Calculated current-time transient

Deposition of mercury at boron-doped diamond (BDD) and platinum electrodes has also been studied [33]. Deposition and oxidation of mercury was performed by cyclic voltammetry from the solution of 1 mM Hg2 ( 104)2 in 1 M Na l04. In order to learn more about this deposition, it was carried out also under chronoamperometric conditions. The results obtained are shown in Fig. 2 in the form of dimensionless current-time transients. Experimental curves obtained at two different overpotentials were compared with the theoretical curves calculated for instantaneous and progressive nucleation. A good agreement of experimental plots with the instantaneous nucleation mechanism was... 

In a detailed investigation of the kinetic behaviour of bases generated from (fluoren-9-ylidene)methane derivatives the problem has been overcome by computer simulation of current-time transients expected for the extended mechanism (including reaction 5), The program used for the comparison of simulated and experimental curves allows both kp and k to vary independently until the RMS deviation between the two i/t curves is minimised. The equilibrium constant for reproportionation (kf/kj) is calculable from values of Ep, (1) and Ep (2). It is important to realise that there may be any number of pairs of values of k and k which can give a good fit between experimental and simulated i/t curves. 

Refinements of the above volume diffusion concept have been made by a model that includes a contribution of surface-diffusion processes to the dissolution reaction of the more active component at subcritical potentials. By adjustment of different parameters, this model allows for the calculation of current-time transients and concentration-depth profiles of the alloy components [102]. In addition to this, mixed control of the dissolution rate of the more active component by both charge transfer and volume diffusion has been discussed. This case is particularly interesting for short polarization times. The analysis yields, for example, the concentration-depth profile and the surface concentration of the more noble component, c, in dependency on the product ky/(t/D), where is a kinetic factor, t is the polarization time, and D is the interdiffusion coefficient. Moreover, it predicts the occurrence of different time domains in the dissolution current transients [109]. 

The theoretical calculations have shown that after an initial rise, the current-time transient passes through several oscillations, and then after the deposition of several layers, steadies toward a final value, ioo (Figure 13). This... 

A model which considers circular crystallites on a circular substrate at low nucleation rate would lead to the growth of a complete planar layer following the formation of a single nucleus, the growth of which would be limited only by the boundaries of the substrate. When the current-time transients were calculated, a statistical analysis of the transients alone was compared with experiments at low overpotentials. The moments of the transients alone consolidated the model and showed that nucleation was uniform over the substrate. The power spectral density of the whole experiment provided the steady-state nucleation rate and showed that, in a stationary state, nucleation could be adequately described as a Poisson process. 

Fig. 12.87. A decay transient from the work of Minevski et al. The current observed is an anodic current arising from the dissolution of hydrogen from palladium after an earlier saturation of the palladium during H2 evolution. Note the unusual bump on the current-time line. Analysis of this area of the transient gave rise to a calculation of the equivalent amount of hydrogen it represents. Thus, knowing the voltage from which it came and using appropriate equations of state, it is possible to calculate the pressure in the voids from which it originated. (Reprinted from Z. Minevski, dissertation, Texas A M University, 1995.)...

Such ring current measurements are of necessity not steady state, as the reaction involves the reduction of a redox-active film of finite thickness coated on the disc electrode. However, the ring response to a transient disc current has also been calculated. The time-dependent collection efficiency Nr has been derived analytically for a galvanostatic (constant current and hence constant flux) step on the disc electrode [20] as a function of the dimensionless time... 

Surface area of third group of concrete slabs Hot metal surface area Cold metal surface area Current time Containment atmosphere temperature before accident Temperature of the external air Time after rupture at which transient calculation is terminated Initial temperature of the containment mixture after efflux Hot metals initial temperature Internal free volume of the containment... 

Evidently the effect of the label change will be to increase the number density of labeled particles in the primary cell near the x=L boundary relative to that near the x = 0 boundary. In the long time limit, it is expected that the system will approach a one-dimensional steady state, in which a self-diffusion current ji of labeled particles will flow in the —e direction independent of r and t. The calculation depends on the establishment of this steady state and is to be contrasted with the use of an initial nonequilibrium ensemble in which one might study the number density and current as transients. Here the number density and current are to be evaluated as time averages, beginning at such a time that initial transients have vanished. 

