  We develop a mathematical framework to analyze electrochemical impedance
spectra in terms of a distribution of diffusion times (DDT) for a parallel
array of random finite-length Warburg (diffusion) or Gerischer
(reaction-diffusion) circuit elements. A robust DDT inversion method is
presented based on Complex Nonlinear Least Squares (CNLS) regression with
Tikhonov regularization and illustrated for three cases of nanostructured
electrodes for energy conversion: (i) a carbon nanotube supercapacitor, (ii) a
silicon nanowire Li-ion battery, and (iii) a porous-carbon vanadium flow
battery. The results demonstrate the feasibility of non-destructive "impedance
imaging" to infer microstructural statistics of random, heterogeneous
materials.
