Levi and Aurbach

FIGURE 10.11 CVs at different sweep rates for composite graphite electrode immersed into 1 M Li Ply in 1 3 ethylene carbonate plus drmethylcarbonate mixture. The region between 0.080 and 0.085 V corresponds to the almost complete Lii u —> LiC interphase transition. (From Levi and Aurbach, 2007. J. Solid State Electrochem. 11, 1031-1042, with permission from Springer.)... 

FIGURE 10.12 Schematic representation for the advance of a moving boundary during the electrochemical conversion of l.itV into LiQ following the description of Levi and Aurbach (2007). It is assumed that lithium ions from the electrolyte (left) intercalate into the LiCV phase. 

Here Vuct is the voltage across resistor Ra is the linearized value obtained from the fit of impedance spectra K is RTIF and P is the transfer coefficient that can be assumed to have a value of 0.5. A more exact treatment allows one to derive a similar relationship using the Framkin isotherm (Levi and Aurbach [1999]). 

The dEldc dependence on state of charge of battery materials is also significantly different from Nemstian but is often weU described by the Frumkin isotherm which takes into account attractive or repulsive interactions of adsorbed species, as reviewed by Levi and Aurbach [1999]. The actual dE/dc in the case of any particular material can be obtained by discharge/relaxation experiments, and knowledge of its value can significantly assist quantitative analysis of impedance spectra, as will be shown in the section on battery-spedfic improvanent in impedance spectra fitting. 

It is important to note that even if the experimental dEldc dependence is used in Eqs (11), (21)-(24), the estimated diffusion coefficient will still be the chemical diffusion coefficient. However, Levi and Aurbach [1999] have shown that analysis of experimental dEldX (where X is degree of occupation of intercalation sites) allows one to obtain a concentration-independent diffusion coefficient through calculation of the enhancement factor explicitly, as shown below... 

Equations (31) and (32) can be used to analyze impedance spectra without knowledge of structural electrode parameters (thickness, density, etc). However, we need this information in order to transform the ohmic parameters obtained by a fit into specific electrochemical parameters. In particular, this information can be used to calculate the effective surface area of the particles. Particles used in practical batteries can usually be treated either as thin plates (Levi and Aurbach [1997]) or as pseudospherical in shape (Barsoukov [2003]), and have a narrow size distribution due to sieving. Particle size values are provided by material manufacturers. The number of particles in a given volume can be estimated from the ratio of their crystallographic density of particles, Op, to the density of the composite-electrode film, a. This allows one to calculate the electrochemically active surface area for a composite electrode for thin-plate particles as 5 = xAdalUOp] and for spherical particles as 5 = 3xAdal[rCp. Here x is the fraction of active material in the composite A is the geometric area of the electrode d is the thickness of the composite electrode <7 is the density of the composite electrode Op is the true density of particles and I and r are the thickness of the plate and radius of spherical particles, respectively. 

Levi, M.D., and Aurbach, D. 2007. The application of electroanalytical methods to the analysis of phase transitions during intercalation of ions into electrodes. Journal of Solid State Electrochemistry 11, 1031-1042. 

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