Electrochemistry Volume 8

A Review of Recent Literature

By D. Pletcher

The Royal Society of Chemistry

Copyright © 1983 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85186-067-1

Contents

Chapter 1 The Electrochemistry of Porous Electrodes: Flooded, Static (Natural) Electrodes By N.A. Hampson and A. J. S. McNeil, 1,
Chapter 2 Electrode Processes in Molten Salts By J. Robinson, 54,
Chapter 3 The Electrochemistry of Transition-metal Complexes By C.J. Pickett, 81,
Chapter 4 The Electrochemistry of Oxygen By D.J. Schiffrin, 126,
Chapter 5 Organic Electrochemistry – Synthetic Aspects By J. Grimshaw and D. Pletcher, 171,

CHAPTER 1

The Electrochemistry of Porous Electrodes: Flooded, Static (Natural) Electrodes

BY N A. HAMPSON AND A.J.S. McNEIL

1 Introduction

The theory of electrochemistry that is presented in the standard textbooks has been obtained by considerations of ideal electrodes, and generally confirmed by experiments with mercury and amalgam electrodes, which present the nearest approach to the ideal situation. The ideal solid electrode is smooth, of accurately known surface area and crystal orientation, structurally perfect, and strain-free. The practical electrochemistry that is encountered in industry is concerned with electrodes that are rough and which present a large number of differently oriented crystal faces to the electrolyte solution. Often, these surfaces are fissured, and they may even contain phase demarcations. The need to present the maximum surface area to the reacting electrode/electrolyte interface has inevitably resulted in the development of quite porous electrodes, such as are commonly found in the electrical storage-battery industry. The lead-acid cell and the Leclanché cell, the two best known to commerce, both contain porous electrodes. Indeed, the authors do not recall a single example of a storage cell consisting of two non-porous solid electrodes.

Porous electrodes can be subdivided into five distinct classes:

(i) a prefatory class of rough electrodes, in which the surface area is somewhat increased over the projected, geometric area of the electrode; all ‘plane’ electrodes are rough to some extent;

(ii) porous or granular electrodes, produced by a specialized process of electrodeposition;

(iii) hydrophobic gas electrodes, whose operation depends critically upon the establishment of a three-component (solid-liquid-gas) interphase;

(iv) flow-through electrodes, with forced input of reactants; and

(v) ‘natural’, flooded, static porous electrodes.

Rangarajan has presented a brief review of the theory and operations of porous electrodes, with a classification of the operating models. The authors are not aware of any review of gas electrodes [class (iii)] that has been made since the survey of fuel cells by Bockris and Srinivasan in 1969. Newman and Tiedemann have on two occasions reviewed the subject of flow-through electrodes [class (iv)]. Work on both of these classes of porous electrodes over the past decade will be reviewed in the next volume in this series.

Although the basic theories have remained unchanged since the review in 1966 by de Levie, a substantial amount of theoretical development and confirmatory experimental work has been carried out in connection with the major porous electrodes of electrotechnology. It is timely, then, to review the progress made in understanding the fundamental electrochemistry of flooded porous electrodes [class (v)]. Although the majority of the papers discussed in this Report are from the main scientific sources, any of the most important contributions that appear uniquely in the published proceedings of symposia have also been included.

It is worthwhile, at this point, to indicate briefly the major avenues of approach that have been followed in studies of porous electrodes. The more important electrochemical relationships are noted, since these are not found in standard works on electrode kinetics.

There are, fundamentally, two approaches which can be taken in order to deal with the porous electrode. First of all, porous electrodes can be considered as extensions of planar electrodes of known electrode-kinetic behaviour. This is the discrete-pore-model approach; historically, it provided the first ex planation of the behaviour of porous electrodes. It was developed to a high degree, notably by Frumkin, Winsel, and de Levies (the review by de Levies contains a thorough account of the early work). Differences from the relationships for the plane electrode arise because, in the ideal porous electrode (i.e. with circular pores), the current, instead of arriving normally to the plane of the electrode, arrives parallel to it. This consideration engenders the concept of the penetration depth, the interlinking of ohmic, concentration, and activation polarizations, and the ‘halving’ of the time–dependent (transient) responses.

The other approach (which, if anything, has been more successful than the pore model) is the macrohomogeneous model. This was first effectively used by Newman and Tobias; the porous electrode was considered to be an ‘average’ of the solid electrode and the electrolyte. Thus, the effective conductance of the porous electrode was the weighted volume average of the respective conductances; diffusion coefficients were similarly averaged, and so on. In this case, however, the electrochemistry cannot be taken from that of the plane electrode; the potential-current relationship must be obtained from the porous electrode by measurement. This is the least satisfying aspect of the macrohomogeneous approach. A possible solution might be to use the pore model to establish the electrochemistry; in general, however, it is clearly better to use experimental methods. The macrohomogeneous model will clearly be the more useful for the electrochemical engineer; however, the single-pore approach is still proving useful in developing our understanding of porous electrodes.

