Interfacial Rate Constants

Stationary microwave electrochemical measurements can be performed like stationary photoelectrochemical measurements simultaneously with the dynamic plot of photocurrents as a function of the voltage. The reflected photoinduced microwave power is recorded. A simultaneous plot of both photocurrents and microwave conductivity makes sense because the technique allows, as we will see, the determination of interfacial rate constants, flatband potential measurements, and the determination of a variety of interfacial and solid-state parameters. The accuracy increases when the photocurrent and the microwave conductivity are simultaneously determined for the same system. As in ordinary photoelectrochemistry, many parameters (light intensity, concentration of redox systems, temperature, the rotation speed of an electrode, or the pretreatment of an electrode) may be changed to obtain additional information. 

These three equations (11), (12), and (13) contain three unknown variables, ApJt kn and sr The rest are known quantities, provided the potential-dependent photocurrent (/ph) and the potential-dependent photoinduced microwave conductivity are measured simultaneously. The problem, which these equations describe, is therefore fully determined. This means that the interfacial rate constants kr and sr are accessible to combined photocurrent-photoinduced microwave conductivity measurements. The precondition, however is that an analytical function for the potential-dependent microwave conductivity (12) can be found. This is a challenge since the mathematical solution of the differential equations dominating charge carrier behavior in semiconductor interfaces is quite complex, but it could be obtained,9 17 as will be outlined below. In this way an important expectation with respect to microwave (photo)electro-chemistry, obtaining more insight into photoelectrochemical processes... 

The combination of photocurrent measurements with photoinduced microwave conductivity measurements yields, as we have seen [Eqs. (11), (12), and (13)], the interfacial rate constants for minority carrier reactions (kn sr) as well as the surface concentration of photoinduced minority carriers (Aps) (and a series of solid-state parameters of the electrode material). Since light intensity modulation spectroscopy measurements give information on kinetic constants of electrode processes, a combination of this technique with light intensity-modulated microwave measurements should lead to information on kinetic mechanisms, especially very fast ones, which would not be accessible with conventional electrochemical techniques owing to RC restraints. Also, more specific kinetic information may become accessible for example, a distinction between different recombination processes. Potential-modulation MC techniques may, in parallel with potential-modulation electrochemical impedance measurements, provide more detailed information relevant for the interpretation and measurement of interfacial capacitance (see later discus-... 

Figure 13. Numerically calculated PMC potential curves from transport equations (14)—(17) without simplifications for different interfacial reaction rate constants for minority carriers (holes in n-type semiconductor) (a) PMC peak in depletion region. Bulk lifetime 10" s, combined interfacial rate constants (sr = sr + kr) inserted in drawing. Dark points, calculation from analytical formula (18). (b) PMC peak in accumulation region. Bulk lifetime 10 5s. The combined interfacial charge-transfer and recombination rate ranges from 10 (1), 100 (2), 103 (3), 3 x 103 (4), 104 (5), 3 x 104 (6) to 106 (7) cm s"1. The flatband potential is indicated.

Figure 14. PMC potential dependence, calculated from analytical formula (18) for different interfacial rate constants for minority carriers S = 1 cm, minority carrier flux toward interface I,- 1 cm-2s 1, a= 780enr1, L = 0.01 cm, 0=11.65 cmV, Ld = 2x 0"3cm), (a) sr = 0 and different charge-transfer rates (inserted in the figures in cm s 1), (b) Constant charge-transfer rate and different surface recombination rates (indicated in the figure).

Figure 15. Effect of interfacial rate constants on PMC behavior and on the photocurrent (/0 = 1 cm-2), (a) Fast interfacial charge-transferrate, and (b) low interfacial charge-transfer rate.

The theoretically derived formula (21) relating PMC measurements to the surface concentration of minority carriers and interfacial rate constants contains a proportionality constant, S, the sensitivity factor. This factor depends on both the conductivity distribution in the semiconductor elec-... 

For an electrode with high interfacial rate constants, for example, relation (28) can be plotted, which yields the flatband potential. It allows determination of the constant C, from which the sensitivity factor S can be calculated when the diffusion constant D, the absorption coefficient a, the diffusion length L, and the incident photon density I0 (corrected for reflection) are known ... 

The fact that a potential-dependent lifetime peak for PMC transients has been found which coincides with the stationary PMC peak in the depletion region near the onset of photocurrents (Fig. 22) is very relevant since the stationary PMC peak is determined by the interfacial rate constants of charge carriers (Figs. 13 and 14) this should also be the case for the transient PMC peak. To demonstrate this correlation, the following formalism can be developed10 ... 

It is important to note that there may be at least two reasons for obtaining deviations from a purely exponential behavior for a PMC transient. These are a too high excess carrier generation, which may cause interfacial rate constants that are dependent on carrier concentration, and an interfacial band bending AU, which changes during and after the flash. For fast charge transfer, a more complicated differential equation has to be solved. 

Equation (40) relates the lifetime of potential-dependent PMC transients to stationary PMC signals and thus interfacial rate constants [compare (18)]. In order to verify such a correlation and see whether the interfacial recombination rates can be controlled in the accumulation region via the applied electrode potentials, experiments with silicon/polymer junctions were performed.38 The selected polymer, poly(epichlorhydrine-co-ethylenoxide-co-allyl-glycylether, or technically (Hydrine-T), to which lithium perchlorate or potassium iodide were added as salt, should not chemically interact with silicon, but can provide a solid electrolyte contact able to polarize the silicon/electrode interface. 

