Potential step perturbation, diffusion

A potential step is subsequently applied to the UME in phase 1, sufficient to electrolyze Red] at the tip, at a diffusion-controlled rate. This perturbs the interfacial equilibrium, inducing the transfer of the target species across the interface, from phase 2 to phase 1, as shown in Fig. 10. 

Equation (69) holds universally, but eqn. (70) applies only to a potential step mean perturbation in the case of semi-infinite linear diffusion. For other mean perturbations or other types of (diffusional) mass transport, eqn. (70) should be replaced by the appropriate expression for F(tm). A survey of such expressions was given in a recent review by Sluyters-Rehbach and Sluyters [53], Unfortunately, most of them are of uncomfortable complexity. Therefore it may be preferable to make use of the less rigorous, but more simple, F(tra ) function that can... 

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... 

The conditions defined to this point apply for any situation in which the solution is uniform before the experiment begins and in which the electrolyte extends spatially beyond the limit of any diffusion layer. The final condition defines the experimental perturbation. In the present case, we are considering a large-amplitude potential step, which drives the surface concentration of O to zero at the electrode surface after t = Q. 

Building on the results of Ref. [61], the Unwin group proposed a family of equilibrium perturbation-based approaches to study lateral surface diffusion [62]. The theory based on numerical solution of a time-dependent diffusion problem was developed to describe a triple potential step transient experiments [62a], chronoamperometric experiments, and steady-state approach curves [62b]. An approximate analytical expression for the approach curves produced by the combination of bulk diffusion in the tip/substrate gap with heterogeneous ET and lateral charge transport on the substrate surface is also available [63],... 

Fig. 10.10 Voltammetry with periodical renewal of the diffusion layer (a) perturbation waveform (b) sequence of chronoamperometric signals recorded at each potential step (c) i vs. E response...

The galvanostatic intermittent titration technique (GITT) has been first proposed by Weppner and Huggins in 1977 [22], This method is of particular interest for the measurement of ion transport properties in solid intercalation electrodes, used in lithium-ion batteries, for instance [18]. The determination of the diffusion constants relies on Fick s law. The GITT method records the transient potential response of a system to a perturbation signal a current step (/s) is applied for a set time xs, and the change of the potential (E) versus time (0 is recorded (Figure 1.11) [18,22],... 

When an ionic single crystal is immersed in solution, the surrounding solution becomes saturated with respect to the substrate ions, so, initially the system is at equilibrium and there is no net dissolution or growth. With the UME positioned close to the substrate, the tip potential is stepped from a value where no electrochemical reactions occur to one where the electrolysis of one type of the lattice ion occurs at a diffusion controlled rate. This process creates a local undersaturation at the crystal-solution interface, perturbs the interfacial equilibrium, and provides the driving force for the dissolution reaction. The perturbation mode can be employed to initiate, and quantitatively monitor, dissolution reactions, providing unequivocal information on the kinetics and mechanism of the process. 

As explained earlier, in transient electrochemical methods an electrical perturbation (potential, current, charge, and so on) is imposed at the working electrode during a time period 0 (usually less than 10 s) short enough for the diffusion layer 8 (2D0) to be smaller than the convection layer (S onv imposed by natural convection. Thus the electrochemical response of the system investigated depends on the exact perturbation as well as on the elapsed time. This duality is apparent when one considers a double-pulse potentiostatic perturbation applied to the electrode as in the double-step chronoampero-metric method. 

Let us consider, for example, the simple nernstian reduction reaction in Eq. (221) and a solution containing initially only the reactant R. Before any electrochemical perturbation the electrode rest potential Ej is made largely positive to E . At time zero the potential is stepped to a value E2, sufficiently negative to E , so that the concentration of R is close to zero at the electrode surface. After a time 6, the electrode potential is stepped back to El, so that the concentration of P at the electrode surface becomes zero. When this potentiostatic perturbation, represented in Fig. 21a, is applied in a steady-state method, the R and P concentration profiles are linear and depend only on the electrode potential but not on time, as shown in Fig. 20a (for k 0). Yet when the same perturbation is applied in transient methods, the concentration profiles are curved and time dependent, as evidenced in Fig. 21b. Thus it is seen from this figure that a step duration at Ei, much longer than the step duration 0 at E2, is needed for the initial concentration profiles to be restored. This hysterisis corresponds to the propagation of the diffusion perturbation within the solution, which then keeps a memory of the past perturbation. This information is stored via the structuring of the concentrations in the space near the electrode as a function of the elapsed time. 

There are many instances in electrochemistry when we find it very difficult to obtain an explicit relationship between current, potential, and time. Either the system itself is intrinsically complex (e.g., a quasireversible charge transfer involving adsorbed and diffusing reactant species) or the experimental conditions are less than ideal (e.g., step experiments carried out on a time domain so short that the rise time of the potentiostat is not negligible). It is usually true in these and other cases that much simpler relationships exist in the Laplace domain between the perturbations and the observables. Thus it can be useful to transform the data and carry out the analysis in transform space (39-42). 

By stepping the electrode potential from a value where no electrode reactions occur to one where the solution component of the adsorbate (H+ in Figure 13.1) is ranoved at a diffusion-controlled rate, H is locally depleted in the solution between the tip and substrate surface. This process perturbs the adsorption/desorption equilibrium, inducing proton desorption from the surface and promoting the diffusion of protons from the surrounding solution into the tip/substrate gap. Additionally, since the desorption process depletes the concentration of adsorbed protons on the part of the substrate... 

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