J. Ross Macdonald

Simplified impedance/frequency‐response results for intrinsically conducting solids and liquids

The small‐signal ac response is considered of a system containing a single species of positive charge and a single species of negative charge. The charge carriers may be of many different types (ions, electrons, vacancies, etc.) and are assumed to have arbitrary mobilities and valences. Quite general boundary conditions are considered which encompass the range from complete blocking to zero blocking (infinite reaction rate at the electrodes) for positive and negative charges separately. The present paper deals primarily with approximations to an earlier exact solution of the problem which, in general, lead to an equivalent circuit made up of three parallel R C combinations in series. The elements of one of these parallel circuits, associated only with bulk effects, are frequency independent, and those of another, which are associated with nonzero blocking, may often be well approximated as independent. The third R C section arises from diffusion effects, involves frequency dependent elements, and exhibits...

Theory of space‐charge polarization and electrode‐discharge effects

A combined, general treatment of intrinsic and extrinsic conduction in a liquid or solid is presented. Positive and negative species of mobile charge of arbitary valences and mobilities are assumed present, together with homogeneous immobile charge in the extrinsic case. The general equations are specialized to a one‐dimensional situation and then to that where both position‐dependent static and much smaller sinusoidally time‐varying components of charge, field, and current are simultaneously present. Sufficiently general boundary conditions are used that any condition from complete blocking to free discharge of positive and negative mobile carriers separately can occur at the electrodes. For the flat‐band condition (zero static field; in the binary electrolyte case, coincidence of the zero charge potential and the equilibrium potential) exact equivalent circuits and an exact expression for the small‐signal impedance are obtained. Relatively simple, closed‐form expressions for the zero‐frequency limiting ...

Theory of Double‐Layer Differential Capacitance in Electrolytes

A theory of the double layer in uni‐univalent unadsorbed electrolytes is developed and used to analyze Grahames experimental measurements of differential capacitance for NaF in water at 0° to 85°C and KF in methanol at 25°C. Excellent agreement with experiment is obtained except in the region of strong anodic polarization; this disagreement is tentatively ascribed to specific adsorption of anions, an effect not quantitatively considered in the present work. Although the quantities calculated herein relate to the entire double layer as, of course, do Grahames data, the Gouy—Chapman theory of the diffuse part of the double layer (without dielectric saturation) is adequate in the present situation for all concentrations considered and has been used throughout. Consequently, the degree of agreement between theory and experiment found reflects primarily upon the applicability of the present theory of the inner layer. In the absence of specific adsorption this region is taken to be a hexagonally close‐packed ...

BINARY ELECTROLYTE SMALL-SIGNAL FREQUENCY RESPONSE

Abstract The frequency response is considered of a two-electrode linearized system containing a single positively charged species and a single negatively charged species. These species may have arbitrary valences and mobilities and may individually react at the electrodes. The results follow from a detailed solution of the equations of charge motion given earlier. Normalized response is exhibited for this unsupported, intrinsic-conduction situation for a wide range of mobility ratios, valence number ratios, and reaction rate ratios. Results are given in the form of specific formulas, impedance-plane plots, and the dependences on normalized frequency of series and parallel resistive and capacitative components of the normalized total impedance of the system. Impedance-plane plots exhibit from one to three connected arcs, depending on the specific situation. Approximate Warburg frequency response appears for the “interface” impedance over a certain frequency region when normalized reaction rate parameters differ, but it only shows up strongly in the total impedance when the mobility ratio departs appreciably from unity as well. Under such conditions, a plateau region, where the total parallel capacitance remains essentially independent of frequency over a wide frequency range, may appear at frequencies just above the Warburg region. The plateau capacitance is close to but not identical to the conventional double-layer capacitance present when both species of charge are completely blocked. In incomplete blocking cases, however, this double-layer capacitance only makes a significant appearance in the approximate equivalent circuit under slow reaction conditions; it is thus not present when one of the reaction rate constants is infinite. In general, the system can show ω−m frequency response for the parallel capacitance over a wide frequency range with 0⩽m⩽2, and with the experimentally common regions where m≌0, 0.5, 1.5, and 2 especially likely. Particular attention is given to deviations from ideal Warburg behavior which led to a combined charge-transfer and heterogeneous chemical reaction resistance. Results are compared to those from conventional supported treatments and show both important similarities and differences. Finally, several new equivalent circuits are presented which are pertinent in various frequency ranges for the unsupported situation.

