Spherical, diffusion electrodes

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

Diffusion of electroactive species to the surface of conventional disk (macro-) electrodes is mainly planar. When the electrode diameter is decreased the edge effects of hemi-spherical diffusion become significant. In 1964 Lingane derived the corrective term bearing in mind the edge effects for the Cotrell equation [129, 130], confirmed later on analytically and by numerical calculation [131,132], In the case of ultramicroelectrodes this term becomes dominant, which makes steady-state current proportional to the electrode radius [133-135], Since capacitive and other diffusion-unrelated currents are proportional to the square of electrode radius, the signal-to-noise ratio is increased as the electrode radius is decreased. 

Spherical diffusion has peculiar properties, which can be utilized to measure fast reaction rates. The diffusion current density of a species i to a spherical electrode of radius ro is given by ... 

Spherical Diffusion. If, as it might happen, the electrode is spherical rather than planar (e.g. using a hanging drop mercury electrode), See Figure 19, Fick s second law should be integrated by corrective terms accounting for the sphericity, or the radius r, of the electrode ... 

Figure 19 (a) Typical electrode for spherical diffusion (b) parameters of the spherical... 

Some of these stability issues can be addressed by the use of protective barrier membranes, at the risk of aggravating another fundamental challenge reactant mass transfer. Typical reactants present in vivo are available only at low concentrations (glucose, 5 mM oxygen, 0.1 mM lactate, 1 mM). Maximum current density is therefore limited by the ability of such reactants to diffuse to and within bioelectrodes. In the case of glucose, flux to cylindrical electrodes embedded in the walls of blood vessels, where mass transfer is enhanced by blood flow of 1—10 cm/s, is expected to be 1—2 mA/cm. ° Mass-transfer rates are even lower in tissues, where such convection is absent. However, microscale electrodes with fiber or microdot geometries benefit from cylindrical or spherical diffusion fields and can achieve current densities up to 1 mA/cm at the expense of decreased electrode area. ... 

Eor spherical diffusion, 1 = 2nFDQd, with parameters as defined in ref 30 and electrode diameter d = 10 gm. 

At a spherical electrode, one must consider a spherical diffusion field as discussed in Sect. 2.4. Fick s second law is then written... 

Other explicit examples will not be treated here. Instead, it is preferred to indicate the route by which many cases may be elaborated, as proposed by Reinmuth [28] and more extensively by Guidelli [75], whose reasonings will be followed here. In fact, Guidelli s treatment has been set up more generally and is applicable in the case of linear diffusion, spherical diffusion, and diffusion towards the expanding planar electrode (i.e. the dropping mercury electrode). Here, we confine ourselves to linear diffusion. 

Ultramicro electrodes are electrodes of any shape with dimensions of the order of a few micrometers and possessing a different electrochemical behaviour compared with its millimetre shaped counterparts. The reason is a different mechanism for the transport of electroactive species towards the electrode (Fig. 1.10). For ultramicro electrodes, the main fraction of electroactive species is transport through a spherical diffusion, where it is a... 

Converting the small currents measured at ultramicro electrodes results in higher current densities compared with the values obtained at micro electrodes. This is also caused by the spherical diffusion path. This effect improves on the sensitivity and the detection limit for analytical purposes, and the small size of the electrodes in combination with application of high scan rates allow measurement to be performed in vivo and with a minimal (negligible) disturbance of the conditions of the system to be analysed. 

In the first case, when only species O is initially present in the electrolytic solution (Fig. 2.13a), it is observed that the amalgamation of species R leads to a shift of the wave to more negative potential values, and this shift is greater the more spherical the electrode, i.e., when the duration of the experiment increases or the electrode radius decreases. In the second case (Fig. 2.13b), both species are initially present in the system so we can study the anodic-cathodic wave. In the anodic branch of the wave, the amalgamation produces a decrease in the absolute value of the current. As is to be expected, the null current potential, crossing potential, or equilibrium potential ( Eq) is not affected by the diffusion rates (D0 and Z)R), by the electrolysis time, by the electrode geometry (rs), nor by the behavior of species R... 

Compared to conventional (macroscopic) electrodes discussed hitherto, microelectrodes are known to possess several unique properties, including reduced IR drop, high mass transport rates and the ability to achieve steady-state conditions. Diamond microelectrodes were first described recently diamond was deposited on a tip of electrochemically etched tungsten wire. The wire is further sealed into glass capillary. The microelectrode has a radius of few pm [150]. Because of a nearly spherical diffusion mode, voltammograms for the microelectrodes in Ru(NHy)63 and Fe(CN)64- solutions are S-shaped, with a limiting current plateau (Fig. 33a), unlike those for macroscopic plane-plate electrodes that exhibit linear diffusion (see e.g. Fig. 18). The electrode function is linear over the micro- and submicromolar concentration ranges (Fig. 33b) [151]. 

