A potential step

The results we obtained in preceding sections can help us to analyze quantitatively some other quantum mechanical problems, not having analogues in classical physics. We will consider further a potential step and a potential barrier. 

The problem can be solved analytically in the framework of Schrodinger s equation and the standard conditions for ij/ix), in particular using the requirement to be continuous at the step boundaries (at JC = 0). As in preceding cases, divide the problem into two parts X 0 (area 1) and x 0 (area 11). In area 1 the particle motion is infinite it can be described by the periodic wavefunction i/ j(x) (7.4.2) with constant probability of finding a particle in any point of this area. 

Considering this problem, we ignore the possibility of a wave reflection from the potential step boundary this does not change the essence of the solution and is not important for us at the moment. 

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Au(m) nanoclusters with heights of up to nm form at-FlOO mV vs. AI/AICI3 (a) a typical height profile is shown in (b). Upon a potential step to -fITOO mV vs. AI/AICI3 the clusters dissolve immediately and leave holes in the surfaces as well as small Au islands (c) alloying between Al and Au is very likely. 

Au(in) a potential step from -f500 mV vs. Cu/Cu (a) to -f450 mV results in the growth of small Cu islands at the steps of the gold terraces (b) (picture from [66] - with permission of the Peep owner societes). 

The electrode process at -500 mV on this potential scale is correlated to the growth of 250 20 pm high islands. They grow immediately upon a potential step from the open circuit potential to -500 mV (arrow in Figure 6.2-13). 

FIGURE 1-2 Concentration profiles for different times t after the start of a potential-step experiment. 

Alternately, for potential-step experiments (e.g., chronoamperometry, see Section 3-1), the charging current is die same as that obtained when a potential step is applied to a series RC circuit ... 

Example 2-4 A potential-step spectroelectrochemistry experiment using a reactant concentration of 2 mM generated a product with an absorbance (sampled after 25 s) of 0.8. Calculate the reactant concentration that yielded an absorbance of 0.4 upon sampling at 16 s. 

Derive the Cottrell equation by combining Fick s first law of diffusion with the tune-dependent change of the concentration gradient during a potential-step experiment. 

In order to relax 1 mol of compacted polymeric segments, the material has to be subjected to an anodic potential (E) higher than the oxidation potential (E0) of the conducting polymer (the starting oxidation potential of the nonstoichiometric compound in the absence of any conformational control). Since the relaxation-nucleation processes (Fig. 37) are faster the higher the anodic limit of a potential step from the same cathodic potential limit, we assume that the energy involved in this relaxation is proportional to the anodic overpotential (rj)... 

Figure 37. Lateral section of a polymeric film during the nucleation and growth of the conducting zones after a potential step. (Reprinted from T. F. Otero, H.-J. Grande, and J. Rodriguez, A new model for electrochemical oxidation of polypyrrole under conformational relaxation control. /. Electroanal. Chem. 394, 211, 1995, Figs. 2-5. Copyright 1995. Reprinted with permission from Elsevier Science.)...

The overall charge (Qt) consumed to oxidize the film by a potential step from Ec to E has two components the charge consumed to relax the compact structure, which will be called the relaxation charge (Qr), and the charge consumed under diffusion control to complete the oxidation, called the later diffusion charge (Qd)- The following equation is obeyed ... 

These two equations quantify the evolution of the relaxation current and the relaxation charge as a function of the polarization time when the conducting polymer is submitted to a potential step from Ec to E. They are the relaxation chronoamperogram and the relaxation chronocoulogram,... 

These equations describe the full oxidation of a conducting polymer Submitted to a potential step under electrochemically stimulated confer-mational relaxation control as a function of electrochemical and structural variables. The initial term of /(f) includes the evolution of the current consumed to relax the structure. The second term indicates an interdependence between counter-ion diffusion and conformational changes, which are responsible for the overall oxidation and swelling of the polymer under diffusion control. 

Nal concentration, up to 1 X 10 They also obtained steady-state data using a potential step method. The applied potential was initially set at -0.9 V, where no iodide absorption occurs. The electrode was stepped to various positive potentials and A/(ad) was recorded until equilibrium values were obtained. Similar measurements were done in an electrolyte without Nal, and A/(no-ad) was recorded as a reference. The net change in frequency was determined as IAA/I = A/(ad) -A/(no-ad). Figure 27.22 shows I AA/I vs. E plots for various Nal concentrations. Analysis of the data showed that iodide adsorption occurred in accordance with the Temkin isotherm. 