Erratum: Electrical Response of Materials Containing Space Charge with Discharge at the Electrodes

The system considered consists of a solid or liquid material containing mobile positive and negative charges between two identical plane electrodes separated by a distance l. The results obtained apply also for a single working electrode, without specific adsorption, and an indifferent electrode. Uni‐univalent, nonrecombining positive and negative charges, usually of equal mobility, are assumed. Frequency and transient responses are compared in the linear regime for two different boundary conditions. One condition requires complete blocking of positive and negative charges at the electrodes; the other requires complete blocking for charges of one sign but allows free discharge of those of opposite sign. The working electrode is thus reversible, or Ohmic, for charges of one sign. The method of solution includes electromigration effects, does not require electroneutrality anywhere, and leads to results satisfying Poissons equation exactly everywhere for all times and frequencies. These results apply to sol...

Accelerated Convergence, Divergence, Iteration, Extrapolation, and Curve Fitting

This paper discusses some applications of the epsilon algorithm (EA), a sequential procedure for calculating Pade approximants. The EA may be used to: (1) accelerate the convergence of slowly converging series and iterations; (2) obtain useful results from divergent series and iterations; (3) obtain the limits of iterated vector and matrix sequences; (4) aid in the solution of differential and integral equations; (5) carry out numerical integration in a new way; (6) extrapolate; (7) fit a curve to a polynomial or to a constant plus sum of exponentials.As an illustration of curve fitting and extrapolation, we present results obtained with exact polynomial data plus random noise combined additively or proportionately. For such nonstationary data, the results are comparable, and in some cases superior, to least squares in yielding good estimates of the exact polynomial coefficients. One important advantage of the EA is that it builds up polynomials whose lower‐order coefficients are independent of higher‐ord...

Static Space Charge and Capacitance for a Single Blocking Electrode

Static space‐charge distributions in materials having one charge blocking and one ohmic electrode are considered with special emphasis on the situation where charge carriers of only one sign are mobile but which may recombine bimolecularly with fixed charges of opposite sign. The dependence of potential, charge, and electric field on distance from the blocking electrode cannot be obtained exactly in closed form but various simple approximate relations are obtained and are compared with accurate digital computer solutions of the exact relation between potential and distance. Comparison is most significant when the distance scale is normalized by the effective Debye length, a quantity which is shown to depend on recombination ratio when charges of only one sign are mobile. The dependence of total space‐charge and differential and static space‐charge capacitance on applied potential and recombination is obtained in closed form, and it is shown that recombination can lead to peaks in the curves of static and ...

The impedance of a galvanic cell with two plane-parallel electrodes at a short distance

Summary A comparison is made, for frequency and electrode-separation dependence of total cell impedance and “interface” impedance, between electrolyte situations with and without a supporting indifferent electrolyte. When no indifferent electrolyte is present, a binary electrolyte situation, two boundary conditions are considered: that of ideally polarized electrodes (complete blocking), and that where charges of one sign are blocked at the electrodes and those of opposite sign are able to pass freely in and out of the electrodes. Further, electroneutrality is not assumed. The usual solution when an indifferent electrolyte is present assumes the presence of reactions of charges of both signs at the electrodes and electroneutrality everywhere. All electrode reactions are taken to be infinitely fast and no specific adsorption is present. Further, no applied static overpotential is assumed, so the results apply most closely at the equilibrium potential. Comparison of the results for the various cases indicates both important similarities and differences. It is shown that when there is no indifferent electrolyte, particular care must be taken in analyzing the total impedance to obtain an appropriate “interface” impedance. Finally, equivalent circuits pertinent for the various situations are considered in detail, and an important equality between structurally different circuits is pointed out.

Static Space‐Charge Effects in the Diffuse Double Layer

Exact solutions of the static Debye‐Huckel space‐charge equations for one and for two blocking electrodes are compared analytically and graphically. It is found that the simpler one‐electrode solution may be employed to characterize the diffuse double layer with high accuracy in most experimental situations in place of the less tractable two‐electrode solution which involves Jacobian clliptic functions. Some consequences of the strong nonlinear voltage dependence of the exact solutions are considered and the application of the solutions to physical situations is discussed.

Static Space Charge and Capacitance for Two‐Blocking Electrodes

A one‐dimensional treatment is presented of space‐charge effects in materials having two charge‐blocking electrodes. Especial attention is given to the situation where univalent, mobile charge carriers of only one sign can recombine bimolecularly with fixed charges of opposite sign; however, the case where noncombining charges of both signs are mobile is also included. Space‐charge potential distributions and differential capacitance cannot be obtained explicitly in the general case but have been accurately calculated using a digital computer. Potential vs distance and capacitance curves are presented which illustrate dependence on applied potential, separation of electrodes, and recombination ratio. The results show features often observed experimentally for a wide variety of materials such as photoconductors, semiconductors, and insulators. The treatment is applicable for much higher applied potentials than is the case for the previously considered one blocking electrode situation with charge accumulati...