Donald R. Franceschetti

Theory of small‐signal ac response of solids and liquids with recombining mobile charge

An exact, small‐signal theory of the impedance of an electrode/material/electrode system under quite general conditions is presented. The system, assumed flat band, consists of a slab of material between two identical plane‐parallel electrodes. The material may be a nondegenerate electronic semiconductor or an ionic conductor. Solid ionic conductors considered are Schottky and Frenkel defect materials, possibly containing neutral defect pairs and/or aliovalent impurities, and fast ion conductors such as Na‐β‐alumina. Liquid ionic conductors treated include unsupported strong, weak or potential electrolytes, and possibly fused salts and oxides. Both intrinsic and extrinsic conduction conditions are included, with a single species of negative mobile charge of arbitrary valence and mobility and a single species of positive mobile charge of arbitrary valence and mobility assumed present. Intrinsic and extrinsic equilibrium and dynamic generation and recombination processes are taken into account. The boundary...

Electrode kinetics, equivalent circuits, and system characterization: Small-signal conditions

Abstract The small-signal steady-state response around the point of zero charge of an electrode/material system is examined for an unsupported electrolyte (material) with two species of charge carrier of arbitrary mobilities and valence numbers and with arbitrary intrinsic/extrinsic conduction character, taking full account of bulk, electrode reaction, sequential adsorption, and diffusion processes. The exact solution of the transport equations of the problem for generalized Chang-Jaffe single-point boundary conditions is compared with the responses of a variety of plausible equivalent circuits, using a complex least squares fitting technique. A hierarchical circuit is found which closely reproduces the exact results when charge of one sign is completely blocked without adsorption, except for some of the cases in which diffusion and reaction effects interfere with each other. The circuit is composed of frequency-independent lumped capacitances and resistances separately identified with bulk, reaction, and adsorption/reaction processes and a single, finite-length, Warburg-like impedance for diffusion effects. Relations between the circuit elements and microscopic electrode/material parameters are found and apply irrespective of the time-frequency overlap between bulk, reaction, and adsorption processes. It is also found that the reaction and adsorption resistances and the adsorption capacitance are all strongly interrelated. The circuit may be used with simultaneous non-linear least squares fitting of the real and imaginary parts of experimental impedance data to obtain estimates of the values of circuit elements and thus of the values of the microscopic parameters characterizing the electrode/material system. The relationship of small-signal response for overpotential-dependent electrode kinetics to that obtained for Chang-Jaffe boundary conditions is then considered. The reaction resistance and adsorption capacitance are found to be formally identical for Butler-Volmer (or Butler-Volmer-like) and Chang-Jaffe conditions. In the d.c. limit these quantities are unchanged, for the boundary conditions just mentioned, when a supporting electrolyte is added. A transformation of variables method is described which permits one to determine the smallsignal impedance of an electrode/material system with general overpotential-dependent firstorder electrode reaction kinetics from a compact-layer model for the small-signal overpotential and the small-signal response obtained for Chang-Jaffe boundary conditions. Exact impedance results are given for the case of a single species of mobile charge carrier. Analysis of this case indicates that the overpotential dependence of the boundary conditions has negligible effect on the small-signal response unless the cell is of microscopic thickness or the Debye length is comparable to the compact layer thickness and the electrode reaction is slow.

Diffusion of neutral and charged species under small-signal a.c. conditions*

Abstract Several of the ways in which diffusion of an electroactive species may affect the small-signalresponse of an electrochemical system are examined, with particular attention to cases in which the electrode reaction produces or consumes a neutral species whose concentration at the electrode surface is determined by diffusion through the electrode. The conventional (time domain) rate and diffusion equations may be expressed in the frequency domain through the use of complex, frequency-dependent rate constants, whose form reflects the sequence of events in the overall reaction, including possible adsorption steps, and leads directly to equivalent-circuit representations of the pertinent parts of the system response. The complex rate constant formalism also allows the immediate generalization of existing exact treatments of unsupported systems to include such diffusion effects.

Numerical analysis of electrical response: Statics and dynamics of space‐charge regions at blocking electrodes

The steady‐state and transient electrical properties of a material containing one or two species of mobile charge carrier in contact with blocking electrodes are examined. The systems treated are 2, 20, and 40 Debye lengths in extent and exemplify the transition from thin‐film or membranelike behavior to the more usual case in which well‐defined space‐charge layers form at each electrode. The static capacitance of the electrode/material/electrode system is examined and the response of the system to a step‐function applied potential difference is obtained by numerical simulation. The simulation results show clearly the role of the system length and charge carrier mobilities in determining the system response. The decay of total current following the potential step is numerically fitted to a sum of exponential decays. The nonlinear character of system response becomes apparent when the transient current associated with the formation of space‐charge layers in response to a potential step is compared with tha...

