Polynomial electrodes

Figure 9. (a) Image showing pDEP of 557 nm diameter latex spheres on polynomial electrodes, (b) Image showing nDEP of 557 nm diameter latex spheres on polynomial electrodes (Green et al. [33]). 

In the limiting case where we have a thin-gap thin-ring electrode and consequently normal convection may be neglected, this problem can, in principle, be solved analytically. However, inversion from Laplace space is difficult and polynomial expansion is necessary except for small k,. If we define... 

The possibly peculiar spacing of the collocation points, crowding close both at the electrode and at the outer diffusion limit, does not matter too much, and seems unnecessary. For example, using only five internal points (that is, five apart from zero and unity), they are placed at the values 0.047, 0.231, 0.5, 0.769, 0.953, a series that is symmetrical about the midway point at 0.500. This spacing has been circumvented by Yen and Chapman [580], using Chebyshev polynomials that open out towards the outer limit. Their work has apparently not been followed up. 

There is an inherent coupling of the behavior of the micro-scale variables to the behavior of macro-scale variables. This in itself presents complications when simrrlating these models. A few researchers have tried to address this problem of couphng of scales in these models. The solid state concentration term defined by the micro scale diffusion equation need to be coupled with the governing equations for the macro-scale to predict electrochemical behavior. Wang and co-workers used volume averaged equations and a parabolic profile approximation for solid-phase concentration. Subramanian et al. developed approximations assuming that the solid-state concentration inside the spherical electrode particle can be expressed as a polynomial in the spatial direction. 

Newman2°° 2° modeled the transient response of a disk electrode to step changes in current. The solution to Laplace s equation was performed using a transformation to rotational elliptic coordinates and a series expansion in terms of Lengendre polynomials. Antohi and Scherson expanded the solution to the transient problem by expanding the number of terms used in the series expansion. ... 

In packed-bed, flow-through electrodes, concentration and potential variation within the bed can also give more than one steady state. The convective transport equation with axial dispersion, coupled with Ohm s law for the electrode potential, was solved recently (418) by polynomial expansion and orthogonal collocation within the bed, to determine multiplicity regions. 

The model parameters are determined in the following manner the functional relations =/( — 8) and Rincorporated into B.C. of Equation (5.21), are obtained by the polynomial regression analysis of the electrode potential curves and the Kceii versus E curves determined from the Jin, versus AE plots, respectively. It should be borne in mind that (1 — 8) does not represent the average lithium content in the electrode, but the lithium content at the surface of the electrode. In other words, the electrode potential (t) in Equation (5.21) is the potential at the electrode surface. As the relationship —/(I — 8) includes information about the phase transition, we can consider the effect of phase transition on the theoretical CT with the functional relationship =fil — 8), without taking any of the intercalation isotherm. 

A numerical study of the influence of the ohmic drop on the evaluation of electrochemical quantities has been conducted, for example, over the AE interval [-20, 20] mV by means of the IRCOM program, which makes use of a polynomial of the sixth degree, considering some experimental polarization curves and taking the values of the electrochemical parameters obtained by the NOLI method. The examples examined have shown that the representation of experimental data by a polynomial of the sixth degree is very good and that the evaluation of the correct order of magnitude of the corrosion current density, in the presence of an ohmic contribution to the electrode potential, requires that the actual values of the Tafel slopes be known. 

II. Electrodes Under Thin-Layer and Semi-infinite Diffusion Conditions and Indirect Coulometric Iterations, William H. Heineman, Fred M. Hawkridge, and Henry N. Blount Polynomial Approximation Techniques for Differential Equations in Electrochemical Problems, Stanley Pons Chemically Modified Electrodes, Royce W. Murray... 

Orthogonal collocation in two dimensions has been used to simulate microdisk edge effects. The first paper in a series (5 up till now), by Speiser and Pons (1982) is a formidable tour de force. A two-dimensional set of polynomials is fitted to the grid and it leads, as in one dimension, to an "easily solved" set of ordinary differential equations. In the fifth part of this series of papers, Cassidy et al (1985), applied the method to electrode ensembles. This is obviously not for the occasional simulator, who is advised to use a simple technique and put up with the long computational times or use someone else s program but the method undoubtedly makes two-dimensional simulations efficient and accurate. 

This is one of the variants of the FEMs. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi or Chebyshev polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their uneven spacing, crowding closer both at the electrode and (perhaps strangely) at the outer limit, and the fact that the outer limit is always unity. This is discussed below. 

The essence is that, if the concentration profile simulated is smooth (which it normally is), then the polynomials will be well behaved in between points and no such problems will be encountered. As is seen below, implicit boundary values can easily be accommodated, and by the use of spline collocation [179-181], homogeneous chemical reactions of very high rates can be simulated. This refers to the static placement of the points. Having, for example, the above sequence of points for five internal points, the point closest to the electrode is at 0.047. This will be seen, below, to be in fact further from the electrode than it seems, because of the way that distance X is normalised so that, for very fast reactions that lead to a thin reaction layer, there might not be any points within that layer. Spline collocation thus takes the reaction layer and places another polynomial within it, while the region further out has its own polynomial. The two polynomials are designed such that they join smoothly, both with the same gradient at the join. This will not be described further here. 

A recent paper by Bieniasz [6] presented a method for computing machine-precision current values for the ultramicroband electrode, using MATLAB, but not the values themselves. The method requires heavy computing, and a follow-up paper provides equally accurate polynomial approximations in the form of coefficients [7],... 

Thallium(l) chloride and bromide do not form stable complexes with typical anions and do not undergo disproportionation thus electrodes using these salts are characterized by good performance and low temperature hysteresis [153, 154]. Upon galvanostatic polarization they exhibit slight hysteresis, however, lower than for calomel electrode. Electrode potentials are reversible with respect to temperature changes and the dependence of potential on temperature is described by third-degree polynomial functions [153, 154] ... 

Fig. 8 Top variation of the average surface charge a) = Q)/A with potential, for a supercapacitor composed of a l-butyl-3-methylimidazolium hexafluorophosphate ionic liquid electrolyte and graphite electrodes. The points are raw data extracted from CGMD simulations while the lines are different polynomial fits of the data. Bottom Surfacic differential capacitance, which is either calculated by differentiating a = f(A ) (the colors match with the top panel plots), or from the fluctuations of the charge, using importance sampling methods (WHAM). °...

Such an example is shown on Fig. 8, which corresponds to the case of a butyl-methylimidazolium hexafluorophosphate ionic liquid (with graphite electrodes). The polynomial fits lead to differential capacitances which vary very diffently with potential from one case to another. Several peaks are observed, but their positions change markedly depending on the fit. 

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