The electrochemical impedance spectrum of asymmetric electrolytes across low to moderate frequencies

We develop new mathematical expressions for the electrochemical impedance spectrum of an asymmetric electrolyte. Specifically, we consider the Poisson-Nernst-Planck (PNP) equations governing ion transport in a binary asymmetric electrolyte with unequal valences and diffusivities, flanked by planar, parallel, blocking electrodes. The electrodes are subject to an oscillating voltage of small amplitude comparable to the thermal voltage scale. We further consider the limit practically applicable to liquid electrolytes of thin Debye layers epsilon = 1/ (kappa L) (sic)1, where kappa(-1) is the Debye screening length and L is the length of the half cell. The impedance is then obtained from the current in the external circuit, defined as the rate of change of electric field at either electrode. We identify two distinct asymptotic regimes for the frequency (omega) in the limit epsilon-, 0: low frequencies, on the order of the inverse bulk diffusion time omega= O(D-A/L-2); and moderate frequencies, on the order of the inverse RC charging time scale omega = O( D-A kappa/L). Here D-A = (z(+)+z_)D+D_/(z(+)D(+) +z_D_) is the ambipolar diffusion coefficient of the salt, z(+ )are the ionic valences, and D+ are the ionic diffusivities. Our analysis provides new, concise expressions for the impedance at low and moderate frequencies, that account for ionic valence asymmetry. Further, in the moderate frequency analysis, higher order contributions (in epsilon) to the impedance reveal that the real part (i.e. the resistance) appears to diverge as omega-1/2 in the limit omega L/ (kappa DA)-, 0, akin to a semi-infinite Warburg element. We determine that this occurs due to the existence of oscillating diffusion layers, O epsilon(1)/2L ( ) in width, that bridge the ion transport between the O (epsilon L) wide Debye layers adjacent to each electrode and the O (L) wide bulk electro-neutral electrolyte. In the low frequency regime, however, these diffusion layers relax, and effectively merge across the bulk of the cell; consequently, the resistance spectrum attains a plateau in the limit of zero frequency omega L-2/D-A-, 0. Thus, our asymptotic analysis discerns the electrochemical processes that dictate the impedance of asymmetric electrolytes at the low and moderate frequency regimes.