Electrode-solution interface, diffusion

Fig. 7. (a) Simple battery circuit diagram where represents the capacitance of the electrical double layer at the electrode—solution interface, W depicts the Warburg impedance for diffusion processes, and R is internal resistance and (b) the corresponding Argand diagram of the behavior of impedance with frequency, for an idealized battery system, where the characteristic behavior of A, ohmic B, activation and C, diffusion or concentration (Warburg... 

Transport of a species in solution to and from an electrode/solution interface may occur by migration, diffusion and convection although in any specific system they will not necessarily be of equal importance. However, at the steady state all steps involved in the electrode reaction must proceed at the same rate, irrespective of whether the rate is controlled by a slow step in the charge transfer process or by the rate of transport to or from the electrode surface. It follows that the rate of transport must equal the rate of charge transfer ... 

In a cathodic process, removal of ions from solutions will result in a decrease in their concentration at the electrode/solution interface compared to that in the bulk solution, and this in turn will cause a concentration gradient and consequent diffusion. Furthermore, the potential gradient... 

Concentration (diffusion or transport) Overpotential change of potential of an electrode caused by concentration changes near the electrode/solution interface produced by an electrode reaction. 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

Similar to those observed with the cysteine-modified electrode in Cu, Zn-SOD solution [98], CVs obtained at the MPA-modified Au electrode in phosphate buffer containing Fe-SOD or Mn-SOD at different potential scan rates (v) clearly show that the peak currents obtained for each SOD are linear with v (not v 1/2) over the potential scan range from 10 to 1000 mVs-1. This observation reveals that the electron transfer of the SODs is a surface-confined process and not a diffusion-controlled one. The previously observed cysteine-promoted surface-confined electron transfer process of Cu, Zn-SOD has been primarily elucidated based on the formation of a cysteine-bridged SOD-electrode complex oriented at an electrode-solution interface, which is expected to sufficiently facilitate a direct electron transfer between the metal active site in SOD and Au electrodes. Such a model appears to be also suitable for the SODs (i.e. Cu, Zn-SOD, Fe-SOD, and Mn-SOD) with MPA promoter. The so-called... 

If electron transport is fast, the system passes from zone R to zone S+R and then to zone SR. In the latter case there is a mutual compensation of diffusion and chemical reaction, making the substrate concentration profile decrease within a thin reaction layer adjacent to the film-solution interface. This situation is similar to what we have termed pure kinetic conditions in the analysis of an EC reaction scheme adjacent to the electrode solution interface developed in Section 2.2.1. From there, if electron transport starts to interfere, one passes from zone SR to zone SR+E and ultimately to zone E, where the response is controlled entirely by electron transport. 

Figure 6.3 Plots of concentration against distance from the electrode solution interface ( concentration profiles ) as a function of time during the chronoamperometry experiment for (a) the concentration of Tl (as reactant) remaining in solution (b) the concentration of Tl + (as product). Movement of the material through the solution is by diffusion, i.e. a convection-free situation.

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. 

We said at the start of this chapter that convection is a much more efficient form of mass transport than diffusion, to the extent that diffusion can be ignored. For mass transport through the solution bulk this is unreservedly true, so movement of the analyte-containing solution to the electrode solution interface is controlled by convective flow. 

Equilibrium is reached when the driving force for the diffusion (the concentration gradient) is compensated for by the electric field (the potential gradient). Under these equilibrium conditions, there is an equilibrium net charge on each side of the junction and an equilibrium potential difference d< >e. This process is analogous to the way charge transfer across a nonpolarizable electrode/solution interface results in the establishment of an equilibrium potential difference across the interface. 

Since there is no net diffusion under equilibrium conditions, then- p hole current is equal to the p —> n hole current. These equilibrium currents are analogous to the equilibrium exchange currents at an electrode/solution interface. They represent the exchange of holes across the junction between the n- and p-types of material and will be designated by the symbol i0fl. This i0 will now be examined more carefully. 