Despite the reported agreement with the theory, the steady state parts of the transients as seen from Figure 13 lie in a range well below any of the theoretical curves. The most important fact, however, is that the first maximum of the experimental transients lies in a range around or below the maximum of the current-time curve for first layer formation as calculated from Jv obtained from the initial part of the same transient. This is obviously impossible within the model of homogeneous nucleation. One possible explanation is to assume that nucleation proceeds on nucleation centers. There is much evidence that active sites play a role in this believed-to-be very simple case of metal deposition. 

This objective can be achieved by considering the ratio of the transient to steady-state current contributions (equations (6.1.4.3) and (6.1.4.4), respectively). This analysis gives a dimensionless parameter (TtDty Vr that can be used to calculate a lower time limit at which the steady-state contribution will dominate the total current to a specified extent. For example, one can calculate the time required for the steady-state current contribution, jjj, to be 10 times larger than the transient component, Taking a typical value of D as 1 X 10 cm sec for an aqueous solution, for an electrode of radius 5 mm, the experimental timescale must be longer than 80 sec. Therefore, steady state is not observed for... 

The classical electrochemical methods are based on the simultaneous measurement of current and electrode potential. In simple cases the measured current is proportional to the rate of an electrochemical reaction. However, generally the concentrations of the reacting species at the interface are different from those in the bulk, since they are depleted or accumulated during the course of the reaction. So one must determine the interfacial concentrations. There axe two principal ways of doing this. In the first class of methods one of the two variables, either the potential or the current, is kept constant or varied in a simple manner, the other variable is measured, and the surface concentrations are calculated by solving the transport equations under the conditions applied. In the simplest variant the overpotential or the current is stepped from zero to a constant value the transient of the other variable is recorded and extrapolated back to the time at which the step was applied, when the interfacial concentrations were not yet depleted. In the other class of method the transport of the reacting species is enhanced by convection. If the geometry of the system is sufficiently simple, the mass transport equations can be solved, and the surface concentrations calculated. 

Fig. 29. Electrodeposition of Ag from 0.017 M AgCN + 0.92 M KCN + 0.11 M K2CO3 solution dimensionless analysis of experimental potentiostatic current transients (/, and tm are the current and time corresponding to the maximum on the current transient curve, respectively). Upper curve calculated for the instantaneous nucleation mechanism lower curve, for the progressive nucleation mechanism. Different symbols/experimental points relating to different potentials [136], Reproduced by permission of The Electrochemical Society, Inc.

Due to their separation in time, it is possible to analyze the metastable pit transients in terms of the electrochemical processes occurring. If each transient is considered to be from a single, hemispherical pit, the dissolution current density in that pit can be calculated ... 

Figure 94 (a) The SCL transient currents for various normalized trapping times (R = Ttrap/t0) as calculated from theory (see Ref. 26) R = oo denotes the trap-free case is the steady-state current without trapping, (b) t trap-free SCL transient current injected from ITO under a positive step voltage applied to an IT0/PPV/TPD PC/A1 device jScl corresponds to in part (a). Bottom TOF photocurrent transient for holes generated by a light pulse at the A1/(TPD PC) interface (the negative polarity applied to ITO). (From Ref. 428). 

For heterogeneous propellants, the current situation is much less satisfactory. The complexity of the combustion process was discussed in Section 7.7. To employ a result like equation (66) directly is questionable, although attempts have been made to evaluate parameters like A and B of equations (67) and (68) from complicated combustion models for use in response-function calculations [81], [82]. Relatively few theories have been addressed specifically to the acoustic response of heterogeneous propellants [82]. Applications of time-lag concepts to account for various aspects of heterogeneity have been made [60], [83], a simplified model—including transient variations in stoichiometry—has been developed [84], and the sideways sandwich model, described in Section 7,7, has been explored for calculating the acoustic response [85], There are reviews of the early studies [7] and of more recent work [82],... 

Application to Polymeric Materials.—There have been several notable examples of the application of slow-response time-domain methods to dielectric measurements of polymers. - Williams used a step-up and step-down method and calculated e"(to), via the Hamon approximation, from the average of the transient charging and discharging currents. Later usage has led to improved precision. ... 

A calculated transient current response to a 10 mV step in potential, introduced at time to, is presented in Figure 7.1 for the electrical circuit inserted in the figure. The time constants for the circuit tmder the conditions of tiie simulation were Ti = 0.0021 s (76 Hz) and T2 = 0.02 s (8 Hz). The potentiail dependence of parameter l i is consistent with the behavior of the charge-transfer resistance described in Chapter 10. 

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