The current-potential relation

To derive the equation connecting the current with the overpotential we consider the dissolution of a metal the reaction is 

The reverse reaction is the metal-deposition reaction. Within the metal, the equilibrium (metal) + ze (metal) M(metal) 

We consider first the hypothetical case in which there is no electrical potential difference between the metal and the solution. Then the Gibbs energy of activation is 

The anodic and cathodic current densities are given by i+ = zFkfC,, / = -zFk,c  

The rate of dissolution is therefore equal to the difference of the rates of the forward and reverse reactions as in ordinary kinetics. When the electrochemical reaction is at 

---

In general, the effects of the process variables on electrocodeposition are often interdependent and therefore, are ill understood. Often a slight change of one variable can sometimes lead to a dramatic change in the amount of particle incorporation. For specific systems, the current density at which maximum incorporation occurs seems to be related to a change in the slope of the current-potential relation-... 

Fig. 7.24. The current-potential relation at a p-n semiconductor junction differs from that of an electrode/solution interface by being totally asymmetrical.

The Current-Potential Relation at a Semiconductor/Electrolyte Interface (Negligible Surface States)21... 

Equation (7.7), in which cR is the concentration of an intermediate, may give an erroneous impression that the current-potential relation is completely determined by the exponential term in d< ). However, species R was the result of a series of charge-transfer mechanisms, and thus its concentration, as shown below, is also potential dependent. To unravel this dependence, it will be recalled that all steps preceding and following the rds can often be assumed to be at equilibrium. Then, one can equate their forward and backward rates, e.g., for the first step A + e B,... 

In many electrochemical techniques, we measure current-potential curves for electrode reactions and obtain useful information by analyzing them. In other techniques, although we do not actually measure current-potential curves, the current-potential relations at the electrodes are the basis of the techniques. Thus, in this section, we briefly discuss current-potential relations at the electrode. 

The current-potential relation for process (2) and that for the whole process [(1) to (3)] are discussed in the following sections. 

Figure 5.3 shows the current-potential relations for a=0.25 and 0.50.2) At E=Eeq, the net current (i) is equal to zero but currents of the same magnitudes (i0) flow in opposite directions. i0 is called the exchange current. The net current is an oxidation current (anodic current) at E>Eeq and a reduction current (cathodic current) at E[Pg.113]

From the current-potential relation, we can determine the current at a given potential and, inversely, the potential at a given current. 

The current-potential relation in Eq. (5.9) is shown by curves 1 to 3 in Fig. 5.6. Curve 2 is for CRed=0 and curve 3 is for COx=0. The curves are S-shaped and the currents at potentials negative enough and positive enough are potential-indepen-dent, being equal to ig and igi, respectively. These currents are called limiting currents and are proportional to the bulk concentrations of Ox and Red, respectively [Eqs (5.7) and (5.8)]. The potential at i = (ig + ibi)/2 is equal to E1/2 in Eq. (5.10) and is called the half-wave potential. Apparently from Eq. (5.10), the half-wave potential is independent of the concentrations of Ox and Red and is almost equal to the standard redox potential E°, which is specific to each redox system. From the facts that the limiting current is proportional to the concentration of the electroactive species and that the half-wave potential is specific to the redox system under study, the current-potential relation can be used both in quantitative and qualitative analyses. 

Curves 4 and 4 in Fig. 5.6 show an example of the current-potential relation obtained for an irreversible electrode process. For a reversible electrode process, the reduction wave appears at the same potential as the oxidation wave, giving an oxidation-reduction wave if both Ox and Red exist in the solution (curves 1, 2 and 3 in Fig. 5.6). For an irreversible process, however, the reduction wave (curve 4) is clearly separated from the oxidation wave (curve 4 ), although the limiting currents for the two waves are the same as those in the reversible process. The cur-rent-potential relation for the irreversible reduction process can be expressed by... 

In electrogravimetry [19], the analyte, mostly metal ions, is electrolytically deposited quantitatively onto the working electrode and is determined by the difference in the mass of the electrode before and after the electrolysis. A platinum electrode is usually used as a working electrode. The electrolysis is carried out by the con-trolled-potential or the controlled-current method. The change in the current-potential relation during the process of metal deposition is shown in Fig. 5.33. The curves in Fig. 5.33 differ from those in Fig. 5.31 in that the potentials at i=0 (closed circles) are equal to the equilibrium potential of the M +/M system at each instant. In order that the curves in Fig. 5.33 apply to the case of a platinum working electrode, the electrode surface must be covered with at least a monolayer of metal M. Then, if the potential of the electrode is kept more positive than the equilibrium potential, the metal (M) on the electrode is oxidized and is dissolved into solution. On the other hand, if the potential of the electrode is kept more negative than the equilibrium potential, the metal ion (Mn+) in the solution is reduced and is deposited on the electrode. 

Professor S. Srinivasan and his team have studied the effect of pressure and characteristics of the current-potential relations in a hydrogen-oxygen fuel cell with a proton exchange membrane (Y. W. Rho, O. A. Velev, S. Srinivasan, and Y. T. Kho,./. Electrochem. Soc. 141 2084, 2089, 1994). In this problem, it is proposed to study the applicability of the theoretical dependence of the cell potential as a function of pressure. The temperature is 25 °C and it may be assumed that the pressure of the gas in each of the compartments, i.e., the anodic compartment (hydrogen) and the cathodic compartment (oxygen), are the same, Pn =Po P- For the formation of water in its standard state, the relevant thermodynamic quantities are ... 

