Linear diffusion, semi-infinite

For many adsorbates, especially organic substances, the concept of semi-infinite linear diffusion can give us some ideas on the time necessary for an adsorbate to be adsorbed. The number of mols adsorbate, n, diffusing to a unit area of a surface per second, is proportional to the bulk concentration of adsorbate, c ... 

In previous sections we have implicitly assumed that diffusion occurs perpendicular to the electrode surface (semi-infinite linear diffusion). If we decrease the size of the electrode to values roughly in the order of the size of diffusion layers, this assumption becomes invalid. Now, additional... 

Sometimes, there is no linear portion to a Randles-Sevdik graph, and the data yield a curved plot. The derivation of equation (6.13) assumes that diffusion is the sole means of mass transport. We also assume that all diffusion occurs in one dimension only, i.e. perpendicular to the electrode, with analyte arriving at the electrode solution interface from the bulk of the solution. We say here that there is semi-infinite linear diffusion. 

The method relies on the relationship between the surface concentration of the electroactive species O (which yields R according to O + ne = R), Co 0,t), having a bulk concentration Co, and the current, i. Under semi-infinite linear diffusion conditions and independently of the particular electrochemical method employed, Co(0, t) can be expressed by (22)... 

Laplace transformation, 1215 Nemst s equation and. 1217 non-steady, 1254 as rate determining step, 1261 Schlieren method, 1235 semi-infinite linear, 1216, 1234, 1255 in solution and electrodeposition, 1335 spherical. 1216. 1239 time dependence of current under, 1224 Diffusion control, 1248... 

For example, the treatment of diffusion that is to follow is solely restricted to semi-infinite linear diffusion, i.e., diffusion that occurs in the region between x = 0 and x —> +oo, to a plane of infinite area. Thus, diffusion to a point sink—called spherical diffusion—is not treated, though it has been shown to be relevant to the particular problem of the electrolytic growth of dendritic crystals from ionic melts. 

Fig. 7.141. Cyclic voltammogram (semi-infinite-linear diffusion conditions) in the system Ag(poly)/5 x 10 2 M Pb(CI04)2 + 5 x 10 1 M NaCI04+5x10-3 M HCI04 with dEdft = 10 mV s-1 at T = 298 K. (Reprinted from E. Budevski, G Staikov, and W. J. Lorenz, Electrochemical Phase Formation and Growth, p. 49, copyright 1996 John Wiley Sons. Reproduced by permission of John Wiley Sons, Ltd.)...

In a typical spectroelectrochemical measurement, an optically transparent electrode (OTE) is used and the UV/vis absorption spectrum (or absorbance) of the substance participating in the reaction is measured. Various types of OTE exist, for example (i) a plate (glass, quartz or plastic) coated either with an optically transparent vapor-deposited metal (Pt or Au) film or with an optically transparent conductive tin oxide film (Fig. 5.26), and (ii) a fine micromesh (40-800 wires/cm) of electrically conductive material (Pt or Au). The electrochemical cell may be either a thin-layer cell with a solution-layer thickness of less than 0.2 mm (Fig. 9.2(a)) or a cell with a solution layer of conventional thickness ( 1 cm, Fig. 9.2(b)). The advantage of the thin-layer cell is that the electrolysis is complete within a short time ( 30 s). On the other hand, the cell with conventional solution thickness has the advantage that mass transport in the solution near the electrode surface can be treated mathematically by the theory of semi-infinite linear diffusion. 

The current i(t) that flows at time t due to the semi-infinite linear diffusion is expressed by i (t) = nFCADA1/2/(nt)1/2 for unit area of the smooth OTE. Thus, the quantity of electricity Q(t) that flows by time t is expressed by Q t) = 2nFCADA1/2t1/2jn1/2. Compare with Eq. (9.3). 

In the case of mass transport by pure diffusion, the concentrations of electroactive species at an electrode surface can often be calculated for simple systems by solving Fick s equations with appropriate boundary conditions. A well known example is for the overvoltage at a planar electrode under an imposed constant current and conditions of semi-infinite linear diffusion. The relationships between concentration, distance from the electrode surface, x, and time, f, are determined by solution of Fick s second law, so that expressions can be written for [Ox]Q and [Red]0 as functions of time. Thus, for... 

