Transmissive boundary

The problem to be solved is described by a system of two differential equations, Eq. (4.26), with the general solution described by Eq. (4.28). Let us assume that the concentration at the exit plane is equal to zero. Four constants (A, A, B, B ) must be obtained from the boundary conditions for the transmissive boundary  

Cr = Aexp(—%/) + Bexp sRl) = 0. These give the following constants  

Using these relations the mass transfer impedance is obtained  

Assuming that the diffusion coefficients of Ox and Red are the same, Zw becomes 

The complex plane plots obtained in this case are shown in Fig. 4.12. The faradaic impedance displays a straight line at 45° at high frequencies where the ac diffusion, that is, the oscillations of the concentration, are limited to the zone around the electrode and diffusion behaves in a semi-infinite linear manner. At higher frequencies oscillations of the concentrations arrive at the back wall and a 

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By definition, the transmissive boundary between phases a and /3 allows the passage of a -+faradaic current owing to the interfacial transfer of the corresponding... 

Transfer of electroactive species is possible at x = /, and C(/) = 0, but dC l)/dx 0. This is the conducting or transmissive boundary. It is observed, for example, in a rotating disk electrode, where the diffusion layer thickness is determined by the rotation rate. 

This is denoted as diffusion for the transmissive boundary condition. The corresponding complete impedance is [5]... 

Fig. II.5.5 Nyquist impedance plot due to finite length diffusion with a transmissive boundary condition...

Fig. 4.11 Concentration profiles in two cases of finite-length linear diffusion left -transmissive boundary, right - reflective boundary...

A case of finite diffusion length with transmissive boundary conditions has also been considered in the literature [277, 278]. It corresponds to the case where hydrogen diffuses across a membrane and is oxidized on the other side. The same Eq. (7.74) is obtained but with tanh replacing coth in Eq. (7.75). 

Figure 2.1.13. Complex plane representations of the impedance due to a finite-length diffusion process with (a) reflective, (b) transmissive boundary conditions at x = 1.

The diffusion impedance of a bulk electrolyte can be described by a finite length Warburg impedance with transmissive boundary (Eq. (7)). A transmissive boundary is appropriate because an ion produced at the cathode is consumed at the anode and vice versa during battery electrochemical processes. A more precise treatment, using... 

Considering the surface roughness and inhomogeneities of Sn electrodes, capacitances and Cji were replaced by the respective CPEs and Qj[. To account for diffusive mass transport, we tested two elements the Warburg impedance W that represents a semi-infinite diffusion and the element O that follows from the description of the so-called transmissive boundary [106]. It is supposed in the latter case that diffusion occurs in the 6-thick layer. Their impedances are, respectively [106] ... 

In the "best-fit" model, the low-frequency portion of the impedance was represented by a series of complex adsorption and finite transmission boundary diffusion, placed in parallel with the charge transfer and the double layer CPE,... 

An example of this approach is a study of corrosion of stainless-steel anodes in an aggressive hydrochloric-acid environment [47]. Two capacitive semicircles were distinguished— high-frequency impedance related to the formation and growth of the corrosion film and a low-frequency feature related to the film-electrolyte charge-transfer resistance coupled with finite transmission boundary diffusion of ions (protons) through the film The circuit can be represented as (R + Over the... 

In this section we consider the basic properties of the scattered field as they are determined by energy conservation and by the propagation properties of the fields in source-free regions. The results are presented for electromagnetic scattering by dielectric particles, which is modeled by the transmission boundary-value problem. To formulate the transmission boundary-value problem we consider a bounded domain Di (of class with boimdary S and exterior D, and denote by n the unit normal vector to S directed into Ds (Fig. 1.8). The relative permittivity and relative permeability of the domain Dt are et and /it, where t = s,i, and the wave number in the domain Dt is kt = where ko is the wave number in free space. The imbounded... 

It should be emphasized that for the assumed smoothness conditions, the transmission boundary-value problem possesses an unique solution [177]. 

The transmission boundary-value problem for homogeneous and isotropic particles has been formulated in Sect. 1.4 but we mention it in order for our analysis to be complete. We consider an homogeneous, isotropic particle occupying a domain D with boundary S and exterior (Fig. 2.1). The imit normal vector to S directed into is denoted by n. The exterior domain Ds is assumed to be homogeneous, isotropic, and nonabsorbing, and if t and jM. are the relative permittivity and permeability of the domain Ht, where t = s, i, we have s > 0 and ps > 0. The wave number in the domain Dt is kt = ko, /etPt, where ko is the wave number in the free space. The transmission boundary-value problem for a homogeneous and isotropic particle has the following formulation. 

The null-field method leads to a nonsingular integral equation of the first kind. However, in the framework of the surface integral equation method, the transmission boundary-value problem can be reduced to a pair of singular integral equations of the second kind [97]. These equations are formulated in terms of two surface fields which are treated as independent unknowns. In order to elucidate the difference between the null-field method and the surface integral equation method we follow the analysis of Martin and Ola [155] and review the basic boundary integral equations for the transmission boundary-value problem. We consider the vector potential Aa with density a... 

The problem of scattering by isotropic, chiral spheres has been treated by Bohren [16], and Bohren and Huffman [17] using rigorous electromagnetic field-theoretical calculations, while the analysis of nonspherical, isotropic, chiral particles has been rendered by Lakhtakia et al. [135]. To accoimt for chirality, the surface fields have been approximated by left- and right-circularly polarized fields and the same technique is employed in our analysis. The transmission boundary-value problem for a homogeneous and isotropic, chiral particle has the following formulation. 

The solution of the transmission boundary-value problem for each particle. 

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