Unfortunately, the to-electrode precipitation required for conventional (photo)electrochemical measurements on colloidal semiconductors necessarily perturbs the (assumed) spherical diffusion fields and surface adsorption equilibria that obtain at particles in the free solution state, phenomena which are instrumental in determining the dynamic and static charge transfer characteristics of the semiconductor. Consequently, there is a requirement for photoelectrochemical techniques capable of in situ, non-per-turbative investigations of the mechanistic details and catalytic properties of colloidal semiconductors in solution conditions typical of their intended ultimate application. Two such techniques are photoelectrophoresis and the Optical Rotating Disc Electrode (ORDE, developed by Albery et al.). As mentioned above, the former technique has already been reviewed by this author elsewhere [47]. Thus, the remainder of this review will concentrate on measurements that can be made with the latter... 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

Oct. 14, 1922, Kromeriz, then Czechoslovakia - Aug. 10, 2005, Berlin, Germany) Koutecky was a theoretical electrochemist, quantum chemist, solid state physicist (surfaces and chemisorption), and expert in the theory of clusters. He received his PhD in theoretical physics, was later a co-worker of -> Brdicka in Prague, and since 1967 professor of physical chemistry at Charles University, Prague. Since 1973 he was professor of physical chemistry at Freie Universitat, Berlin, Germany. Koutecky solved differential equations relevant to spherical -> diffusion, slow electrode reaction, - kinetic currents, -> catalytic currents, to currents controlled by nonlinear chemical reactions, and to combinations of these [i-v]. For a comprehensive review of his work on spherical diffusion and kinetic currents see [vi]. See also Koutecky-Levich plot. 

The situation at a miniature disc microelectrode embedded in a flat insulator surface (such as an RDE of very small size) can be approximated by spherical symmetry, obtained for a small sphere situated at the center of a much larger (infinitely large, in the present context) spherical counter electrode. How will the change of geometry influence the diffusion-limited current density This is shown qualitatively in Fig. 18L. As time progresses, the diffusion layer thickness increases, causing, in the planar case, a proportional decrease in the diffusion current density. In the spherical configuration the electroactive... 

Fig. 20L Development of the diffusion field near the surface of an ensemble of micro electrodes, (a) planar diffusion-, (b) spherical diffusion with no overlap-, (c) spherical diffusion with substantial overlap-, (d) total overlap, equivalent to planar diffusion to the whole surface.

Equation 45L corresponds to spherical diffusion to each ultramicro electrode, with negligible overlap between the individual electrodes. This is the region in which the total limiting current is nearly independent of time, as seen in Fig. 21L. Whether such a region is observed in practice depends on the design of the ensemble, that is, on... 

We have considered so far only disc-shaped microelectrodes, for which spherical diffusion can be applied, to a good approximation. Other forms have been used, mainly because they might be easier to fabricate. Most noted among these is the linear or strip niicroelec-trode, which is macroscopic in length but microscopic in width. The diffusion field at such electrodes can be approximated. satisfactorily by diffusion to a cylinder. The enhancement of diffusion is less than that... 

Interestingly, no and Do can also be obtained from a single CA experiment if an ultramicroelectrode is used [6]. Following the potential step, planar diffusion will initially dominate, as the diffusion layer is smaller than the radius of the electrode. The current thus follows the usual Cottrel equation (Eq. 39). Later in the CA experiment, however, the diffusion layer grows larger than the radius of the ultramicroelectrode, and spherical diffusion will now dominate. This results in a time-independent current, given by Eq. 48 if the electrode is disc shaped, or Eq. 49 for microsphere geometry. 

The development of ultramicroelectrodes with characteristic physical dimensions below 25 pm has allowed the implementation of faster transients in recent years, as discussed in Section 2.4. For CA and DPSC this means that a smaller step time x can be employed, while there is no advantage to a larger t. Rather, steady-state currents are attained here, owing to the contribution from spherical diffusion for the small electrodes. However, by combination of the use of ultramicroelectrodes and microelectrodes, the useful time window of the techniques is widened considerably. Compared to scanning techniques such as linear sweep voltammetry and cyclic voltammetry, described in the following, the step techniques have the advantage that the responses are independent of heterogeneous kinetics if the potential is properly adjusted. The result is that fewer parameters need to be adjusted for the determination of rate constants. 

