A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 46
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More precisely, the linear treatment is adequate as long as the second (constant)term of (5.2.18) is small compared to the Cottrell term. For accuracy within a%,nFAD0C%^^nFADfC%or-2With a = 10% and Do = 10~5 cm2/s, ^/2/r0 ^ 18 s1/2/cm. A typical mercury drop mightbe 0.1 cm in radius; hence the linear treatment holds within 10% for about 3 s.The numerator of (5.2.26) is the thickness of the diffusion layer; thus the importanceof the steady-state term, which manifests spherical diffusion, depends mainly on the ratioof that thickness to the radius of the electrode. When the diffusion layer grows to a thickness that is an appreciable fraction of r0, it is no longer appropriate to use equations forlinear diffusion, and one can expect the steady-state term to contribute significantly to themeasured current.5.23Microscopic and Geometric AreasIf the electrode surface is strictly a plane with a well-defined boundary, such as anatomically smooth metal disk mounted in a glass mantle, the area A in the Cottrell equation is easily understood.
On the other hand, real electrode surfaces are not smoothplanes, and the concept of area becomes much less clear. Figure 5.2.2 helps to definetwo different measures of area for a given electrode. First there is the microscopic area,which is computed by integrating the exposed surface over all of its undulations,crevices, and asperities, even down to the atomic level. An easier quantity to evaluateoperationally, is the geometric area (sometimes called the projected area). Mathemati-5.2 Potential Step Under Diffusion Control167- Geometric areaFigure 5.2.2 Electrode surfaceand the enclosure formed byprojecting the boundary outward inparallel with the surface normal.The cross-section of the enclosure isthe geometric area of the electrode.Projected enclosure >Rough electrode surface -cally, it is the cross-sectional area of the enclosure formed by projecting the boundaryof the electrode outward in parallel with the mean surface normal.
The microscopic areaAm is, of course, always larger than the geometric area Ag, and the roughness factor p isthe ratio of the two:P = AJA%(5.2.27)Routinely polished metal electrodes typically have roughness factors of 2-3, but singlecrystal faces of high quality can have roughness factors below 1.5. Liquid-metal electrodes (e.g., mercury) are often assumed to be atomically smooth. One can estimate themicroscopic area by measuring either the double-layer capacitance (Section 13.4) or thecharge required to form or to strip a compact monolayer electrolytically from the surface.For example the true areas of platinum and gold electrodes are often evaluated from thecharge passed in removal of adsorbed films under well-defined conditions.
Thus, the truearea of Pt can be estimated from the charge needed to desorb hydrogen (210 /xC/cm2) andthat of Au from the reduction of a layer of adsorbed oxygen (386 fxC/cm2) (4). Uncertainty in this measurement arises because it can be difficult to subtract contributions fromother faradaic processes and double-layer charging and because the charge for desorptiondepends upon the crystal face of the metal (see Figure 13.4.4).The area to be used in the Cottrell equation, or in other similar equations describingcurrent flow in electrochemical experiments, depends on the time scale of the measurements. In the derivation of the Cottrell equation (Section 5.2.1), the current is defined bythe flux of species diffusing across the plane at x = 0. The total rate of reaction in molesper second, giving the total current in amperes, is the product of that flux and the crosssectional area of the diffusion field, which is the area needed for the final result.In most chronoamperometry, with measurement times of 1 ms to 10 s, the diffusionlayer is several micrometers to even hundreds of micrometers thick.
These distances aremuch larger than the scale of roughness on a reasonably polished electrode, which willhave features no larger than a small fraction of a micrometer. Therefore, on the scale ofthe diffusion layer, the electrode appears flat; the surfaces connecting equal concentrations in the diffusion layer are planes parallel to the electrode surface; and the area of thediffusion field is the geometric area of the electrode. When these conditions apply, as inFigure 5.2.3a, the geometric area should be used in the Cottrell equation.Let us now imagine a contrasting situation involving a much shorter time scale, perhaps 100 ns, where the diffusion layer thickness is only 10 nm. In this case, depicted inFigure 523b, much of the roughness is of a scale larger than the thickness of the diffusion layer; hence the surfaces of equal concentration in the diffusion layer tend to followthe features of the surface.
They define the area of the diffusion field, which is generallylarger than the geometric area. It approaches the microscopic area, but might not be quiteas large, because features of roughness smaller in scale than the diffusion length tend tobe averaged within the diffusion field.168 • Chapter 5. Basic Potential Step MethodsElectrodeElectrode(a)(b)Figure 5.2.3 Diffusion fields at (a) long and (b) short times at a rough electrode.
Depicted here isan idealized electrode where the roughness is caused by parallel triangular grooves cut on linesperpendicular to the page. Dotted lines show surfaces of equal concentration in the diffusion layer.Vectors show concentration gradients driving the flux toward the electrode surface.Similar considerations are needed to understand chronoamperometry at an electrode that is active only over a portion of a larger area, as in Figure 5.2.4. Such a situation can arise when arrays of electrodes are fabricated by microelectronic methods, orwhen the electrode is a composite material based on conducting particles, such asgraphite, in an insulating phase, such as a polymer.