Based on electrochemical experiments combined with ex situ analysis by AES, LEED, and RHEED, Wang et al. (2001) suggested the formation of a (2 x 2) (2CO + O) adlayer on Ru(OOOl) at = 0.2 V in CO-samrated HCIO4, similar to the phase formed in UHV after CO adsorption on a (2 x 2)0-covered surface [Schiffer et al., 1997]. Erom the total charge density transferred after a potential step to 1.05 V in a CO-free electrolyte, they concluded that only 60% of the CO content in such an adlayer can be oxidized under these conditions [Wang et al., 2001]. 

Figure 15.9 Normalized current transients for COads monolayer electro-oxidation in 0.1 M H2SO4 after a potential step from 0.1 V vs. RHE to various electro-oxidation potentials for Pt/GC electrodes with different particle sizes. The CO adsorption potential was 0.1 V vs. RHE, T = 298K. (Reprinted from Maillard et al. [2007b], Copyright 2007, with permission from Elsevier.)...

Figure 16.11 Titania (TiO c)-suppoited (a) and niobium-doped titania (Nbo.o8Tio.920j /Au)-suppoited (b) Au particle-size-dependent specific activities for—CO oxidation at a potential step of 0.5 V vs. RHE. The current densities have been corrected for the TEM-deteimined...

A further spectroscopic approach was employed in the work by Ding et al. [43], in which the absorption transient is recorded after a potential step. For the case of TCNQ, the absorption transient is given by... 

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. 

Fig. 2.8. (a)On-line mass spectroscopic detection of HD (m/e = 3) produced by application of a potential step to 0.97 V vs. PdH to a Pt electrode in 10 2 M CD3OH/10 4 M/HClO4/0.1 M NaC104/H20 during 0.5 s, followed by a potential step to —0.44 V vs. Pd-H. For comparison HD signal without methanol oxidation when switching the potential from open circuit to H2 evolution at — 0.44 V (b). Upper part recording of current. 

Fig. 4.7. Current (a) and mass intensity (b) voltammograms for methanol adsorbate oxidation without tin (dashed line) and with tin (full line). Base electrolyte dotted line. The voltammograms were recorded after applying a potential step to E0, = 0.475 V during 15 min.

Alternatively, one may control the electrode potential and monitor the current. This potentiodynamic approach is relatively easy to accomplish by use of a constant-voltage source if the counterelectrode also functions as the reference electrode. As indicated in the previous section, this may lead to various undesirable effects if a sizable ohmic potential drop exists between the electrodes, or if the overpotential of the counterelectrode is strongly dependent on current. The potential of the working electrode can be controlled instead with respect to a separate reference electrode by using a potentiostat. The electrode potential may be varied in small increments or continuously. It is also possible to impose the limiting-current condition instantaneously by applying a potential step. 

Consider first the diffusion-limited regime. The simplest experiment to perform is a chronoamperometric measurement, i.e. to monitor the current after a potential step to a value where an electroactive species will undergo electron transfer. This effectively allows us to monitor the rate of reaction, v, as a function of time, through the relationship ... 

The Au-NPs were electrodeposited on the carbon fiber microelectrodes from 0.5 M H2S04 solution containing l.OmM Na AuCl4] by applying a potential step from 1.1 V to 0V for 30 s. Cysteine-modified Au-NPs-electrodeposited CFMEs were prepared by... 

Fig. 11. X-t scan with an STM, showing the structure transition from ( /3 x /3) / 30" to (1 x 1) within a Cu adlayer on Au(lll) in 0.05 M H2SO4 + 1 mM CUSO4, induced by a potential step from +0.12 to +0.04 V vs. Cu/Cu++ at t — 1 s. The transition is seen to start from a monoatomic high step [46].

The principle of this method is quite simple The electrode is kept at the equilibrium potential at times t < 0 at t = 0 a potential step of magnitude r) is applied with the aid of a potentiostat (a device that keeps the potential constant at a preset value), and the current transient is recorded. Since the surface concentrations of the reactants change as the reaction proceeds, the current varies with time, and will generally decrease. Transport to and from the electrode is by diffusion. In the case of a simple redox reaction obeying the Butler-Volmer law, the diffusion equation can be solved explicitly, and the transient of the current density j(t) is (see Fig. 13.1) ... 

Figure 13.1 Current transient for a potential step, where A is a constant given by ...

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