Numerical analysis of electrical response: Biased small-signal a.c. response for systems with one or two blocking electrodes

Abstract The small-signal a.c. response of systems with two blocking electrodes, or one blocking and one ohmic electrode, subject to an external steady voltage bias is examined. The electrolyte is unsupported and may contain one or two mobile charge carrier species. The small-signal response displays features which are readily associated with the redistribution of charge within the system caused by the external bias and the injection of charge into a system with one non-blocking electrode. A procedure is given for approximating the small-signal response of a system containing many Debye lengths from the small-signal response of a thinner system. In a addition to describing the impedance and admittance of an initially flat-band system subject to an external bias, the results obtained are also applicable to a limiting case of an unbiased system with intrinsic space charge (Frenkel) layers. An equivalent circuit is found which usually provides an adequate approximate representation of the system response, and the physical interpretation of the circuit parameters is discussed.

COMPACT AND DIFFUSE DOUBLE LAYER INTERACTION IN UNSUPPORTED SYSTEM SMALL-SIGNAL RESPONSE *

Abstract Previous exact results for the small-signal impedance of an unsupported electrode/material/electrode system which include effects of the finite size of charge carriers are simplified and discussed. The material contains non-recombining charges of opposite sign with the positive one immobile and uniformly distributed. General boundary conditions which encompass the range from no electrode reaction to ohmic electrode behavior are employed. In the presence of an electrode reaction, the interaction of the compact and diffuse double layers leads to considerably more complexity in the equivalent circuit than might appear in simple treatments of the supported case, in which the diffuse double layer capacitance is neglected or the compact double layer and diffuse double layer capacitances are placed in series. Two different approximate equivalent circuits made up of frequency-independent elements are found which yield remarkable agreement with the exact results over the entire frequency range of interest. The first involves the ordinary approximate circuit (OAC) previously found in the absence of compact layer effects plus a series compact layer contribution involving a parallel resonant circuit with quality factor at resonance which may approach unity. Pseudo-inductance effects are found to be extremely significant in this representation. The second approximate equivalent circuit, simpler and almost as accurate as the first, has the same form as the original OAC but with its reaction element values altered by the presence of the compact layer. For non-Butler-Volmer electrode kinetics an upper limit is found for the experimentally determinable apparent reaction rate constant, a feature of practical importance for thin films or membranes. The response of thin films and membranes, including compact layer effects, can very readily be erroneously confused with pure bulk response, yielding entirely incorrect values for the geometrical capacitance and bulk resistance of the material.

Iterative extended Hückel studies of some pyridine-Fe(II)-porphin complexes

Iterative extended Hückel self-consistent charge calculations were performed for a number of conformations of Fe(II) porphin, a pyridine complex of Fe(II) porphin, and a pyridine-Fe(II)-porphin CO complex. For Fe(II) porphin, spin zero and spin 2 states were found to have their energy minimum in a planar conformation. Addition of the pyridine ligand results in an out-of-plane minimum energy conformation in the spin 2 case. Ligand bond formation is readily interpreted in terms of conventional chemical bonding theory. An energy-bond length curve was obtained for CO binding to pyridine-Fe(II)-porphin and found to be qualitatively reasonable.

On generation recombination and trapping kinetics in theories of small‐signal electrical response

It is shown that small‐signal electrical response results obtained for generation‐recombination or trapping rate laws appropriate to a neutral‐center dissociation‐association mechanism encompasses the electrical response to be expected for other frequently encountered generation‐recombination or trapping rate laws under a wide range of conditions. The rate constants appearing in such kinetic expressions are related to a number of differently defined charge‐carrier lifetimes employed in the semiconductor and electrochemical literature. The ranges of charge‐carrier lifetime characteristic of semiconductors, defect solid ionic conductors, and electrolyte solutions are briefly discussed.

Semi-empirical study of a reduced rubredoxin analogue☆

Abstract Molecular orbitals were determined for an analogue of reduced rubredoxin in an iterative extended Huckel (self-consistent charge) calculation. Zero field splittings and Mossbauer spectral parameters were then computed. The results are compared with available experimental data.

Comment on ’’Theory and analyses of the ac characteristics of defect thin‐film insulators’’

A theory of metal‐insulator‐metal thin films proposed by Nadkarni and Simmons is examined in relation to earlier treatments. Attention is drawn to approximations inherent in the theory and physical processes not considered by the authors.