Stationary mass transport

Vieltisch and co-workers [29] and Schmickler and co-workers [30] have recently used a ring electrode under pipe turbulent flow in order to achieve higher stationary mass transport conditions and therefore study very fast electrode kinetics (k° > 1cm s 1). 

Fig. 1. Voltaimrogram observable with stationary mass transport to the electrode. A and B, oxidi2able compounds C, reducible compound f, diffusion or limiting current ff, half-vrave potential.

Though in the general case, mathematical expressions of the Nernst model are more complicated than of those semi-infinite diffusion, stationary mass transport is described by a rather simple Eq. (3.12). In this connection, there occurs an interesting possibility to use superposition of both models, which is convenient to apply when i is the periodic time function. Perturbation signals of this type are considered in the theory of electrochemical impedance spectroscopy. In this case, i(t)... 

An alternative to the rotating disk method in a quiescent fluid is a stationary disk placed in a rotating fluid. This method, like the rotating disk, is based on fluid mechanics principles and has been studied using benzoic acid dissolving into water [30], Khoury et al. [31] applied the stationary disk method to the study of the mass transport of steroids into dilute polymer solutions. Since this method assumes that the rotating fluid near the disk obeys solid body rotation, the stirring device and the distance of the stirrer from the disk become important considerations when it is used. A similar device was developed by Braun and Parrott [32], who used stationary spherical tablets in a stirred liquid to study the effect of various parameters on the mass transport of benzoic acid. 

Lichtner, P. C., 1988, The quasi-stationary state approximation to coupled mass transport and fluid-rock interaction in a porous medium. Geochimica et Cosmochimica Acta 52, 143-165. 

When the two liquid phases are in relative motion, the mass transfer coefficients in either phase must be related to the dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive transfer to the Schmidt number. Another complication is that such a boundary cannot in many circumstances be regarded as a simple planar interface, but eddies of material are transported to the interface from the bulk of each liquid which change the concentration profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most industrial circumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass transfer model must therefore be replaced by an eddy mass transfer which takes account of this surface replenishment. 

Figure 2. Release of organotin from polymer matrix, nonsteady state mass transport in (a) polymer matrix and stationary state mass transport in boundary layer and (b) both media...

The experiment is carried out under stationary conditions (i.e. the solution is kept unstirred) in order to ensure that the mass transport is purely diffusive. 

Reducing the Effect of Resistance to Mass Transport in the Stationary and Mobile Phases... 

One of several possibilities to classify elec-troanalytical methods is based on the quantity that is controlled in the experiment, that is, current or potential. Alternatively, since diffusion is an important mode of mass transport in most experiments, we distinguish techniques with stationary or nonstationary diffusion. Finally, transient methods are different from those that work in an exhaustive way. 

UMEs decrease the effects of non-Earadaic currents and of the iR drop. At usual timescales, diffusional transport becomes stationary after short settling times, and the enhanced mass transport leads to a decrease of reaction effects. On the other hand, in voltammetry very high scan rates (i up to 10 Vs ) become accessible, which is important for the study of very fast chemical steps. For organic reactions, minimization of the iR drop is of practical value and highly nonpolar solvents (e.g. benzene or hexane [8]) have been used with low or vanishing concentrations of supporting electrolyte. In scanning electrochemical microscopy (SECM [70]), the small size of UMEs is exploited to locahze electrode processes in the gm scale. 

In Section 7.2, we looked at electroanalytical systems where the electrode rotates while the bulk of the solution remained still. In this present section, we will reverse this experimental concept by considering the case where it is the solution which flows - this time past a stationary electrode. Here, we shall be looking at flow ceils and channel electrodes. The principal mode of mass transport in both cases is convection, since the solution moves relative to the electrode. 

The fifth stationary phase architecture involves chemically derivatized polymeric substrates (CMS). This type of material tends to involve proprietary chemistry, so the actual chemistry used for the derivatization reaction is usually unknown. In general, materials of this sort are of rather substantial capacity, so they have come into vogue in recent years with the general shift toward materials of increasing capacity. The critical difficulty with such materials is the requirement that the derivitization must be constrained to the surface. Reactions that take place beneath the surface in the dense polymer matrix of the substrate will exhibit sluggish mass transport and relatively poor chromatographic performance. Early examples of this stationary phase architecture exhibited relatively poor performance but newer materials such as Showa Denko s IC SI-52 4E illustrate that high-performance materials can indeed be constructed in this manner. 

Microcylindrical electrodes are easier to constract and maintain than microdisk electrodes [37]. Mass transport to a stationary cylinder in quiescent solution is governed by axisymmetrical cylindrical diffusion. For square-wave voltammetry the shape and position of the net current response are independent of the extent of cyhn-drical diffusion [38]. The experiments were performed with the ferri-ferrocyanide couple using a small platinum wire (25 pm in diameter and 0.5 -1.0 cm in length) as the working electrode [37]. 

The rotating disc electrode is constructed from a solid material, usually glassy carbon, platinum or gold. It is rotated at constant speed to maintain the hydrodynamic characteristics of the electrode-solution interface. The counter electrode and reference electrode are both stationary. A slow linear potential sweep is applied and the current response registered. Both oxidation and reduction processes can be examined. The curve of current response versus electrode potential is equivalent to a polarographic wave. The plateau current is proportional to substrate concentration and also depends on the rotation speed, which governs the substrate mass transport coefficient. The current-voltage response for a reversible process follows Equation 1.17. For an irreversible process this follows Equation 1.18 where the mass transfer coefficient is proportional to the square root of the disc rotation speed. 

The mass transport rate coefficient, kd, for a RDE at the maximum practical rotation speed of 10000 per min"1 is approximately 2 x 10-2 cms-1 [28], which sets a limit of about 10 3 cms 1 for the electrode reaction kinetics. For the study of very fast electrode processes, such as some outer sphere redox reactions on noble metal electrodes under stationary conditions, higher mass transport rates in the solution adjacent to the electrode must be employed. 

The invention of the dropping mercury electrode in 1922 by Heyrovsky [1] led to the development and the extensive use of polaro-graphy, which must be considered to be the first linear sweep voltammetry method. In the period from 1947 to 1959, the theory and practice of voltammetry at solid stationary electrodes were developed [2—20]. Due to the significant differences in the mode of mass transport to the two types of electrode, the response and the range of utility differ markedly. Thus, the techniques are sufficiently different that they must be treated separately. The generally accepted convention is that polaro-graphy refers to measurements at the dropping mercury electrode, while measurements at stationary electrodes are referred to as linear sweep voltammetry (LSV). 

After perturbation, the surface concentrations of the reactants, intermediates, and products attain new values. In a genuine stationary technique, these new values are maintained by providing a constant rate of mass transport (hydrodynamic techniques). The surface concentration, c[ of a species i is related to its flux, J, by... 

J.W. Cahn and R.W. Balluffi. Diffusional mass-transport in polycrystals containing stationary or migrating grain boundaries. Scripta Metall. Mater., 13(6) 499-502, 1979. 

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). 

Table 1.4 Mass transport coefficients m,, for different experimental conditions. The values of m, correspond to the application of a constant potential. The expressions corresponding to the Rotating Disc Electrode (convective mass transport) under stationary conditions and to Dropping Mercury Electrode with the expanding plane model (diffusive-convective mass transport) have also been included...

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

The first applied potential is set at a value E at a stationary spherical electrode during the interval 0 < t < i. The diffusion mass transport of the electroactive species toward or from the electrode surface is described by the following differential equation system ... 

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. 

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