Infinite thickness, diffusion layer

For a semi-infinite diffusion process at cathode represented by Warburg impedance, the Nyquist plot appears as a straight line with a slope of 45°. The impedance increases linearly with decreasing frequency. The infinite diffusion model is only valid for infinitely thick diffusion layer. For finite diffusion layer thickness, the finite Warburg impedance converges to infinite Warburg impedance at high frequency. At low frequencies or for small... 

The Kubelka-Munk theory treats the diffuse reflectance of infinitely thick opaque layers [4], a situation achieved in practice for UV/VIS spectroscopy through the use of powder path lengths of at least several millimeters. In this instance, the Kubelka-Munk equation has the form... 

The generally accepted theory of diffuse reflectance was developed originally by Kubelka and Munk [43,44] for application to infinitely thick, opaque layers. 

This equation gives the ratio of absorption and scattering coefficients in the case of R diffuse reflectance in an infinitely thick, opaque layer. In the presence of a sample with A molar absorptivity and c molar concentration, the Kubelka-Munk equation takes the following form ... 

This function has become the fundamental law of diffuse reflectance spectroscopy. It relates the diffuse reflectance R of an infinitely thick, opaque layer and the ratio of the absorption and scattering coefficients K/S. Since the scattering coefficient is virtually invariable in the presence of a chromatographic band, the Kubelka-Munk equation can be written in the form ... 

This simple relationship between incident and transmitted light is well known as the Boguert-Lambert-Beer law. This expression renders positive values for Ij < Iq. In case of scattering material like TLC plates, a part of the scattered light is emitted as reflectance J from the plate surface to the top. For the hrst approximation of a parallel incident light beam with the intensity /g, some radiation may be scattered inside the layer and some radiation may be absorbed either by the sample or by the layer itself. According to the Schuster equations and with the abbreviation R (the diffuse reflectance of an infinitely thick layer). 

We consider, then, two media (1 for the cell-wall layer and 2 for the solution medium) where the diffusion coefficients of species i are /),yi and 2 (see Figure 3). For the planar case, pure semi-infinite diffusion cannot sustain a steady-state, so we consider that the bulk conditions of species i are restored at a certain distance <5,- (diffusion layer thickness) from the surface where c, = 0 [28,45], so that a steady-state is possible. Using just the diffusive term in the Nernst-Planck equation (10), it can be seen that the flux at any surface is ... 

If instead of semi-infinite diffusion, some distance (5m acts as an effective diffusion layer thickness (Nernst layer approximation), then a modified expression of equation (63) applies where ro is substituted by 1 / (1 /Vo + 1 /<5m ) (see equation (38) above). For some hydrodynamic regimes, which for simplicity, are not dealt with here, the diffusion coefficient might need to be powered to some exponent [57,58],... 

The voltammetric features of a reversible reaction are mainly controlled by the thickness parameter A = The dimensionless net peak current depends sigmoidally on log(A), within the interval —0.2 < log(A) <0.1 the dimensionless net peak current increases linearly with A. For log(A )< —0.5 the diSusion exhibits no effect to the response, and the behavior of the system is similar to the surface electrode reaction (Sect. 2.5.1), whereas for log(A) > 0.2, the thickness of the layer is larger than the diffusion layer and the reaction occurs under semi-infinite diffusion conditions. In Fig. 2.93 is shown the typical voltammetric response of a reversible reaction in a film having a thickness parameter A = 0.632, which corresponds to L = 2 pm, / = 100 Hz, and Z) = 1 x 10 cm s . Both the forward and backward components of the response are bell-shaped curves. On the contrary, for a reversible reaction imder semi-infinite diffusion condition, the current components have the common non-zero hmiting current (see Figs. 2.1 and 2.5). Furthermore, the peak potentials as well as the absolute values of peak currents of both the forward and backward components are virtually identical. The relationship between the real net peak current and the frequency depends on the thickness of the film. For Z, > 10 pm and D= x 10 cm s tlie real net peak current depends linearly on the square-root of the frequency, over the frequency interval from 10 to 1000 Hz, whereas for L <2 pm the dependence deviates from linearity. The peak current ratio of the forward and backward components is sensitive to the frequency. For instance, it varies from 1.19 to 1.45 over the frequency interval 10 < //Hz < 1000, which is valid for Z < 10 pm and Z) = 1 x 10 cm s It is important to emphasize that the frequency has no influence upon the peak potential of all components of the response and their values are virtually identical with the formal potential of the redox system. 

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. 

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]. 

At the other boundary, bulk concentration of A must be maintained at some finite distance from the electrode, while the concentration of B will be zero at the same point. This distance may be regarded as the diffusion layer thickness. In terms of the simulation, the establishment of the semi-infinite boundary condition requires the determination of the number of volume elements making up the diffusion layer. This will be a function of the number of time iterations that have taken place up to that point in the simulation. At any time in the physical experiment, the diffusion layer thickness is given by 6(Dt)1/2. This rule of thumb may be combined with Equation 20.7 to calculate Jd, the number of volume elements in the diffusion layer ... 

