Diffusion layer problems

A solution to the diffuse layer problem was proposed independently by Gouy and by Chapman ... 

The effect known either as electroosmosis or electroendosmosis is a complement to that of electrophoresis. In the latter case, when a field F is applied, the surface or particle is mobile and moves relative to the solvent, which is fixed (in laboratory coordinates). If, however, the surface is fixed, it is the mobile diffuse layer that moves under an applied field, carrying solution with it. If one has a tube of radius r whose walls possess a certain potential and charge density, then Eqs. V-35 and V-36 again apply, with v now being the velocity of the diffuse layer. For water at 25°C, a field of about 1500 V/cm is needed to produce a velocity of 1 cm/sec if f is 100 mV (see Problem V-14). 

Examples of such irreversible species (12) include hydroxjiamine, hydroxide, and perchlorate. The electrochemistries of dichromate and thiosulfate are also irreversible. The presence of any of these agents may compromise an analysis by generating currents in excess of the analytically usehil values. This problem can be avoided if the chemical reaction is slow enough, or if the electrode can be rotated fast enough so that the reaction does not occur within the Nemst diffusion layer and therefore does not influence the current. 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

Diffusion in a convective flow is called convective diffusion. The layer within which diffnsional transport is effective (the diffnsion iayer) does not coincide with the hydrodynamic bonndary layer. It is an important theoretical problem to calcnlate the diffnsion-layer thickness 5. Since the transition from convection to diffnsion is gradnal, the concept of diffusion-layer thickness is somewhat vagne. In practice, this thickness is defined so that Acjl8 = (dCj/ff) Q. This calcniated distance 5 (or the valne of k ) can then be used to And the relation between cnrrent density and concentration difference. 

The constancy of the diffusion layer over the entire surface and thus the uniform current-density distribution are important features of rotating-disk electrodes. Electrodes of this kind are called electrodes with uniformly accessible surface. It is seen from the quantitative solution of the hydrodynamic problem (Levich, 1944) that for RDE to a first approximation... 

This book seeks essentially to provide a rigorous, yet lucid and comprehensible outline of the basic concepts (phenomena, processes, and laws) that form the subject matter of modem theoretical and applied electrochemistry. Particular attention is given to electrochemical problems of fundamental significance, yet those often treated in an obscure or even incorrect way in monographs and texts. Among these problems are some, that appear elementary at first glance, such as the mechanism of current flow in electrolyte solutions, the nature of electrode potentials, and the values of the transport numbers in diffusion layers. 

At present it is impossible to formulate an exact theory of the structure of the electrical double layer, even in the simple case where no specific adsorption occurs. This is partly because of the lack of experimental data (e.g. on the permittivity in electric fields of up to 109 V m"1) and partly because even the largest computers are incapable of carrying out such a task. The analysis of a system where an electrically charged metal in which the positions of the ions in the lattice are known (the situation is more complicated with liquid metals) is in contact with an electrolyte solution should include the effect of the electrical field on the permittivity of the solvent, its structure and electrolyte ion concentrations in the vicinity of the interface, and, at the same time, the effect of varying ion concentrations on the structure and the permittivity of the solvent. Because of the unsolved difficulties in the solution of this problem, simplifying models must be employed the electrical double layer is divided into three regions that interact only electrostatically, i.e. the electrode itself, the compact layer and the diffuse layer. 

The diffuse layer is formed, as mentioned above, through the interaction of the electrostatic field produced by the charge of the electrode, or, for specific adsorption, by the charge of the ions in the compact layer. In rigorous formulation of the problem, the theory of the diffuse layer should consider ... 

A rigorous solution of this problem was attempted, for example, in the hard sphere approximation by D. Henderson, L. Blum, and others. Here the discussion will be limited to the classical Gouy-Chapman theory, describing conditions between the bulk of the solution and the outer Helmholtz plane and considering the ions as point charges and the solvent as a structureless dielectric of permittivity e. The inner electrical potential 0(1) of the bulk of the solution will be taken as zero and the potential in the outer Helmholtz plane will be denoted as 02. The space charge in the diffuse layer is given by the Poisson equation... 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

The problem may be a semantic one because OH- does not bind very strongly to cationic micelles (Romsted, 1984) and competes ineffectively with other ions for the Stern layer. But it will populate the diffuse Gouy-Chap-man layer where interactions are assumed to be coulombic and non-specific, and be just as effective as other anions in this respect. Thus the reaction may involve OH- which is in this diffuse layer but adjacent to substrate at the micellar surface. The concentration of OH- in this region will increase with increasing total concentration. This question is considered further in Section 6. 

Calculation stability implies that At/Ay2 <0.5. The fulfillment of this condition may become a problem when fast reactions, or more precisely, large values of the kinetic parameter, are involved since most of the variation of C then occurs within a reaction layer much thinner than the diffusion layer. Making Ay sufficiently small for having enough points inside this layer thus implies diminishing At, and thus increasing the number of calculation lines, to an extent that may rapidly become prohibitive. This is, however, not much of a difficulty in a number of cases since the pure kinetic conditions are reached before the problem arises. This is, for example, the case with the calculation alluded to in Section 2.2.5, where application of double potential step chronoamperometry to various dimerizations mechanisms was depicted. In this case the current ratio becomes nil when the pure kinetic conditions are reached. 

