Diffusion limited steady-state currents

The diffusion-limited steady-state current at UMEs may be written generally as... 

On the other hand, if the voltammetry is based on steady-state currents, i is half of the diffusion-limited steady-state current for a disk, which is InFDoCQrQ, and the condition... 

The transient and steady-state voltammetric responses of GNEs have been analyzed by simulation, theory, and experiment (1, 2). An approximate analytical expression for the diffusion-limited steady-state current at a GNE, t,i , is given by equation (6.3.11.3) ... 

The equation for the diffusion limiting steady-state current to the orifice of a silanized pipette is... 

There are, however, obvious limitations. It is not possible to make a very small spherical electrode, because the leads that connect it to the circuit must be even much smaller lest they disturb the spherical geometry. Small disc or ring electrodes are more practicable, and have similar properties, but the mathematics becomes involved. Still, numerical and approximate explicit solutions for the current due to an electrochemical reaction at such electrodes have been obtained, and can be used for the evaluation of experimental data. In practice, ring electrodes with a radius of the order of 1 fxm can be fabricated, and rate constants of the order of a few cm s 1 be measured by recording currents in the steady state. The rate constants are obtained numerically by comparing the actual current with the diffusion-limited current. 

Finding rigorous analytical expressions for the single potential step voltammograms of these reaction mechanisms in a spherical diffusion field is not easy. However, they can be found in reference [63, 64, 71-73] for the complete current-potential curve of CE and EC mechanisms. The solutions of CE and EC processes under kinetic steady state can be found in references [63, 64] and the expression of the limiting current in reference [74], Both rigorous and kinetic steady state solutions are too complex to be treated within the scope of this book. Thus, the analysis of these processes in spherical diffusion will be restricted to the application of diffusive-kinetic steady-state treatment. 

Unlike macroelectrodes which operate under transient, semi-infinite linear diffusion conditions at all times, UMEs can operate in three diffusion regimes as shown in the Figure for an inlaid disk UME following a potential step to a diffusion-limited potential (i.e., the Cottrell experiment). At short times, where the diffusion-layer thickness is small compared to the diameter of the inlaid disc (left), the current follows the - Cottrell equation and semi-infinite linear diffusion applies. At long times, where the diffusion-layer thickness is large compared to the diameter of the inlaid disk (right), hemispherical diffusion dominates and the current approaches a steady-state value. 

Concentration resistance, Rconc- The third factor determining the nature of the deposit is mass transport. The corresponding resistance is referred to as the concentration resistance, Rconc, which results from the depletion of the electro-active species at the cathode surface, caused by mass-transport limitation. The mechanism of mass transport of the electro-active species (either charged or uncharged) could be diffusion, convection or migration, or some combination of these mechanisms. For the simple onedimensional case (corresponding to semi-infinite linear diffusion) at steady state, the rate of mass transport, expressed as the current density, can be written as... 

The electrochemical processes on microelectrodes in bulk solution can be under activation control at overpotentials which correspond to the limiting diffusion current density plateau of the macroelectrode. The cathodic limiting diffusion current density for steady-state spherical diffusion, /l,sphere is given by ... 

Fig. 2. Influence of the radial diffusion on the current-time response under diffusion-limited conditions. (A) at the static spherical electrode of the radius ro = 0.5mm (B) at the disc microelectrode of the radius a = 5 m. Simulated currents in dimensionless scales ipi = 1/ (nFTrrocXD ) and I/(nF7racXD ), respectively 1, currents according to the Cottrell equation 2, currents corrected for the radial diffusion 3, steady state currents. (A) = 2000, (B)...

On the other hand, when the thickness of the diffusion layer becomes sufficiently thicker than the radius of the electrode, the diffusion layer spreads hemispherically around the electrode and the mode of diffusion becomes radiative. As a result, as a steady-state diffusion layer of the solute concentration that is inversely proportional to the distance from the electrode surface is formed, the current also becomes steady state. This steady-state current value (limiting current value) is expressed as follows ... 

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

Similarly to the response at hydrodynamic electrodes, linear and cyclic potential sweeps for simple electrode reactions will yield steady-state voltammograms with forward and reverse scans retracing one another, provided the scan rate is slow enough to maintain the steady state [28, 35, 36, 37 and 38]. The limiting current will be detemiined by the slowest step in the overall process, but if the kinetics are fast, then the current will be under diffusion control and hence obey the above equation for a disc. The slope of the wave in the absence of IR drop will, once again, depend on the degree of reversibility of the electrode process. 

A more rigorous treatment takes into account the hydrodynamic characteristics of the flowing solution. Expressions for the limiting currents (under steady-state conditions) have been derived for various electrodes geometries by solving the three-dimensional convective diffusion equation ... 

Fig. 3. Steady state concentration profiles of catalyst and substrate species in the film and diffusion layer for for various cases of redox catalysis at polymer-modified electrodes. Explanation of layers see bottom case (S + E) f film d diffusion layer b bulk solution i, limiting current at the rotating disk electrode other symbols have the same meaning as in Fig. 2 (from ref.

In many cases the concentration of a substance can be determined by measuring its steady-state limiting diffusion current. This method can be used when the concentration of the substance being examined is not very low, and other substances able to react in the working potential range are not present in the solution. 

