Vl. Kobelev et al., Fractal theory for the temperature dependence of the constant-phase element of solid electrolytes, RUSS J ELEC, 35(3), 1999, pp. 294-302
The temperature dependence of the constant-phase element (CPE) is described
in terms of fractal representations of grain boundaries of a polycrystalli
ne electrolyte, both inside it and at the electrode-sample interface. The s
imple expression alpha(i) = d(i) is used to relate the fractal dimensionali
ty d(i) of the boundary to CPE alpha(i) of the ith crystallite. CPE of a sa
mple comprising a large number of crystallites is determined by averaging a
lpha(i) from the density of distribution of fractal dimensionalities n(d(i)
) of boundaries of individual crystallites (normal distribution). Selecting
, as an example, beta = beta(0)T(-1)exp(-Delta b x 1000/T) as a parameter o
f the normal distribution beta(T) allows one to make comparisons with the e
xperiment using two parameters, beta(0) and Delta b. To determine beta(T),
beta is also represented through the diffusion coefficient, capacitance, re
sistance, and average area of boundaries of crystallites.