Electrochemical Impedance Imaging via the Distribution of Diffusion Times

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 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 nondestructive ``impedance imaging'' to infer microstructural statistics of random, heterogeneous materials.