VII. OXIDES AND SENSITIZATION CELLS 1. Potential Dependence of Interfacial Rate Constants... 

Since the potential-dependent photocurrent and the potential-dependent PMC signal were measured and potential-dependent interfacial rate constant kr can be determined. It turns out that it increases exponentially with the electrode potential applied... 

In this chapter we have attempted to summarize and evaluate scientific information available in the relatively young field of microwave photoelectrochemistry. This discipline combines photoelectrochemical techniques with potential-dependent microwave conductivity measurements and succeeds in better characterizing the behavior ofphotoinduced charge carrier reactions in photoelectrochemical mechanisms. By combining photoelectrochemical measurements with microwave conductivity measurements, it is possible to obtain direct access to the measurement of interfacial rate constants. This is new for photoelectrochemistry and promises better insight into the mechanisms of photogenerated charge carriers in semiconductor electrodes. 

The schemes in Figs. 44 and 45 may serve to summarize the main results on photoinduced microwave conductivity in a semiconductor electrode (an n-type material is used as an example). Before a limiting photocurrent at positive potentials is reached, minority carriers tend to accumulate in the space charge layer [Fig. 44(a)], producing a PMC peak [Fig. 45(a)], the shape and height of which are controlled by interfacial rate constants. Near the flatband potential, where surface recombination... 

From top to bottom the curves are for first-order interfacial rate constants of 1 x 10 1 x 10... 

In this recently developed device shown in Fig. 5.25 [30], two immiscible liquid streams converge in a micro-channel and then separate. A well-defined flow has been demonstrated, with no entrainment of the opposed flows, for flowrates from 10 4 to 1.0 cm3 s 1. The highest flow rate corresponds to solution contact times around 1 ms, corresponding to a kL of the order of 0.1 cm s 1. Thus, the confluence microreactor has the potential to measure liquid-liquid interfacial rate constants much higher than those accessible by current methods, and interesting applications are anticipated. The device itself has been patented [31]. 

We will see that in the steady state of the blocking cells, we can extract partial conductivities, and from the transients chemical diffusion coefficients (and/or interfacial rate constants). Cell 7 combines electronic with ionic electrodes here a steady state does not occur but the cell can be used to titrate the sample, i.e., to precisely tune stoichiometry. Cell 1 is an equilibrium cell which allows the determination of total conductivity, dielectric constant or boundary parameters as a function of state parameters. In contrast to cell 1, cell 2 exhibits a chemical gradient, and can be used to e.g., derive partial conductivities. If these oxygen potentials are made of phase mixtures212 (e.g., AO, A or AB03, B203, A) and if MO is a solid electrolyte, thermodynamic formation data can be extracted for the electrode phases. 

The complex formation proceeded almost completely at the interface. The rate constant of k=5.3xl02M 1 s 1 was determined by a stopped-flow spectrometry in the region where the formation rate was independent of pH. The conditional interfacial rate constants represented by k[ = k k2 [HL] / (k2 + k i[H + ]) were larger in the heptane-water interface than the toluene-water interface, regardless of metal ions. The molecular dynamics simulation of the adsorptivities of 5-Br-PADAP in heptane-water and toluene-water interfaces suggested that 5-Br-PADAP could be absorbed at the interfacial region more closely to the aqueous phase, but 5-Br-PADAP in the toluene-water... 

Albery et al. [16] have used Marcus Theory [16] interfacial processes to calculate that an interfacial rate constant of 2p,ms-1 constitutes a free energy barrier of 44 kJ mol-1. The slowest rate here (2,4-D) gives 49kJmol-1, and rates >100jxms-1 have barriers of 34kJmol-1, still rather larger than that of diffusion itself (20 kJ mol-1). However, the RDC method is not suitable for accurate measurement of k > 20 xm s-1. 

It has been shown that the magnitude of the rate constant for crossing the octanol-water interface makes the energy barrier significantly larger than the diffusional barrier. It has also been shown that for compounds with log Pow less than =1.2, the overall rates are faster and the interfacial kinetics term more important. However, detailed development of a model would be needed to understand what the relative importance of diffusion and interfacial terms (such as cuticle or membrane permeation) are in vivo. No clear dependence of interfacial rate constants on log Pow was seen, but the initial emphasis of such a study should be on the intermediate... 

To investigate the relationship between the interfacial rate constant and the diffiision layer thickness, the protonation reaction rate was measured by the CLM method. The schematic diagram of the CLM apparatus is shown in Figure 10.2. In the CLM method, the thickness of the dodecane phase in the aqueous hydrochloric acid phase can be controlled by the initial volume of each. When the thickness of the dodecane phase was changed from 53 to 132 pm, the observed interfacial diprotonation rate constant decreased from 9.9 x 10 s to 1.5 x 10 s". The experimental results were well reproduced by Equation (1), substituting 3 with the organic phase thickness [10]. 

Time-resolved photoinduced microwave conductivity measurements can be made as a function of applied potential. It has been shown that the measured minority-carrier lifetime r for moderately fast or slow interfacial charge transfer depends not only on the interfacial rate constant and surface recombination Ukc, but also on the energy band bending (AE) and the Debye length Ld (Tributsch, 1999). 

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