Stagnant Diffusion Layer

Following equation (11.15), the expression of the convective-diffusion impedance can be found to be 

In stagnant environments, if natural convection can be ignored, the convective-diffusion equation is reduced to 

No steady-state solution is possible. As shown in Example 2.5, the concentration is given in terms of a similarity variable as  

In principle, impedance measurements are possible only for stationary systems, i.e., those systems for which a steady solution is possible. However, after a sufficient time, the concentration profile near the electrode can be considered to be stationary wifii respect to the time required for the impedance measurement. 

Substitution of the definition for concentration (equation (11.44) into the expression for conservation of species i (equation (11.41)) yields 

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A stagnant diffusion layer is often assumed to approximate the effect of aqueous transport resistance. Figure 4 shows a membrane with a diffusion layer on each side. The bulk solutions are assumed to be well mixed and therefore of uniform concentrations cM and cb2. Adjacent to the membrane is a stagnant diffusion layer in which a concentration gradient of the solute may exist between the well-mixed bulk solution and the membrane surface the concentrations change from Cm to c,i for solution 1 and from cb2 to cs2 for solution 2. The membrane surface concentrations are cml and cm2. The membrane has thickness hm, and the aqueous diffusion layers have thickness ht and h2. 

If q is negligibly small a stagnant diffusion layer in aqueous phase is a rate-limiting step. This case is often the most useful from an analytical point of view. The well-known equation for a reversible polarographic wave can be obtained as ... 

Every solid catalyst in solution is surrounded by a "stagnant diffusion layer which reactants must cross in order to reach the surface. The resulting concentration profile is sketched in Fig. 6. The rate of the reactant s arrival at the solid/liquid interface is determined by its concentration gradient at that interface, (dc/dx)x=0. The diffusion layer therefore has the same effect on the rate as does the simplified layer shown by the dotted lines [63]. The thickness of this so-called Nernst layer is designated 5. It follows from Fick s first law of diffusion that the number of moles of reactant A, nA, that reach the surface in unit time is given by... 

Peskin [49] used the Galerkin finite-element method to compute current distribution and shape change for electrodeposition into rectangular cavities. A concentration-dependent overpotential expression including both forward and reserve rate terms was used, and a stagnant diffusion layer was assumed. An adaptive finite-element meshing scheme was used to redefine the problem geometry after each time step. 

Figure 11.3 Oscillating concentration profiles for a finite stagnant diffusion layer a) imaginary part b) real part.

Figure 11.6 Oscillating concentration as a function of time with position as a parameter for a finite stagnant diffusion layer.

Plot, on an impedance plane format, the impedance obtained for a Nernst stagnant diffusion layer and the impedance obtained for a rotating disk electrode under assumption of an infinite Schmidt number. Show that, while the behaviors of the two models at high and low frequencies are in agreement, the two models do not agree at intermediate frequencies. Explain. 

A Nemst stagnant-diffusion-layer model was used to accovmt for the diffusion impedance. This model is often used to account for mass transfer in convective systems, even though it is well known that this model caimot ac-coimt accurately for the convective diffusion associated with a rotating disk electrode. 

We saw above that the concentration gradient at an electrode will be linear with respect to the spatial coordinate perpendicular to the electrode surface if the anode/cathode cell were operated at a constant current density and if the fluid velocity were zero. In actuality, there will always be some bulk liquid electrolyte stirring during current flow, either an imposed forced convection velocity or a natural convection fluid motion due to changes in the reacting species concentration and fluid density near the electrode surface. In electrochemical systems with fluid flow, the mass transfer and hydrodynamic fluid flow equations are coupled and the solution of the relevant differential equations is often a formidable task, involving complex mathematical and/or numerical solution techniques. The concept of a stagnant diffusion layer or Nemst layer parallel and adjacent to the electrode surface is often used to simplify the analysis of convective mass transfer in... 

To avoid the uncertainty in determining the stagnant diffusion layer thickness S, the following semi-empirical engineering model is often used... 

The simplest model of mass transport is the bulk diffusion of growth units through a stagnant diffusion layer adjacent to the crystal surface with complete mixing beyond. For a diffusion layer of thickness 6, the linear growth rate G can be derived by integrating Fick s law as... 

The extracted species diffuses across the stagnant diffusion layer at the membrane/feed solution interface where it reacts with the carrier, and the reacted carrier species is replaced by another carrier species from the bulk of the membrane. 

The magnitude of the average limiting currents observed at a glass-mounted Pt disc electrode is proportional to the concentration of reactant q,uik. and to the area of the electrode. A, for a range of electrode diameters from 0.25 to 6.0 mm consistent with Eqn. (2). However, the effect of the diffusion coefficient, D, on the observed limiting current has been shown [25-30] to reveal the subtleties of the nature of mass transport observed. A linear relationship, a D, is indicative of a truly stagnant diffusion layer, in the presence of 20 kHz, 7 i j is... 

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