The geometry shown here corresponds to a semi-infinite planar diffusion. Other geometries (e.g., radial geometries) typical for microsensors can be used. The enzyme-containing layer is usually a hydrogel, whose optimum thickness depends on the enzymatic reaction, on the operating pH, and on the activity of the enzyme (i.e., on the Km). Enzymes can be used with nearly any transduction principle, that is, thermal, electrochemical, or optical sensors. They are not, however, generally suitable for mass sensors, for several reasons. The most fundamental one is the fact... 

Fick s first and second laws (Equations 6.15 and 6.18), together with Equation 6.17, the Nernst equation (Equation 6.7) and the Butler-Volmer equation (Equation 6.12), constitute the basis for the mathematical description of a simple electron transfer process, such as that in Equation 6.6, under conditions where the mass transport is limited to linear semi-infinite diffusion, i.e. diffusion to and from a planar working electrode. The term semi-infinite indicates that the electrode is considered to be a non-permeable boundary and that the distance between the electrode surface and the wall of the cell is larger than the thickness, 5, of the diffusion layer defined as Equation 6.19 [1, 33] ... 

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 1.20 Complex-plane impedance plot (Nyquist plane) for an electrochemical system, with the mass transfer and kinetics (charge transfer) control regions, for an infinite diffusion layer thickness. 

Figure 7-5 a) Single layer diffusion from a source with a barrier on one side, b) Diffusion from an infinite thick layer represented as coming from infinitely many sources. 

In general, the frequency dependence of the dif-fusional impedance and the geometry of diffusion are correlated. The (ice)-1/2 frequency dependence corresponds to the semi-infinite planar diffusion such a frequency dependence is valid only if the characteristic length, (D/ce)1/2 of diffusion is much shorter than any size of the electrode or the thickness of the electrolyte layer from which diffusion proceeds. Otherwise spherical, or bounded diffusion with different frequency dependence is observed. 

Unlike macroelectrodes which operate under transient, semi-infinite linear diffusion conditions at all times, UMEs can operate in three diffusion regimes as shown in the Figure for an inlaid disk UME following a potential step to a diffusion-limited potential (i.e., the Cottrell experiment). At short times, where the diffusion-layer thickness is small compared to the diameter of the inlaid disc (left), the current follows the - Cottrell equation and semi-infinite linear diffusion applies. At long times, where the diffusion-layer thickness is large compared to the diameter of the inlaid disk (right), hemispherical diffusion dominates and the current approaches a steady-state value. 

The Warburg impedance is only valid if the diffusion layer has an infinite thickness. If the diffusion layer is bounded, the impedance at lower frequencies no longer obeys Equation 4.32. Instead, the bounded Warburg element (BW) should be used to replace the Warburg. The impedance of the series connection between the resistance and the BW, shown in Figure 4.9a, can be calculated by adding their impedances ... 

If there are deviations from the ideal semi-infinite linear diffusion process, the bounded Randles cell can also be modified by replacing the Warburg impedance with a CPE. The structure of the model is shown in Figure 4.20a. This modification is applied when the transport limitations appear in a layer of finite thickness. 

Further experiments have been conducted to confirm whether or not the presumed diffusion layer and its thickness, 8, as estimated from (95) corresponds to physical reality. First AC impedance spectroscopy has been used to find the frequency response of the real and imaginary components of the cell impedance and compared with the theoretical prediction for diffusion across a thinned diffusion layer. At very high AC frequencies, where the AC perturbation had insufficient time to probe to the edge of the diffusion layer, effectively the response expected for semi-infinite diffusion was seen ( Warburgian behaviour ). At lower AC frequencies, as expected, the cell impedance was greatly reduced in the presence of ultrasound. Moreover, not only was the quantitative behaviour as predicted theoretically... 

The situation at a miniature disc microelectrode embedded in a flat insulator surface (such as an RDE of very small size) can be approximated by spherical symmetry, obtained for a small sphere situated at the center of a much larger (infinitely large, in the present context) spherical counter electrode. How will the change of geometry influence the diffusion-limited current density This is shown qualitatively in Fig. 18L. As time progresses, the diffusion layer thickness increases, causing, in the planar case, a proportional decrease in the diffusion current density. In the spherical configuration the electroactive... 

It should also be noted that the time of experiment for a microelectrode is not limited by natural convection, as found in the case of semi-infinite linear diffusion. Natural convection gives rise to a diffusion layer thickness of the order of 0.02 cm, and its effect is not felt, if the radius of the microelectrode is less than about 10 pm. 

The Kubelka-Munk equation is only valid when R corresponds to the diffuse reflectance of an opaque layer of infinite thickness, so that the background is no longer visible (i.e., the difference in intensity between the incident and reflected beams is independent of the thickness of the structure). 

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