The plot of normalized steady-state current vs. tip-interface distance, shown in Fig. 12, demonstrates that as the tip-interface distance decreases the steady-state current becomes more sensitive to the value of Kg. Under the defined conditions the shape of the approach curve is highly dependent on the concentration in the second phase, for Kg values over a very wide range, with a lower limit less than 0.1 and upper limit greater than 50. This suggests that SECMIT can be used to determine the concentration of a target solute in a phase, without the UME entering that phase, provided that the diffusion coefficients of the solute in the two phases are known. 

As might be expected, similar trends to those identified above are observed as y is varied, while maintaining constant and K high and nonlimiting. The transient and steady-state current responses, shown respectively in Figs. 14 and 15 for = 1 and K = 10, vary between a lower limit which is close to the response for an inert interface when y < 0.01, and an upper limit (when y > 1000) which is characteristic of SECM diffusion-control in phase 1 with no resistance from interfacial kinetics or transport in phase 2. 

The effect of increasing y is to increase the diffusion coefficient of the solute in phase 2 compared to that in phase 1. For a given value of this means that when a SECMIT measurement is made, the higher the value of y the less significant are depletion effects in phase 2 and the concentrations at the target interface are maintained closer to the initial bulk values. Consequently, as y increases, the chronoamperometric and steady-state currents increase from a lower limit, characteristic of an inert interface, to an upper limit corresponding to rapid interfacial solute transfer, with no depletion of phase 2. 

Under conditions of nonlimiting interfacial kinetics the normalized steady-state current is governed primarily by the value of K y, which is the relative permeability of the solute in phase 2 compared to phase 1, rather than the actual value of or y. In contrast, the current time characteristics are found to be highly dependent on the individual K. and y values. Figure 16 illustrates the chronoamperometric behavior for K = 10, log(L) = —0.8 and for a fixed value of Kf.y = 2. It can be seen clearly from this plot that whereas the current-time behavior is sensitive to the value of Kg and y, in all cases the curves tend to be the same steady-state current in the long-time limit. This difference between the steady-state and chronoamperometric characteristics could, in principle, be exploited in determining the concentration and diffusion coefficient of a solute in a phase that is not in direct contact with the UME probe. 

The normalized steady-state current vs. tip-interface distance characteristics (Fig. 18) can be explained by a similar rationale. For large K, the steady-state current is controlled by diffusion of the solute in the two phases, and for the specific and y values considered is thus independent of the separation between the tip and the interface. For K = 0, the current-time relationship is identical to that predicted for the approach to an inert substrate. Within these two limits, the steady-state current increases as K increases, and is therefore diagnostic of the interfacial kinetics. 

For comparable diffusion coefficients of the target solute in the two phases and nonlimiting transfer kinetics, systems characterized by different should be resolvable on the basis of transient and steady-state current responses to a value of up to 50 at practical tip-interface separations. If the diffusion coefficient in phase 2 becomes lower than that in phase 1, diffusion in phase 2 will be partly limiting at even higher values of K. On the other hand, as the value of y increases or interfacial kinetics become increasingly limiting, lower values of suffice for the constant-composition assumption for phase 2 to be valid. 

FIG. 24 Steady-state diffusion-limited current for the reduction of oxygen in water at an UME approaching a water-DCE (O) and a water-NB (A) interface. The solid lines are the characteristics predicted theoretically for no interfacial kinetic barrier to transfer and for y = 1.2, Aj = 5.5 (top solid curve) or y = 0.58, = 3.8 (bottom solid curve). The lower and upper dashed lines denote the... 

The driving force for the transfer process was the enhanced solubility of Br2 in DCE, ca 40 times greater than that in aqueous solution. To probe the transfer processes, Br2 was recollected in the reverse step at the tip UME, by diffusion-limited reduction to Br . The transfer process was found to be controlled exclusively by diffusion in the aqueous phase, but by employing short switching times, tswitch down to 10 ms, it was possible to put a lower limit on the effective interfacial transfer rate constant of 0.5 cm s . Figure 25 shows typical forward and reverse transients from this set of experiments, presented as current (normalized with respect to the steady-state diffusion-limited current, i(oo), for the oxidation of Br ) versus the inverse square-root of time. 

FIG. 28 Normalized steady-state diffusion-limited current vs. UME-interface separation for the reduction of oxygen at an UME approaching an air-water interface with 1-octadecanol monolayer coverage (O)- From top to bottom, the curves correspond to an uncompressed monolayer and surface pressures of 5, 10, 20, 30, 40, and 50 mN m . The solid lines represent the theoretical behavior for reversible transfer in an aerated atmosphere, with zero-order rate constants for oxygen transfer from air to water, h / Q mol cm s of 6.7, 3.7, 3.3, 2.5, 1.8, 1.7, and 1.3. (Reprinted from Ref. 19. Copyright 1998 American Chemical Society.)... 

In a typical voltammetric experiment, a constant voltage or a slow potential sweep is applied across the ITIES formed in a micrometer-size orifice. If this voltage is sufficiently large to drive some IT (or ET) reaction, a steady-state current response can be observed (Fig. 1) [12]. The diffusion-limited current to a micro-ITIES surrounded by a thick insulating sheath is equivalent to that at an inlaid microdisk electrode, i.e.,... 

Assuming that the orifice is disk-shaped, one can calculate the steady-state diffusion-limiting current to a pipette from Eq. (1). However, current values about three times higher than expected from Eq. (1) were measured for interfacial IT [18] and ET [5]. The following empirical equation for the limiting current at a pipette electrode was proposed [18bj ... 

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