Now, this quantity impedance (Z) turns out upon detailed analysis to contain within the characteristics of its variation with frequency,48 properties of the reaction occurring at the electrode/solution interface. For example, if a reaction occurring there has as its rale-determining step the electron transfer, then the variation of the impedance with frequency will have certain characteristics different from those shown in the Z — log to plot if the rate-determining step involves instead diffusion in the solution. So, by working out how Z varies with log CD according to a chosen mechanism... 

Electrode-solution interface. The tightly adsorbed inner layer (also called the compact, Helmholtz, or Stem layer) may include solvent and any solute molecules. Cations in the inner layer do not completely balance the charge of the electrode. Therefore, excess cations are required in the diffuse part of the double layer for charge balance. 

Since the electrochemical reduction or oxidation of a molecule occurs at the electrode-solution interface, molecules dissolved in solution in an electrochemical cell must be transported to the electrode for this process to occur. Consequently, the transport of molecules from the bulk liquid phase of the cell to the electrode surface is a key aspect of electrochemical techniques. This movement of material in an electrochemical cell is called mass transport. Three modes of mass transport are important in electrochemical techniques hydrodynamics, migration, and diffusion. 

The important concept in these dynamic electrochemical methods is diffusion-controlled oxidation or reduction. Consider a planar electrode that is immersed in a quiescent solution containing O as the only electroactive species. This situation is illustrated in Figure 3.1 A, where the vertical axis represents concentration and the horizontal axis represents distance from the electrodesolution interface. This interface or boundary between electrode and solution is indicated by the vertical line. The dashed line is the initial concentration of O, which is homogeneous in the solution the initial concentration of R is zero. The excitation function that is impressed across the electrode-solution interface consists of a potential step from an initial value E , at which there is no current due to a redox process, to a second potential Es, as shown in Figure 3.2. The value of this second potential is such that essentially all of O at the electrode surface is instantly reduced to R as in the generalized system of Reaction 3.1 ... 

To impose the diffusion-controlled conversion of O to R as described earlier, the potential E impressed across the electrode-solution interface must be a value such that the ratio Cr/Cq is large. Table 3.1 shows the potentials that must be applied to the electrode to achieve various ratios of C /Cq for the case in which Eq R = 0. For practical purposes, C /C = 1000 is equivalent to reducing the concentration of O to zero at the electrode surface. According to Table 3.1, an applied potential of -177 mV (vs. E° ) for n = 1 (or -88.5 mV for n = 2) will achieve this ratio. Similar arguments apply to the selection of the final potential. On the reverse step, a small C /Cq is desired to cause diffusion-controlled oxidation of R. Impressed potentials of +177 mV beyond the E° for n = 1 (and +88.5 mV for n = 2) correspond to Cr/Cq = 10"3. These calculations are valid only for reversible systems. Larger potential excursions from E° are necessary for irreversible systems. Also, the effects of iR drop in both the electrode and solution must be considered and compensated for as described in Chapter 6. 

Figure 3.37 illustrates the Nernst diffusion layer in terms of concentration-distance profiles for a solution containing species O. As pointed out previously, the concentration of redox species in equilibrium at the electrode-solution interface is determined by the Nernst equation. Figure 3.37A illustrates the concentration-distance profile for O under the condition that its surface concentration has not been perturbed. Either the cell is at open circuit, or a potential has been applied that is sufficiently positive of Eq R not to alter measurably the surface concentrations of the 0,R couple. 

The spatial uniformity of temperature in the cell is difficult to determine, and we are not aware of a careful study of this problem. In most experiments, it is the temperature of the electrode-solution interface or that of the diffusion layer that is relevant. A possible internal thermometer could be created by measuring a temperature-sensitive voltammetric function, for example, the peak separation in the cyclic voltammogram of a reversible reaction, which is 2.22RT/ F. The resolution is not likely to be outstanding, but such a technique would probably allow detection of serious differences between the thermocouple reading and the actual temperature of the electrode-solution interface. 

