Modeling the Effects of Ion Association on Alternating Current Impedance of Solid Polymer Electrolytes

This work presents a rigorous continuum model describing the transport of ions and associated ion pairs in solid polymer electrolytes subjected to small amplitude alternating current ~ac! excitation. The model treats ion association as a reversible reaction among ions and ion pairs. Dimensionless governing equations are developed from component mass balances, flux equations based on dilute solution theory, and the Poisson equation. Assuming reversible electrode reactions and electroneutrality, the model equations have an analytical solution. Further simplifications are possible in limiting cases ~weak and strong association, zero and infinite frequency excitation!, giving expressions consistent with previously published models. We use the model to explore the effect of association/dissociation reaction rates, ion pair diffusivity, and fractional dissociation on ac impedance behavior. We present a scheme for establishing component diffusivities and fractional dissociation from independent experimental data for lithium perchlorate in poly~ethylene oxide!. With no additional adjusted parameters, satisfactory agreement exists between calculated and experimental ac impedance data. © 2002 The Electrochemical Society. @DOI: 10.1149/1.1480018# All rights reserved. However, the mathematical models used to extract transport proper- ties from the data do not generally account for ion association. In- stead, transport properties are interpreted in the context of the usual strong electrolyte model. 5 The impact of this assumption may differ from one technique to another, so accurate, consistent values of ionic diffusion coefficients and transference numbers may be diffi- cult to obtain. Specifically, small signal ac and dc conductance mea- surements will not yield accurate values of transport properties if ion association changes the number and mobility of charge carriers. Pulsed-field-gradient nuclear magnetic resonance ~NMR! spectroscopy 6 only provides values of transport properties represent- ing averages over atoms as free ions and ion pairs. Recognizing this limitation, models of battery cell performance 7 rely on empirical correlations of conductivity data ~again, inter- preted in the context of strong electrolyte! rather than a more fun- damental description. The empirical approach provides the basic data needed to engineer a particular device, but nothing more, no deeper understanding of transport mechanisms, nor any basis for extrapolating to other conditions. Furthermore, the empirical ap- proach is labor-intensive, necessitating many measurements to de- scribe, for example, the complete temperature- and concentration- dependence of ionic conductivity. More sophisticated models of ion transport may be able to ad- dress these concerns. In particular, models of ion transport in poly- mer electrolytes should account for ion association. By reducing the number of experiments needed to characterize electrolyte conductiv- ity, such a model may serve as a phenomenological aid. Moreover, ion association models may provide greater insight into the funda- mental mechanisms of ion transport in polymer electrolytes. To this end, our previous work 8 describes a rigorous theoretical analysis of the effect of ion association on direct current ~dc! con- ductivity, general current-potential behavior, and limiting current density in solid polymer electrolytes. The predictions of the model highlight the effects of the relative diffusion coefficients and dimen- sionless association constant on concentration distributions of simple ions and ion pairs, the limiting current density, and the po- tential drop required to drive a specified current density. The quali- tative trends of these predictions are in accord with experimental observations. However, no single dc measurement ~e.g., conductiv- ity, current vs. potential, or limiting current density !, analyzed in conjunction with the dc model, can be used to quantify the extent of ion association. Additional experimental data are needed to establish values of all of the model's dimensionless parameters. Other obvious sources of data are alternating current ~ac! imped- ance experiments. Much effort has been devoted to developing mod- els of ion transport under ac excitation. The early work of MacDonald 9-12 accounted for ion-ion interactions via the Poisson equation. Ultimately, this model 12 was extended to include the ef- fects of ion association modeled as an equilibrium reaction between ions and neutral, immobile ion pairs. The ac transport model of