Current dependence on time

This equation is analogous to Eq. (5.4.18) or (5.4.19) for the steady-state current density, although the instantaneous current depends on time. Thus, the results for a stationary polarization curve (Eqs (5.4.18) to (5.4.32)) can also be used as a satisfactory approximation even for electrolysis with the dropping mercury electrode, where the mean current must be considered... 

Cu, Pb, Cd and Ni cathodie deposition on silicon proceeds via formation of 3D nuclei. Mechanisms of nucleation were evaluated from DC current dependences on time (chronoamperograms). Examples of chronoamperograms of metal nueleation on Si are shown in Fig. la. The nucleation is typically progressive, so the deposit properties depend on the nucleation stage. 

Adatoms moving along the substrate surface under an STM tip introduce an additional current noise [2, 3]. Random jiunps of adatoms under an STM tip and the current dependence on time, I(t), caused by these jumps, are shown schematically in Fig. 1. The random function 7(r) found experimentally can be used to obtain information about the diffusion process. It is the task of the theory to connect I(t) with the parameters of the system under investigation, in particular with the diffusion coefficients of adatoms and their density. 

Because this relationship contains r, the current depends on time therefore it is not a steady-state limit such as we found for the sphere and the disk. Even so, time appears only as an inverse logarithmic function, so that the current declines rather slowly in the longtime limit. It can still be used experimentally in much the same way that steady-state currents are exploited at disks and spheres. In the literature, this case is sometimes called the quasi-steady state. 

As it can be seen from the presented plots the splash of electric current in the nanotube occurs before the saturation regime is set up. Moreover, the greater phonon relaxation time the greater difference between the peak value of the electric current and its saturation value is observed. The results of our calculations have revealed (see, for example, Fig. 2) that at Fo 1 MV/m and T 100 fs the pronounced damping oscillation of the current dependence on time is even observed. Such a behavior is mainly related to the processes responsible for the energy redistribution between the electron and phonon gases (owing to Eq. (2)). The collisional operator I does not depend on time at t = 0, and I depends on time dramatically at r — +oo). 

In chronoamperometry, after switching on an overpotential, the time dependence of the current is monitored. For purely diffusion-controlled processes, the current depends on time according to the Cottrell equation in Chapter 5, Eq. (5.20). The current decreases proportional to 1/Vt. The combination with charge transfer control leads to the following... 

During current flow, the concentration of reaction products near the surface of the electrode will increase, and a limiting condition may also arise, but for different reasons, which are related to attainment of the solubility limits by given substances. The material precipitating will screen the electrode surface and interfere with a further increase in current. The value of the limiting current will depend on the nature of the deposit formed and is less reproducible than in the previous case specifically, it may depend on time. 

We see that the expression for the current consists of two terms. The first term depends on time and coincides completely with Eq. (11.14) for transient diffusion to a flat electrode. The second term is time invariant. The first term is predominant initially, at short times t, where diffusion follows the same laws as for a flat electrode. During this period the diffusion-layer thickness is still small compared to radius a. At longer times t the first term decreases and the relative importance of the current given by the second term increases. At very long times t, the current tends not to zero as in the case of linear diffusion without stirring (when is large) but to a constant value. For the characteristic time required to attain this steady state (i.e., the time when the second term becomes equal to the first), we can write... 

Fig. 5.46 The dependence on time of the instantaneous current / at a dropping mercury electrode in a solution of 0.08 m Co(NH3)6C13 + 0.1 m H2SO4 + 0.5m K2S04 at the electrode potential where -7 -/d (i.e. the influence of diffusion of the electroactive substance is negligible) (1) in the absence of surfactant (2) after addition of 0.08% polyvinyl alcohol. The dashed curve has been calculated according to Eq. (5.7.23). (According to J. Kuta and I.

Figure 6. Change in content of KCN (1, 2, 3) and hydrogen peroxide (T, 2 , 3 ) in model solution depending on time of plasma treatment under various current strength ...

All these experiments were carried out at such low scan rates that the outside diffusion layer of the cosubstrate (on the order of 105 A) is much larger than the film thickness. An experimental test for knowing whether this condition is fulfilled is that the plateau of S-shaped catalytic current then observed is much larger than the reversible cosubstrate peak observed in the absence of substrate i icat. Under these conditions, the concentration profiles within the film (bottom of Figure 5.30) do not depend on time. 

The steady-state current depended on the concentration of ammonia. A linear relationship was observed between the current decrease and the ammonia concentration below 42 mg l-1 (current decrease 4.7 iA). The minimum concentration for the determination of ammonia was 0.1 mg 1" (signal to noise, 20 reproducibility, 5 %). The current decrease was reproducible within 4 % of relative error when a sample solution containing 21 mg 1 of ammonium hydroxide was employed. The standard deviation was 0.7 mg 1 in 20 exper-ments. The response time of the sensor was within 4 min. 

A sinusoidal voltage or current function that is dependent on time t may be represented by the following expressions ... 

The dependence on time of the current when a constant potential is applied to a plane electrode for different values of the heterogeneous rate constant k° ranging from reversible to totally irreversible processes is shown in Fig. 3.2. 

This different behavior can be explained by considering that for a CE mechanism (the reasoning is similar for an EC one), C species is required by the chemical reaction whose equilibrium is distorted in the reaction layer (whose thickness in the simplified dkss treatment is <5r = jDj(k + 2)) and by the electrochemical reaction, which is limited by the diffusion layer (of thickness 8 = yfnDt). For a catalytic mechanism, C species is also required for both the chemical and the electrochemical reactions, but this last stage gives the same species B, which is demanded by the chemical reaction such that only in the reaction layer do the concentrations of species B and C take values significantly different from those of the bulk of the solution. In summary, the catalytic mechanism can reach a true steady-state current-potential response under planar diffusion because its perturbed zone is restricted to the reaction layer <5r, which is independent of time, whereas the distortion of CE (or EC) mechanism is extended until the diffusion layer 8, which depends on time, and a stationary current-potential response will not be reached under these conditions. 

In the case of a pseudo-first-order EC mechanism, only the half-wave potential depends on time and the equilibrium and rate constants, with the limiting current remaining unaltered with the variation of these parameters—identical to that corresponding to an E mechanism. 

The voltammetric peak reflects the time-dependent concentration gradient of the metal in the mercury electrode during the potential scan. Peak potentials serve to identify the metals in the sample. The peak current depends on various parameters of the deposition and stripping steps, as well as on the characteristics of the metal ion and the electrode geometry. For example, for a mercury film electrode, the peak current is given by... 

We note that under conditions of diffusion control, the current depends on t . Thus, a small change in drop time, resulting from a change in surface tension with potential, does not produce a large difference in the diffusion current. If the error caused by this effect is considered troublesome, it is possible to knock the drops off at fixed intervals, yielding drops of exactly equal size, irrespective of the surface tension. This mode of operation becomes of particular importance for kinetic studies conducted at the foot of the polaro-graphic wave, since the activation-controlled current is proportional to the surface area, which is itself proportional to (the volume increases linearly with time). 

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