2 The Development of Theories

The Single-pore Model. — We summarize the state of the theory of porous electrodes at the time of the review by de Levie in 1966.

In spite of the clearly evident need to consider porosity of the electrode in relation to electrode-kinetic investigations of many types, it is not customary to do so. This arises from two causes. One is that, to a first approximation, porous and rough electrodes behave as smooth electrodes of enhanced surface area. The other is that porosity is difficult to incorporate into electrode kinetics, because of lack of definition of the porous electrode. Simplification of the porous electrode to give, for example, a parallel array of pores of uniform diameter is an obvious first extension from the planar electrode. In considering this model, the pore is essentially one–dimensional and the resistance of the electrolyte is uniformly distributed along its length. The simplest approach is to assume that the pore is of uniform cross–section and completely filled with solution. Frumkin 6 considered the curvature of the equipotential surfaces within the pore; however, by replacing the equipotential surfaces by the mean values which lies in planes perpendicular to the axis of the pore, the problem is avoided, as the model becomes essentially one-dimensional.

de Levie has shown that the transmission-line representation leads to the expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

for the potential drop (E0 – Ex) over the distance x from the mouth of the pore (x = 0) within a pore of length l. The important constant ρ has the form:

ρ = (Rω/RD)½ (2)

where Rω is the ohmic resistance of the solution within the pore for unit pore length (as distinct from that of the bulk solution) and RD is the charge-transfer resistance for unit length. The quantity ρ-1 has the dimensions of length and is called the ‘penetration depth’. The current at the mouth of the pore(total current) is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The pore behaves as a resistance, according to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

If we assume a cylindrical pore, the value of Rω in the pore can be calculated as:

Rω)-1 = πa2κ (5)

for a radius a cm and solution conductance κ [OMETGA]-1 cm-1, and with RD terms of the exchange-current density i0 using:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The penetration depth becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Hence, the penetration depth decreases with decreasing κ and pore radius and with increasing i0. A number of workers have attempted to get exact solutions for the current-potential characteristics of a circular pore, assuming linearity of the characteristic, but, without a detailed knowledge of the structure of a pore, there seems little point in this, particularly as the more complex situations yield solutions which agree with the simple ones to within 5%, provided that the pores are of significant depth.

The simple use of the specific charge-transfer resistance for the impedance of the electrode surface only holds for small polarizations, generally of the order of a few millivolts. For more significant polarizations, when the planar electrode (under charge-transfer control) is expected to follow Erdey-Gruz kinetics, 14 the Tafel equation can be applied in the region for which the overpotential (η) exceeds some tens of millivolts. For a cylindrical pore, the Tafel slope dE/d(ln i) is double that which would characterize the corresponding planar electrode.

The fact that the form of the kinetic equations for an electrode involves the product of both charge-transfer and mass-transport characteristics clearly emphasizes these two as equally important modes of limitation of the current for a porous electrode. Changes of concentration within the pores of the electrode can obviously be just as important as the Tafel behaviour in limiting the current. de Levie catalogues the early work and shows that a limitation due to diffusion results in a current-potential relationship that is similar to the doubled (planar) Tafel behaviour of the charge-transfer-limited system.

The penetration depth is a function of both i0 and the concentration at the opening of the pore; even at the mouth of the pore, the concentration (cx = 0) is different from that of the bulk (cb), and the current into the porous electrode results from Fick’s first law, as:

ix=0 = (cb – cx=0)πa2zFD (8)

where δ is the thickness of the diffusion layer, provided that the front of the electrode can be treated as flat. In terms of porous parameters:

ix=0 = ρcbD’ tanh ρl/(1 + ρδ tanh ρl) (9)

where D’ is an ‘effective’ coefficient for diffusion, expressed as D’ = πa2zFD in terms of the diffusion coefficient, D. When the potential becomes very large, the penetration depth is small and the diffusion-limited current becomes that of a flat electrode of the same dimensions as the projected area of the porous one. This illustrates why the use of flooded porous electrodes is more effective for slow electrode reactions, where transport of mass is less significant.

The case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where Sr and So are reduced and oxidized species, respectively) was considered by Austin and Lerner, who established the expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [varies] is the cathodic charge-transfer coefficient and cr and co refer to concentrations of reduced and oxidized species, respectively, either in the bulk solution (cb) or at the mouth of the pore (cx = 0).

The penetration depth, ρ-1, accordingly has a maximum value at:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

which is at or near the standard potential of the redox system: Dr and Do are diffusion coefficients of the reduced and oxidized species, respectively.

Generally, impedance measurements are made with small perturbations of amplitude, so that the rate equation may be considered linear. Thus, for a frequency ω, the pore exhibits an impedance:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where ZP is the impedance per unit pore length.