Another important case involves anelectrode covered by a blocking layer with pinholes through which the electroactivespecies may access the electrode surface (Section 14.5.1). At short time scales, whenthe diffusion layer thickness is small compared to the size of the active spots, each spotgenerates its own diffusion field (Figure 5.2.4a), and the area of the overall diffusionfield is the sum of the geometric areas of the individual active spots. At longer timescales, the individual diffusion fields begin to extend outside the projected boundariesof the spots, and linear diffusion is augmented by a radial component.
(Figure 5.2.4Z?)At still longer time scales, when the diffusion layer is much thicker than the distancesbetween the active zones, the separated diffusion fields merge into a single larger field,again exhibiting linear diffusion and having an area equal to the geometric area of theentire array, even including that of the insulating zones between the active sites (Figure5.2.4c). Thus, the individual active areas are no longer distinguishable. Molecules diffusing to the electrode come, on the average, from so far away that the added distance(and time) required to reach an active place on the surface becomes negligible. Thisproblem has been treated analytically for cases in which the active spots are uniform insize and situated in a regular array (5), but in the more general case digital simulation isrequired.Since capacitive currents are generated by events occurring within very small distances at an electrode surface (Chapter 13), they always reflect the microscopic area.
Foran electrode made of a polished polycrystalline metal, the area giving rise to a nonfaradaic current may be significantly larger than that characterizing the diffusion field. Onthe other hand, the opposite can be true if one is using an array of small, widely spacedelectrodes embedded in an inert matrix.5.3 DIFFUSION-CONTROLLED CURRENTS ATULTRAMICROELECTRODESEarly in this chapter we anticipated the unusual and advantageous properties of electrodes with very small sizes. Here we explore those properties more carefully, and wegain our first view of some of the experimental methods based on them. We will return5.3 Diffusion-Controlled Currents at Ultramicroelectrodes <4 169Solution phaseInactive surfaceД.1.,r—У---У—.Insulating matrixElectronicallyconducting phase(a)(b)у—чу* f-\f—/\jч/i1 11 1! 11 11 II 11 II \ 1 i|Figure 5.2.4 Evolutionof the diffusion field duringchronoamperometry at an electrodewith active and inactive areas on itssurface.
In this case the electrode isa regular array such that the activeareas are of equal size and spacing,but the same principles apply forirregular arrays, (a) Shortelectrolysis times, (b) intermediatetimes, (c) long times. Arrowsindicate flux lines to the electrode.(c)repeatedly to these UMEs and their applications. Since the first edition, no advance haschanged electrochemical science to a greater degree than the advent of UMEs, whichoccurred principally through the independent work of Wightman and Fleischmann andtheir coworkers about 1980 (6, 7).
These devices have extended electrochemicalmethodology into broad new domains of space, time, chemical medium, and methodology (6-13). In the remainder of this book, we will encounter many illustrations basedon them.It is obvious that UMEs are smaller than "normal" electrodes, which, depending onthe application, might have dimensions of meters, centimeters, or millimeters. At present, there is no broadly accepted definition of a UME, although there is a generalagreement on the essential concept, which is that the electrode is smaller than the scaleof the diffusion layer developed in readily achievable experiments.
Not all applicationsdepend on the development of such a relationship between the diffusion layer and theelectrode, but many do. To understand them, one must recognize the peculiar featuresof such systems and treat them theoretically. Other applications of UMEs rest on thesmall time constants or low ohmic drops that are characteristic of very small electrodes(Section 5.9).In this book, we define a UME operationally as an electrode having at least one dimension (such as the radius of a disk or the width of a band) smaller than 25 /xm.
This170Chapter 5. Basic Potential Step Methodsaspect is called the critical dimension. Electrodes with a critical dimension as small as 0.1fim ( = 1 0 0 nm = 1000 A) can be made. Even smaller critical dimensions, down to a fewnm, have been reported. When the electrode's critical dimension becomes comparable tothe thickness of the double layer or to the size of molecules, one can expect to deal withnew elements of theory and experimental behavior. These considerations place a lowerlimit of about 10 nm (100 A) on the critical dimension of UMEs (12, 14, 15). Electrodessmaller than this limit have been called nanodes in some of the literature, but a definitionof this term based on function is yet to be worked out.5.3.1Types of UltrainicroelectrodesSince only one dimension of an electrode must be small to produce properties characteristic of a UME, there is a good deal of latitude in other physical dimensions and, consequently, a variety of useful shapes.Most common is the disk UME, which can be fabricated by sealing a fine wire in aninsulator, such as glass or a plastic resin, and then exposing and polishing a crosssection of the wire.
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