and m is the number of points considered in the distribution. The usefulness of the method as a tool to help predict the effect of shielding using baffles in chromium plating cells is explored by comparing computed and experimental thickness profiles.

Interpreting magnetic fields of the brain: minimum norm estimates.

The authors have applied estimation theory to the problem of determining primary current distributions from measured neuromagnetic fields. In this procedure, essentially nothing is assumed about the source currents, except that they are spatially restricted to a certain region. Simulation experiments show that the results can describe the structure of the current flow fairly well. By increasing the number of measurements, the estimate can be made more localised. The current distributions may be also used as an interpolation and an extrapolation for the measured field patterns.

Electrochemical Impedance for a large structure in soil

The electrochemical impedance for a two-electrode cell consisting of a buried metallic structure and a small non-polarizable disk electrode is discussed on the basis of a computer simulation considering the distribution of ac signal current. Since interfacial impedance decreases with increasing ac frequency, the electrochemical impedance at high and low frequencies correspond to those for the primary and secondary current distributions, respectively. The complex plane plot of the simulated electrochemical impedance for a buried structure exhibits a deformed semicircle, the intercept of which on the real axis is not proportional to the true transfer resistance. The impedance diagrams for buried structure in the presence of a local corrosion deviate from those in the absence of local corrosion below a certain frequency which is determined mostly by the distance between the local corrosion site and the probe electrode. This break-point frequency serves as a reliable tool for detecting the location of a local corrosion of a buried structure.

Current Distribution in Electrochemical Cells

Knowledge of current distribution is of prime importance to the electroplater, however, its calculation in other than simple geometries is difficult. Primary current density distribution is simply based upon the system geometry and gives a pessimistic solution, however, this is usually sufficient or practical purposes. Resistance analogue methods are not suitable for day to day operations but represent a useful laboratory tool for simple geometries. In calculating primary and secondar current distribution in complex geometries numerical methods are used and the problem solved by computer