Steady state limiting current

Unsteady-state effects and transition times are also significant in forced convection. Whenever the mass-transfer boundary layer is large anywhere on the working electrode surface, the commonly employed experimental time range of a few minutes may not be adequate to reach the steady-state limiting current. 

Experimental data relative to unsteady-state mass transfer as a result of a concentration step at the electrode surface are not available. However, for a linear increase of the current to parallel-plate electrodes under laminar flow, Hickman (H3) found that steady-state limiting-current readings were obtained only if the time to reach the limiting current at the trailing edge of the plate (see Section IV,E), expressed in the dimensionless form of Eq. (18), is... 

The minimum time necessary to obtain steady-state limiting currents by a current ramp was found to be... 

Fig. 9. Logarithmic plot of apparent limiting-current density as a function of current increase rate at a rotating-disk electrode i — apparent limiting current density i, = true steady-state limiting current density di/dt = current increase rate (A cm-2 sec-1) (u = rotation rate (rad sec-1). [From Selman and Tobias (S10).]...

In many cases mass transfer is not the sole cause of unsteady-state limiting currents, observed when a fast current ramp is imposed on an elongated electrode. In copper deposition, in particular, as a result of the appreciable surface overpotential (see Section III,C) and the ohmic potential drop between electrodes, the current distribution below the limiting current is very different from that at the true steady-state limiting current. 

The well-known equation i = 4nFDCr (i limiting current, n number of electrons implied in the electrochemical process, F Faraday constant, D diffusion coefficient, C electroactive specie concentration and r radius of the disk) describes the theoretical steady-state limiting currents of the disk UMEs. This equation is useful to determine the effective radius of a disk UME and to estimate diffusion coefficients. In this sense, the above-mentioned polished carbon disk UMEs have been characterised through the limiting currents obtained in solution with known parameters, i.e. ferrocyanide aqueous solutions (0.05 M and 2M KC1) [118]. The experimental limiting currents were fairly accurately described by this equation ( + 10%). When the effective radius is determined, this equation can be employed to obtain unknown diffusion coefficients. In this way, we have estimated the diffusion coefficients for /i-carotene in several aprotic solvents with different electrolytic concentrations [123]. 

Fig. 2.15 (Solid line) Current-time curves for the application of a constant potential to a spherical electrode calculated from Eq. (2.142). D0 = Dr = 10-5 cm2 s 1, co = cr = 1 mM, rs = 0.001 cm, (E — E ) = -0.2 V, 7=298 K. (Dashed line) Current-time curves for the application of a constant potential to a planar electrode of the same area as the spherical one calculated from Eq. (2.28). (Dotted line) Steady-state limiting current for a spherical electrode calculated from Eq. (2.148). The inner figure corresponds to the plot of the current of the spherical electrode versus j ft...

The time evolution of the cathodic limiting current (Eq. 2.147) has been plotted in Fig. 2.15 together with that obtained for a planar electrode (Eq. 2.28) and the constant steady-state limiting current for a spherical electrode given by... 

Figure 16 shows the steady-state limiting current density, ilim, for the oxygen reduction reaction (ORR) on pure Al, pure Cu, and an intermetallic compound phase in Al alloy 2024-T3 whose stoichiometry is Al20Cu2(Mn,Fe)3 after exposure to a sulfate-chloride solution for 2 hours (43). The steady-state values for the Cu-bearing materials match the predictions of the Levich equation, while those for Al do not. Reactions that are controlled by mass transport in the solution phase should be independent of electrode material type. Clearly, this is not the case for Al, which suggests that some other process is rate controlling. 

For the case of a microdisc electrode convergent diffusion leads to a steady-state limiting current given by (91). 

For hydrodynamic electrodes, in order to solve the convective-diffusion equation analytically for the steady-state limiting current, it is necessary to use a first-order approximation of the convection function(s) (such as the Leveque approximation for the channel). These approximate expressions for the steady-state mass transport limited currents were introduced in Section 4 (see Table 5). 

The ability of the current transducer to amplify low-current signals must be considered when choosing a potentiostat for SECM experiments. As a tip is miniaturized to nanometer dimensions, the accurate and stable measurement of SECM probe current becomes challenging. A 10 /xm diameter disk electrode produces about 2 nA of steady-state limiting current in a 1 mM solution of mediator, but a 10 nm disk electrode would produce only 2 pA. In the... 

This relation is the general response function for a step experiment in a reversible system when the sampling of current occurs in the steady-state regime. The steady-state limiting current, (5.2.21) or (5.3.2), is the special case for the diffusion-limited region, where 6 0. Let us represent this limiting current as and rewrite (5.4.53) as... 

For the early transient regime, 8 1 [see Section 5.4.2(a)], and (5.5.42) becomes (5.2.10), which is inverted to the Cottrell equation, (5.2.11). For the steady-state regime, 8 1 and (5.5.42) collapses to a form that is easily inverted to the relationship for the steady-state limiting current (5.3.2). Actually, (5.5.42) can be inverted directly to the full diffusion-controlled current-time relationship at a sphere, (5.2.18). All of these relationships also hold for a hemisphere of radius tq, which has half of both the area and the current for the corresponding sphere. 

It is noted that the enzyme reaction is nonlinear with respect to both [Mqx] and [S], Eq. (2). When conditions [Mqx] and [S] Ks are fulfilled, Eq. (2) is reduced to Ve = ( ( at/- M)[E][Eox]- The mediated bioelectrocatalysis under such conditions is described by the established theory of the catalytic current [6]. Thus, the steady-state limiting current /j um is written by... 

The above discussion assumes the condition [S] Ks- When the dependence of [S] on the catalytic current is to be studied, the bulk concentration of S, [S], must be lowered. Under such conditions, however, no steady-state current is observed on CVs as shown by curve (b). This is because the substrate depression occurs in the vicinity of the electrode surface, and no steady state is attained. A digital simulation technique [12] would be the most straightforward way to analyze such nonsteady-state currents or [Sj dependence of the catalytic current [13-15]. Substrate depression can be avoided when two enzyme reactions are coupled in mediated bioelectrocatalysis in such a way that S in the first enzyme reaction is regenerated from the product P by the second enzyme reaction to keep the S/P ratio constant [16]. Under such conditions, the steady-state limiting current can be given by [16,17]... 

A steady-state limiting current 7s um is observed under steady-state conditions at 7 E°m... 

FIG. 14 Dependence of the steady-state limiting current 4 on the concentration of o-glucose Cs h was measured at 0.5 V versus Ag/AgCl with film-coated glucose oxidase (180 rg)-BQ (30%)-carbon paste electrodes at a film thickness of (a) 50 /xm (nitrocellulose film), (b) 50 fim (dialysis membrane), and (c) 100 fjm (dialysis membrane). (From Ref. 33.)... 

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