For a semi-infinite pore, the phase angle of Z0 (the impedance of the pore) is half the planar impedance (since Rω is a real quantity), and |Z0| is proportional to ZP½ Thus a ‘squaring’ operation is a simple way of correlating the impedance of a porous electrode with that of the corresponding flat electrode. The phase angle of the impedance of the pore is a function of the depth of the pore, and deep pores clearly contribute more significantly to the impedance of the electrode than do shallow ones.

The part of the impedance of a porous electrode that is due to double-layer charging is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [square root of (-1)] and C is the capacitance per unit length of pore. When the pore is infinite in length it exhibits a phase shift of 45° between current and potential, and it appears as a simple Warburg impedance. The reciprocal penetration depth:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

increases with ω; at sufficiently high values of ω, shallow pores will behave as inifnitely long ones. Using the equation for the impedance (Z) of an electrode reaction that is controlled by double-layer charging and the electrode reaction:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

the charge-transfer resistance is given by :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

and the familiar semicircle in the diagram in the complex plane for the planar electrode becomes lemniscate.

If the diffusion processes that are linked with the electrode reaction can be described in terms of semi-infinite linear diffusion, as was done by Winsel, then a Warburg dependance with a phase angle of 45° for the surface impedance of the pore becomes a phase angle of 22 1/2° for a semi-infinite pore.

The transient responses of porous electrodes are extended in the time domain, and steady states may take a considerable time to achieve. Equations show that the double-layer charging process is time-dependent, with a time constant of l2RωC. The charging process travels down the pore and reaches different depths l at different times; deviations from simple semi-infinite behaviour must be considered where the charging process reaches the bottom end of the pore. The problem of coupling the double-layer charging process with the transfer of charge at an electrode and the attendant diffusion is formidable. Double-layer charging is a much faster process than diffusion, and it has been suggested that the processes are separable. Ksenzhek solved the Fick equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

for a galvanostatic pulse to show that the potential increased linearly with time, at a rate that is proportional to the concentration (c) of electro-active species, providing an effective concentration impedance Z’, as was observed for the oxidation of hydrogen on porous nickel.

The Macrohomogeneous Model. — de Levies primarily considered the single-pore model, thinking of the porous electrode as a combination of single pores, essentially without cross-links. The macroscopic current is then the sum of the contributions of all the single pores. Prediction of the overall performance of the electrode, however, would require detailed knowledge of the distribution of particle sizes, which would be difficult to determine as well as to relate to practical (e.g. sintered) electrodes. In contrast, the macroscopic or continuum approach describes the whole electrode-electrolyte system as two continua (one of the electrode matrix and the other of the solution, filling in all the void fraction of the electrode). Both phases are assumed to be homogeneous, isotropic, and spatially complementary. The avoidance of detailed structural description (e.g. for tortuosity factor) is reflected in mathematical simplicity, although the mathematical expressions are analogous to those of the single-pore model.

Newman and Tobias were the first to consider explicitly the continuum approach, their initiative being followed by Grens and Tobias and by Micka. de Levies gives a brief summary of this new development as it stood in 1966.

The macrohomogeneous model, developed by Newman and Tobias, disregards the actual structural detail of the pores and regards the porous electrode as the superposition of two continua, i.e. the solution and the matrix. The current density in the electrolyte solution that is due to a flux Ni of species i is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

For the material balance,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where ε is porosity, ci is the concentration of species i, and jin is the pore- wall flux of species i, averaged over interfacial area a. For electroneutrality,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where i1 and i2 are current densities in the matrix and in the electrolyte in the pores, respectively.

The electrochemistry is ex pressed by the general relationship:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where αa and αc are transfer coefficients in the anodic and cathodic directions, respectively, and ηs is the surface overpotential.

Transport processes for the matrix phase are introduced, using Ohm’s law:

i1 = -σ[nabla]Φ1 (23)

where σ and Φ1 are the conductivity and electric potential of the matrix, respectively. For a dilute solution, within the pores, the solutes move by processes of diffusion, dispersion, migration, and convection:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

where Da is a dispersion coefficient, ui is the mobility of species i, Φ2 is electric potential in solution, and v is the fluid velocity.

These seven equations (18) — (24) are solved together, after various simplifications and adjustments, as required by the special features of the systems considered.

Theoretical Modelling of Porous Electrode Systems. — Theoretical developments cannot be separated into two distinct streams, springing from either the single-pore or the continuum theory. These approaches are complementary rather than exclusive, and investigators have employed both, in theoretical as well as experimental work. By the early 1970s, as Newman and Tiedemann remarked, the basic theoretical groundwork had been developed to the point where it could begin to be applied to almost any electrode system. What appears to be a distinctive feature of the work of the past decade is that investigators have begun to study specific electrode systems, adapting the general theoretical principles to their particular electrochemical character.

(Continues…)Excerpted from Electrochemistry Volume 8 by D. Pletcher. Copyright © 1983 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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