Current-time behavior, Cottrell equation

The product D0 (dCo/dx)x=0 t is the flux or the number of moles of O diffusing per unit time to unit area of the electrode in units of mol/(cm2 s). (The reader should perform a dimensional analysis on the equations to justify the units used.) Since (3Co/3x)x=01 is the slope of the concentration-distance profile for species O at the electrode surface at time t, the expected behavior of the current during the chronoamperometry experiment can be determined from the behavior of the slope of the profiles shown in Figure 3. IB. Examination of the profiles for O at x = 0 reveals a decrease in the slope with time, which means a decrease in current. In fact, the current decays smoothly from an expected value of oo at t = 0 and approaches zero with increasing time as described by the Cottrell equation for a planar electrode,... 

Additionally, a slope equal to Vi confirms the validity of Eq. (2.28). Note that the Cottrell equation considers planar diffusion as the sole form of mass transport. However, in practice, purely planar diffusion is only achieved with very large or shielding electrodes. Deviations from cottrellian behavior at short times (doublelayer current) and long times (convection,3 edge effects4) would restrict the applicability of the above equation (see Fig. 2.3). 

This result indicates that as l decreases the current takes a very high value at times close to zero and it diminishes very fast (i.e., species O is consumed instantaneously). In order to check the values of parameter A for which the response evolves from the cottrellian behavior to that given by Eq. (2.199), the behavior of the ratio (ITLV/Id,c) versus A has been plotted in Fig. 2.24. From this figure, it can be seen that for A < 0.46 the current is identical to that obtained with the Cottrell equation (with a maximum relative difference of 5 %). For higher values of A, the ratio decreases until for A > 1.21 the current behaves in line with Eq. (2.199) (with a maximum relative difference of 5 %). In practice, that means that the current will be similar to that obtained at a semi-infinite cell for times close to zero and it will decrease with time faster, the smaller the value of l (i.e., the faster A increases) due... 

We describe here that the redox oligomer wires fabricated with the stepwise coordination method show characteristic electron transport behavior distinct from conventional redox polymers. Redox polymers are representative electron-conducting substances in which redox species are connected to form a polymer wire.21-25 The electron transport was treated according to the concept of redox conduction, based on the dilfusional motion of collective electron transfer pathways, composed of electron hopping terms and/or physical diffusion.17,18,26-30 In the characterization of redox conduction, the Cottrell equation can be applied to the initial current—time curve after the potential step in potential step chronoamperometry (PSCA), which causes the redox reaction of the redox polymer film ... 

If charge diffusion is significantly slower so that the distance of charge transport, L, (=2(Dt) ) is clearly smaller than the thickness of the lamina, 5, the electrochemical response will be equivalent to that recorded when reactants freely diffuse from an infinite volume of solution to the electrode. This situation, often termed as thick-layer behavior, corresponds to semi-infinite boundary conditions, and concentration profiles such as that shown in Figure 2.5c are then predicted. Accordingly, Cottrell-type behavior is observed, for instance, in cyclic voltammetry (CV) and chronoamperometry (CA). In this last technique, a constant potential sufficiently cathodic for ensuring diffusion control in the reduction of Ox to Red is applied. The resulting current-time (i-t) curves should verify the Cottrell equation presented in the previous chapter (Equation (1.3)). 

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