Diffusion impedance reflecting boundary

The complete expression for the Warburg impedance corresponding to finite diffusion with reflective boundary condition is [5]... 

FIGURE 5-12 Transmission-line representation of diffusion impedance A. simple diffusion with reflecting boundary B. reflecting boundary diffusion coupled with homogeneous reaction C. simple diffusion with absorbing boundary D. absorbing boundary diffusion coupled with homogeneous reaction... 

Fig. II.5.6 Nyquist impedance plot due to finite-length diffusion with a reflective boundary condition...

A finite length diffusion layer thickness cannot only be caused by constant concentrations of species in the bulk of the solution but also by a reflective boundary, that is, a boundary that cannot be penetrated by electroactive species (dc/dr = 0). This can happen when blocking occurs at the far end of the diffusion region and no dc current can flow through the system, for example, a thin film of a conducting polymer sandwiched between a metal and an electrolyte solution [6]. The impedance in this case can be described with the expression... 

Fig. 3.7 The complex-plane impedance plot representation (also called the Argand diagram or Nyquist diagram) of the ideal impedance spectra in the case of reflective boundary conditions. Effect of the ratio of the film thickness (L) and the diffusion coefiicient (D). L/D (7) 0.005 (2) 0.1 (2) 0.2 4) 0.5 and (5) 1 s /. i o = 2 Q, 7 ct = 5 Q, (7 = 50 cm fis / Cdi = 20 pFcm. The smaller numbers refer to frequency values in Hz...

We will exemplify this approach for the finite-length-diffusion case. The impedance function for this case is described in Section 2.1.3, Eq. (135) for a reflective boundary and Eq. (136) for a transmissive one. The first case corresponds to diffii-... 

The case of interest for DSC and solar cells in general is the diffusion-recombination impedance with a reflecting boundary condition at the end of the... 

FIGURE 5-7 Complex plots of the impedance model for ordinary diffusion in a iayer of thickness L A. reflecting boundary condition dC/dX = 0 at X = L B. absorbing boundary condition C = 0 at X = L... 

The problem of combining various types of diffusion processes, including diffusion with finite boundaries and homogeneous reactions were addressed earlier [34,36]. One of the studied cases was finite reflecting boimdary diffusion, which also shows a capacitive dispersion [37]. A total expression for diffusion impedance was derived as a rearrangement of Eq. 5-45 for a case where the capacitive dispersion can be described by a constant phase element... 

In a broad sense a parallel combination of charge transfer resistance and CPE elements, in series with finite diffusion element typically represent the circuit. When potential modulation is introduced, charge-transfer-related impedances decrease with increases in electrochemical potential and capacitance for the metal-polymer interface. The capacitance is usually nonideal due to film or electrode porosity [13] and typically is represented by the CPE element. If the film is formed as a reflective boundary, the angle is sometimes different from -90° because of inhomogeneity of the film and distributed values for diffusion coefficients. If two films are formed on the electrode, two RI CPE semicircles are often observed. 

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.

Ions diffuse toward the center of particles, therefore their diffusion path is limited, e.g. the boundary condition is reflective. Impedance analysis of the finite length diffusion for different electrode geometries and boundary conditions is summarized by Jacobsen and West [1995]. Particle geometries occurring in battery materials are thin plate (planar), spherical, and cylindrical. Below are equations for corresponding geometries, modified so that parameters are expressed in electrical terms. 

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