In most cases, Do is about 10 cm s, and it then follows that the characteristic radius of an UME is smaller than 6 [im. However, it is important to notice that Do is sometimes appreciably smaller than 10 cm s if O is a very large molecule, or, more likely, if a viscous or glass-like medium is used planar diffusion would then prevail even at a radius of 6 pm. Note also that a microelectrode with a radius of 100 pm experiences a substantial contribution from spherical diffusion. If the geometry of the electrode differs from the simple disk or microsphere shape, the definition is less clear. In general, the characteristic behavior of a UME is observed if at least one of the dimensions is in the micrometer range. For instance, a band electrode with a width of less than 10 pm behaves as an UME even if it is several mm long. 

The early applications of fast CV mainly focused on the measurement of the peak-potential separation, AEp, for the reduction or oxidation process of aromatic compounds, to obtain the pertinent standard heterogeneous rate constant k° from the relationship given in Table 2 [22]. The largest k° values of about 4 cm s were found for the reduction of aromatic hydrocarbons such as anthracene at a gold electrode in acetonitrile. The peak-potential separation increased from the 58 mV expected for a reversible process at low v to about 100 mV on going to v values of 10 kV s . This also shows that there is no real need for employing extremely large sweep rates in the determination of k° for the majority of compounds. Rather, it is important to ensure that the measurements at the lower sweep rates are not hampered by the contribution from spherical diffusion if a too small UME is used. 

The scanning electrochemical microscopy (SECM) technique introduced in recent years by Allen Bard is another area where the smallness of the electrode is essential [38]. The principle in SECM is a mobile UME inserted in an electrolyte solution. The UME is normally operated in a potentiostatic manner in an unstirred solution so that the current recorded is controlled solely by the spherical diffusion of the probed substance to the UME. The current can be quantified from Eqs. 48, 49, or 89 as long as the electrode is positioned far from other interfaces. However, if a solid body is present in the electrolyte solution, the diffusion of the substance to the UME is altered. For instance, when the position of the UME is lowered in the z direction, that is, towards the surface of the object, the diffusion will be partially blocked and the current decreases. By monitoring of the current while the electrode is moved in the x-y plane, the topology of the object can be graphed. The spatial resolution is about 0.25 pm. In one investigation carried out by Bard et al, the... 

Electrocatalysts One of the positive features of the supported electrocatalyst is that stable particle sizes in PAFCs and PEMFCs of the order of 2-3 nm can be achieved. These particles are in contact with the electrolyte, and since mass transport of the reactants occurs by spherical diffusion of low concentrations of the fuel-cell reactants (hydrogen and oxygen) through the electrolyte to the ultrafine electrocatalyst particles, the problems connected with diffusional limiting currents are minimized. There has to be good contact between the electrocatalyst particles and the carbon support to minimize ohmic losses and between the supported electrocatalysts and the electrolyte for the proton transport to the electrocatalyst particles and for the subsequent oxygen reduction reaction. This electrolyte network, in contact with the supported electrocatalyst in the active layer of the electrodes, has to be continuous up to the interface of the active layer with the electrolyte layer to minimize ohmic losses. 

An example of cylindrical diffusion is diffusion toward a conducting wire. Solutions for cylindrical electrodes have been given by Fleischmaim et al. and Jacobsen and West. " The methods presented by both groups give the same results however, the latter is simpler. In this case the diffusion equation is similar to that for spherical diffusion [Eq. (70)]. The solution is shown here for the oxidized form only ... 

Ideally, first the measurement modeling should be carried out. The number and the nature of the circuit elements should be identified and then the process modeling should be carried out. Such a procedure is relatively elementary for a circuit containing simple elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation CPE, semi-infinite, finite length, or spherical diffusion, etc. However, many different conditions arise from the numerical calculations (e.g., for correct solution for porous electrodes, for... 

Figure 4.4.5 Types of diffusion occurring at different electrodes. (a) Linear diffusion to a planar electrode, (b) Spherical diffusion to a hanging drop electrode.

If the electrode in the step experiment is spherical rather than planar (e.g., a hanging mercury drop), one must consider a spherical diffusion field, and Pick s second law becomes... 

The numerator of (5.2.26) is the thickness of the diffusion layer thus the importance of the steady-state term, which manifests spherical diffusion, depends mainly on the ratio of that thickness to the radius of the electrode. When the diffusion layer grows to a thickness that is an appreciable fraction of ro, it is no longer appropriate to use equations for linear diffusion, and one can expect the steady-state term to contribute significantly to the measured current. 

---