Fig. 2.74. Schematic representation of the transfer of electrons from metal to the conduction band of water before and after the breakdown. The current-potential relation leadiig to breakdown involves a plateau in which current hardly it> creases as potential approaches breakdown.

Increasing anode potential beyond the limiting current plateau range may cause gas bubbles to form on anode surface. The gas bubbles break anodic layers and thus mass transport controlled condition. Hence, the current-potential relation becomes linear again. Gas bubbles produced on anode lead to a rough surface [4]. 

Eyring and his coworkers studied the kinetic process of CO2 reduction at a Hg electrode mainly on the basis of their current potential measurements. Since the cathodic current corresponds accurately to CO2 reduction in the neutral pH region, the reaction kinetics can be discussed on the basis of the current potential relation. 

The analysis of the kinetics of alloy deposition is complicated by the fact that at least two reactions occur in parallel. Consequently, the current-potential relation observed represents a combination of the contributions of two processes, each having its own overpotential, rate constant and potential dependence of the current density. Thus, any information obtained from the current-potential relation observed is of questionable value in evaluating the mechanism of the formation of the alloy. 

It should not be surprising that an ion such as WO4 cannot be reduced readily all the way to metallic tungsten. Indeed, it is surprising that there are certain conditions under which it can be reduced. Moreover, alloy deposition is often a complex, and quite unpredictable, process. In what is called anomalous deposition we classify processes that behave imexpectedly - the composition of the alloy cannot be predicted from the current-potential relation of the alloying elements studied each by itself. When forming a Ni-Fe alloy it seems that Fe + ions in solution inhibit the rate of deposition of nickel, while NP ions accelerate the rate of deposition of iron. In the deposition of a Ni-Zn alloy the situation is somewhat different. Here, one finds a complete synergistic effect adding either ion to the solution enhances the rate of deposition of the other metal. 

The complete current-potential relation under illumination has already been derived in Section 7.3.3 (Eq. 7.68). In this case it was assumed that the cathodic dark current is only due to the injection of holes into the valence band of an n-type electrode. It was further shown that the current-potential relation could be simplified if the recombination is the rate-determining step (Eq. 7.73). The pre-exponential factor in Eq. (7.73), y o, mainly depends on material parameters such as diffusion constant and length of minority carriers as given by Eq. (7.65). For instance, the recombination is fast if the diffusion length is short, which leads to high /o values and thereby to large cathodic dark currents (Eq. 7.73). As already mentioned, there arc many cases where the photocurrent is due to a hole transfer to occupied states of the redox system but the dark current corresponds to an electron transfer from the conduction band to the empty states of the redox system. In this case the current-potential dependence for an n-type electrode has in principle the same shape... 

Rather than discuss the mechanisms of electrode reactions in further detail, we will describe some general implications of the current-potential relation. 

Stoichiometric numbers may be obtained from the current-potential relation at low overpotential, where... 

A rigorous analysis of the current-potential relation and current distribution in a single pore was carried out taking into account all forms of polarization considering a concentration gradient of only the reactant and in the axial direction of the pore (108). The assumption of a unidirectional concentration gradient is valid under the conditions that the local activation-controlled current density is considerably less than the limiting current density due to radial diffusion (i < iJlO). 

Next let us examine the current-potential relation with increasing lithium intercalation/deintercalation time. For this purpose, the values of current at the times when various amounts of cathodic or anodic charge have passed (open symbols in Figures 9a-e) were obtained as a function of potential step. For example, in order to determine the open circle A in Figure 9(a), the cumulative charge vs. time plot is first... 

If the quantity w (where co1/2 = L/ XD, XD being the diffusion layer thickness) is large enough so that only the first term in Eq. (26) is considered, then the current-potential relation takes the form... 

Pyun et al. started " their exploration of the anomalous current response with the CTs obtained from Lii.sCoOj which is the cathode material of almost all commercially available rechargeable lithium batteries today. They reported that the CTs obtained from U1.SC0O2 composite and thin fitm " electrodes hardly exhibit a typical trend of diffusion controlled lithium transport, i. e. Cottrell behavior. Furthermore, they have found that the current-potential relation obeys Ohm s law during the CT experiments. They thus suggested that lithium transport at the interface of the electrode and the electrolyte is mainly limited by the internal cell resistance, and not by lithium diffusion in the bulk electrode. This concept is called the cell-impedance controlled lithium transport. 

The main difference appears in the exponential term. The expressions at the metal-solution interface contains the factor a in the exponential term, the value of which is less than one, but the similar expression at a semiconductor solution interface does not have an a term. This is because the change in the potential drop in the semiconductor-solution interface has been considered to occur inside the semiconductor (Figure 18). However, in the presence of a large number of surface states (lO cm ) the surface of the metal becomes metallized, and in such a situation the change in potential drop mainly occurs in the Helmholtz layer of the double layer. Hence, for such a metallized semiconductor-solution interface the exponential term of the current potential relation involves the transfer coefficient, a. 

---