What benefits and drawbacks to these problems can one expect from the use of cyclic voltammetry instead of RDEV They are related. In a general case, the application of cyclic voltammetry will be more complicated, because playing with the scan rate, one can make the diffusion layer penetrate the film or remain outside, as is the case with RDEV. We have already seen a fruitful application of the first of these possibilities in the use of cyclic voltammetry to the characterization of electron hopping transport within the redox films (Section 4.3.4). In the second situation, cyclic voltammetry may replace RDEV in a manner similar to what has been seen in Section 4.3.2 Each time a term (1 — ///a) is encountered in the analysis, it suffices to replace it by... 

The diffusion-reaction problem in the more general case occurs in a system containing n — 1 inactivated enzyme layers adjacent to the electrode surface on top of which N — n active layers have been deposited. Table 6.9 lists the equations that govern the fluxes of the two forms of the cosubstrate in such systems. 

However some problems must be addressed prior to the use of Equation (3.66) from the data generated above. The first problem is that the ions in the diffuse layers must be included in the estimation of k. This becomes essential as soon as the particle-particle interactions become significant and so Equations (3.62) to (3.65) should contain a volume fraction dependence of k of the form given by Russel,30 for example for a symmetrical electrolyte ... 

This point has been noted in earlier treatment of this problem (3). The value of this contribution depends on the model we adopt to describe the diffuse layer. 

The presence of the diffuse layer determines the shape of the capacitance-potential curves. For a majority of systems, models describing the double-layer structure are oversimplified because of taking into account only the charge of ions and neglecting their specific nature. Recently, these problems have been analyzed using new theories such as the modified Poisson-Boltzmann equation, later developed by Lamper-ski. The double-layer capacitanties calculated from these equations are... 

C. S. Kong, D.-Y. Kim, H.-K. Lee, Y.-G. Shul, and T.-H. Lee. Influence of pore-size distribution of diffusion layer on mass-transport problems of proton exchange membrane fuel cells. Journal of Power Sources 108 (2002) 185-191. 

The first problem is that any ion needs to be brought from the bulk solution to the electrode surface before it can be discharged. On approaching the electrode, the ion needs to cross a boundary layer called the diffusion layer. This layer is about 0.1 mm (or 100 (tm) thick and is distinguished from the electrode double layer which is 100000 times smaller (Fig. 6.7). 

At Coventry University [20] we have obtained similar results to Tu under conventional (silent) conditions. Our initial thoughts were that if the effect of the tail off was either a consequence of mass transfer to the electrode, or a consequence of some problem with the diffusion layer, then ultrasound might be expected to have an effect and thus improve the plating rate. Investigations in the presence of ultrasound and at various pH values did not significantly affect the plating characteristics i. e. the plateau effect still remained. However, the overall efficiency in the presence of ultrasound was affected (Fig. 6.10). 

One case is where the ions are traveling to the electrode by a process of diffusion. Then the steady-state diffusion problem can be looked at from the diffusion-layer point of view (Section 7.9). The variation of concentration with distance can be approximated to a linear variation, and the linear concentration gradient can be considered to occur over an effective distance of 8, die diffusion-layer thickness. Then the diffusion current is given by (Section 7.9.10). 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Equation (11) is also applicable as a good, or reasonably good, approximation to a number of techniques classified as d.c. voltammetry , in which the response to a perturbation is measured after a fixed time interval, tm. The diffusion layer thickness, 5/, will be a function of D, and tm and the nature of this function has to be deduced from the rigorous solution of the diffusion problem in combination with the appropriate initial and boundary conditions [21—23]. The best known example is d.c. polarography [11], where the d.c. current is measured at the dropping mercury electrode at a fixed time, tm, after the birth of a new drop as a function of the applied d.c. potential. The expressions for 5 pertaining to this and some other techniques are given in Table 1. 

In the solution of mass transport problems, several dimensionless groups are used in order to reduce the number of variables. Diffusion layer thicknesses etc. are expressed in much of the literature in terms of these dimensionless variables. 

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. 

No discussion of digital simulation would be complete without some mention of the methods employed in treating electrogenerated chemiluminescence (ECL). In the most common statement of the problem, an aromatic hydrocarbon (A) is alternately oxidized and reduced at a single electrode to produce its radical anion and cation species (B and D). These species, in turn, react within the diffusion layer to regenerate the hydrocarbon. 

With these assumptions in mind, we now complete the outline of the solution of the diffusion-reaction problem as it applies to the most difficult case, the pH-based enzymatic sensors (potentiometric or optical). We assume only that there is no depletion layer at the gel/solution boundary (7), and that there is no fixed buffer capacity (4). The objective of this exercise is to find out the optimum thickness of the gel layer that is critically important for all zero-flux-boundary sensors, as follows from (2.26). 

The differential equations and the boundary value problem (bvp) which can be fulfilled by rR and c 1 if the thickness of the membrane is greater than the diffusion layer of X+ into the membrane are ... 

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