The membrane system considered here is composed of two aqueous solutions wd and w2, separated by a liquid membrane M, and it involves two aqueous solution/ membrane interfaces WifM (outer interface) and M/w2 (inner interface). If the different ohmic drops (and the potentials caused by mass transfers within w1 M, and w2) can be neglected, the membrane potential, EM, defined as the potential difference between wd and w2, is caused by ion transfers taking place at both L/L interfaces. The current associated with the ion transfer across the L/L interfaces is governed by the same mass transport limitations as redox processes on a metal electrode/solution interface. Provided that the ion transport is fast, it can be considered that it is governed by the same diffusion equations, and the electrochemical methodology can be transposed en bloc [18, 24]. With respect to the experimental cell used for electrochemical studies with these systems, it is necessary to consider three sources of resistance, i.e., both the two aqueous and the nonaqueous solutions, with both ITIES sandwiched between them. Therefore, a potentiostat with two reference electrodes is usually used. 

Diffuse layer capacitance — The diffuse layer is the outermost part of the electrical double layer [i]. The electrical double layer is the generic name for the spatial distribution of charge (electronic or ionic) in the neighborhood of a phase boundary. Typically, the phase boundary of most interest is an electrode/solution interface, but may also be the surface of a colloid or the interior of a membrane. For simplicity, we here focus on the metal/solution interface. The charge carriers inside the metal are electrons, which are confined to... 

Nernst layer -> Diffusion of electroactive species from the bulk solution to the -> electrode surface, or vice versa, takes place in a thin layer of stagnant solution close to the electrode/solution interface when the concentration of electroactive species at this interface, q (x = 0), deviates from the bulk concentration c (see Figure). The concentration gradient at the electrode/solution interface drives the diffusion flux of electroactive species. Generally, there is a linear concentration profile close to the electrode surface and at longer distances from the electrode surface the concentration asymptotically approaches the bulk concentration. The Nernst layer is ob-... 

Instead of attempting a general discussion of the three conditions characterizing a particular diffusion problem, it is best to treat a typical electrochemical diffusion problem. Consider that in an electrochemical system a constant current is switched on at a time arbitrarily designated t=0 (Fig. 4.17). The current is due to charge-transfer reactions at the electrode-solution interfaces, and these reactions consume a species. Since the concentration of this species at the interface falls below the bulk concentration, a concentration gradient for the species is set up and it diffuses toward the interface. Thus, the externally controlled current sets up a diffusion flux within the solution. 

It is convenient from many points of view to assume that the constant value of the flux is unity, i.e., 1 mole of the diffusing species crossing 1 cm of the electrode-solution interface per second. This unit flux corresponds to a constant current density of 1A cm . This normalization of the flux scarcely affects the generality of the treatment because it will later be seen that the concentration response to an arbitrary flux can easily be obtained from the concentration response to a unit flux. 

This then is the fundamental equation showing how the concentration of the diffusing species varies with distance x from the electrode-solution interface and with... 

In the present chapter, the relationship between the electrode potential and the activity of the solution components in the cell is examined in detail. The connection between the Galvani potential difference at the electrode solution interface and the electrode potential on the standard redox scale is discussed. This leads to an examination of the extrathermodynamic assumption which allows one to define an absolute electrode potential. Ion transfer processes at the membrane solution interface are then examined. Diffusion potentials within the membrane and the Donnan potentials at the interface are illustrated for both liquid and solid state membranes. Specific ion electrodes are described, and their various modes of sensing ion activities in an analyte solution discussed. The structure and type of membrane used are considered with respect to its selectivity to a particular ion over other ions. At the end of the chapter, emphasis is placed on the definition of pH and its measurement using the glass electrode. 

Total electrode impedance consists of the contributions of the electrolyte, the electrode solution interface, and the electrochemical reactions taking place on the electrode. First, we consider the case of an ideally polarizable electrode, followed by semi-infinite diffusion in linear, spherical, and cylindrical geometry and, finally a finite-length diffusion. 

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