The theoretical treatment of mass transfer in LSV and CV assumes that only diffusion is operative. Supporting electrolyte concentrations of the order of 0.1 M are generally used at substrate concentrations of the order of 10-3 M, which should preclude the necessity of considering mass transfer by migration. Here, it is assumed that planar stationary electrodes are used under circumstances where diffusion can be considered to be semi-infinite linear diffusion. Other types of electrode may give rise to spherical, cyclindrical or rectangular diffusion and these cases have been treated. 

Equation (69) holds universally, but eqn. (70) applies only to a potential step mean perturbation in the case of semi-infinite linear diffusion. For other mean perturbations or other types of (diffusional) mass transport, eqn. (70) should be replaced by the appropriate expression for F(tm). A survey of such expressions was given in a recent review by Sluyters-Rehbach and Sluyters [53], Unfortunately, most of them are of uncomfortable complexity. Therefore it may be preferable to make use of the less rigorous, but more simple, F(tra ) function that can... 

In the foregoing, the expressions needed to account for mass transport of O and R, e.g. eqns. (23), (27), (46), and (61c), were introduced as special solutions of the integral equations (22), giving the general relationship between the surface concentrations cG (0, t), cR (0, t) and the faradaic current in the case where mass transport occurs via semi-infinite linear diffusion. It is worth emphasizing that eqns. (22) hold irrespective of the relaxation method applied. Of course, other types of mass transport (e.g. bounded diffusion, semi-infinite spherical diffusion, and convection) may be involved, leading to expressions different from eqns. (22). 

The mathematics of semi-infinite linear diffusion were given by eqns. (19a—d). The Laplace transforms of these equations are... 

The derivation is most elegantly performed by the Laplace transform method, starting with the general relationship, valid for semi-infinite linear diffusion. 

As described in the introduction, submicrometer disk electrodes are extremely useful to probe local chemical events at the surface of a variety of substrates. However, when an electrode is placed close to a surface, the diffusion layer may extend from the microelectrode to the surface. Under these conditions, the equations developed for semi-infinite linear diffusion are no longer appropriate because the boundary conditions are no longer correct [97]. If the substrate is an insulator, the measured current will be lower than under conditions of semi-infinite linear diffusion, because the microelectrode and substrate both block free diffusion to the electrode. This phenomena is referred to as shielding. On the other hand, if the substrate is a conductor, the current will be enhanced if the couple examined is chemically stable. For example, a species that is reduced at the microelectrode can be oxidized at the conductor and then return to the microelectrode, a process referred to as feedback. This will occur even if the conductor is not electrically connected to a potentiostat, because the potential of the conductor will be the same as that of the solution. Both shielding and feedback are sensitive to the diameter of the insulating material surrounding the microelectrode surface, because this will affect the size and shape of the diffusion layer. When these concepts are taken into account, the use of scanning electrochemical microscopy can provide quantitative results. For example, with the use of a 30-nm conical electrode, diffusion coefficients have been measured inside a polymer film that is itself only 200 nm thick [98]. 

In the simulation of diffusion-limited semi-infinite linear diffusion, the establishment of the electrode boundary condition is equally straightforward The... 

The simulation of other electrochemical experiments will require different electrode boundary conditions. The simulation of potential-step Nernstian behavior will require that the ratio of reactant and product concentrations at the electrode surface be a fixed function of electrode potential. In the simulation of voltammetry, this ratio is no longer fixed it is a function of time. Chrono-potentiometry may be simulated by fixing the slope of the concentration profile in the vicinity of the electrode surface according to the magnitude of the constant current passed. These other techniques are discussed later a model for diffusion-limited semi-infinite linear diffusion is developed immediately. 

In the case of semi-infinite linear diffusion, the concentration of electroactive species A will be maintained at zero at the electrode surface if the diffu-... 

The exact solution for the time-dependence of the current at a planar electrode embedded in an infinitely large planar insulator, the so-called semi-infinite linear diffusion condition, is obtained. Solving the diffusion equation under the proper set of boundary and initial conditions yields the time-dependent concentration profile. 

In the above discussion, semi-infinite linear diffusion has been assumed. In practical work, this means that the diameter of the working electrode, d, is much larger than the thickness of the diffusion layer, S. The question that then arises is what happens when the diameter of the electrode is allowed to decrease and finally reaches the thickness of the diffusion layer ... 

Figure 3.11 Reversible cyclic voltammogram obtained for an interfacial supramolecular assembly under semi-infinite linear diffusion conditions...

---