Transport kinetics, planar diffusion

As discussed earlier, it is generally observed that reductant oxidation occurs under kinetic control at least over the potential range of interest to electroless deposition. This indicates that the kinetics, or more specifically, the equivalent partial current densities for this reaction, should be the same for any catalytically active feature. On the other hand, it is well established that the O2 electroreduction reaction may proceed under conditions of diffusion control at a few hundred millivolts potential cathodic of the EIX value for this reaction even for relatively smooth electrocatalysts. This is particularly true for the classic Pd initiation catalyst used for electroless deposition, and is probably also likely for freshly-electrolessly-deposited catalysts such as Ni-P, Co-P and Cu. Thus, when O2 reduction becomes diffusion controlled at a large feature, i.e., one whose dimensions exceed the O2 diffusion layer thickness, the transport of O2 occurs under planar diffusion conditions (except for feature edges). 

For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with... 

Thus far we have considered only the case of planar macroelectrodes. Although these are widely used for electrochemical experiments, they have some drawbacks mainly due to the distorting effects arising from their large capacitance and ohmic drop. In addition, mass transport in linear diffusion is quite inefficient such that in the case of fast homogeneous and heterogeneous reactions, the response is diffusion-limited and therefore it does not provide kinetic information. 

Research into this area is dominated by microelectrodes. At short times, the diffusion layer thickness is much smaller than the microelectrode radius and the dominant mass transport mechanism is planar diffusion. Under these conditions, the classical theories, e.g., that of Nicholson and Shain, can be used to extract kinetic parameters from the scan rate dependence of the separation between the anodic and cathodic peak potentials. Using this approach, the standard heterogeneous electron transfer rate constant, k°, may be determined from the published working curves relating AEp to a kinetic parameter The variation of AEp with o is determined and, from this, T is calculated. k° is then determined by the following equation ... 

It follows from Equation 6.12 that the current depends on the surface concentrations of O and R, i.e. on the potential of the working electrode, but the current is, for obvious reasons, also dependent on the transport of O and R to and from the electrode surface. It is intuitively understood that the transport of a substrate to the electrode surface, and of intermediates and products away from the electrode surface, has to be effective in order to achieve a high rate of conversion. In this sense, an electrochemical reaction is similar to any other chemical surface process. In a typical laboratory electrolysis cell, the necessary transport is accomplished by magnetic stirring. How exactly the fluid flow achieved by stirring and the diffusion in and out of the stationary layer close to the electrode surface may be described in mathematical terms is usually of no concern the mass transport just has to be effective. The situation is quite different when an electrochemical method is to be used for kinetics and mechanism studies. Kinetics and mechanism studies are, as a rule, based on the comparison of experimental results with theoretical predictions based on a given set of rate laws and, for this reason, it is of the utmost importance that the mass transport is well defined and calculable. Since the intention here is simply to introduce the different contributions to mass transport in electrochemistry, rather than to present a full mathematical account of the transport phenomena met in various electrochemical methods, we shall consider transport in only one dimension, the x-coordinate, normal to a planar electrode surface (see also Chapter 5). 

The potential corresponding to a current 7plane = (Vpl e +/pl e /2, is called reversible half-wave potential, E 2, in planar geometry. This parameter can deviate from the formal potential because it is affected by the diffusion coefficients of the electroactive couple and also by the electrode geometry and size (i.e., it is affected by the kinetics of the mass transport) see Fig. 2.21. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

In an earlier discussion (Sect. 3.4) it was mentioned that the electrode reactions of proteins could be governed by radial-type diffusion, even at a macroscopically planar electrode surface, because of specificity for electroactive sites having microscopic dimensions. Deviation from linear diffusion behaviour depends upon the size and density of these sites [125,126]. Thus while the differences in waveshape between native and Ru couples that was noted in these studies may have a purely chemical-kinetic origin, the interesting possibility also arises that there exists different effective mass transport behaviour for the electrode reactions of each redox centre contained in a multi-centred macromolecule. Further evidence for this suggestion is presented in Sect. 5.2. 

Kinetics of the mass transfer The rate of mass transfer plays a dominant role in reversible electrode reactions. It is assumed that the concentration of the electroactive reactant at the electrode surface is governed by the imposed potential according to Eq. (5). In this paragraph only diffusive mass transport, i.e., the transport of a substance under the influence of the concentration gradient at the electrode surface will be discussed. For further consideration the simplest possible conditions are chosen the planar electrode of large surface and the diffusion in the direction perpendicular to the electrode. 

This is applicable to various carrier symmetries such as planar, cylindrical and spherical. Here Q is the amount of molecules released per unit exposed area of the carrier, t denotes time and a, b and k are constants. This power-law function is related to the Weibull function that has been suggested as a universal tool for describing release from both Euclidian and fractal systems, and may be considered as a short-time approximation of the latter (Kosmidis et al. 2003). The constant a takes initial delay and burst effects into account, and is a kinetic constant (Jamzad et al. 2005). The power law exponent, k, also called the transport coefficient, characterises the diffusion process and equals 0.5 for ordinary case I (or carrier conttoUed) diffusion in systems for which no swelling of the carrier material occurs, which can be expected for mesoporous material (Ritger and Peppas 1987). Diffusion-controlled release from a planar system, in which the carrier structure is inert, may be described by the Higuchi square-root-of-time law ... 

The general boundary conditions, Eq. (142), also called the kinetic boundary conditions, can be exploited for solving the particle adsorption problem under the pure diffusion transport conditions when Eq. (138) applies. Using the Laplace transformation method, analytical results have been derived for the spherical and planar interfaces, both for irreversible and reversible adsorptions [2,113,114]. The effect of the finite volume also has been considered in an exact... 

FIG. 37 Kinetics of irreversible adsorption under the diffusion controlled transport at a planar surface expressed as the versus (t/tch) dependencies (where t = l/(Sgit ) D is the... 

The action of phosphonic acid substituted calix[4]arenes (701)-(704) on solvent-containing planar bilayer membranes made of cholesterol and egg phosphatidylcholine (egg PC) or synthetic 18-carbon-tail phospholipid DOPC have been investigated in a voltage-clamp mode. A steady-state voltage-dependent transmembrane current has been achieved only after addition of the compound (702) from the side of the membrane the positive potential has been applied to. This current exhibited anion selectivity passing more chloride at negative potentials applied from the side of the membrane to which calix[4]arene (702) has been introduced. The kinetics and temperature-dependence determined for calix[4]arene (702)-mediated ionic transport suggested a carrier mode of facilitated diffusion. 

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