Polymer Electrolytes: Hopping, Domain Structures and Frequency-Dependent Conductivity
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POLYMER ELECTROLYTES: HOPPING, DOMAIN STRUCTURES AND FREQUENCY-DEPENDENT CONDUCTIVITY
MARK A. RATNER AND A. NITZAN* Department of Chemistry and Materials Research Center, Northwestern University, Evanston, Illinois 60208 * Permanent Address: Department of Chemistry, Tel-Aviv University Tel-Aviv, Israel
Abstract The dynamic bond percolation model was developed to deal with dynamic disorder, treating ion mobility by a percolation model in which the assignment of any site-to-site jump as allowed or forbidden changes on a timescale related to the local reorganizational dynamics of the polymer segments (the renewal time). Here we discuss the special cases of highfrequency spectra and partially crystalline electrolytes. At high frequencies, the present hopping model yields unphysical behavior (frequencyindependent response); we trace this back to the incorrect treatment of short-time dynamics, and show how it can be corrected. For partially crystalline materials, we show that a rollover feature in the spectrum, in the microwave range, can be expected when ions are trapped in isolated regions of high conductivity, such as amorphous pockets in largely crystalline PEO.
I.
Introduction
The conductivity process in polymer/salt solvent-free electrolytes has been described in various pictorial, mechanistic, modelistic and conceptual ways.1-6 Perhaps the best classification of these materials is as a 6 concentrated coulomb fluid with what Armand has characterized as an immobile solvent (the polymer host). The two most striking points about these materials are the approximate validity of the Walden relationship Dn constant and the high stoichiometric concentration of ions, varying from 0.1 to 10 molar (Here D, n are respectively the diffusion coefficient and the microscopic viscosity). The validity of the Walden relationship is but one of the experimental evidences that in these materials ion diffusion 7 or8 Angell's statement ' conductivity is strongly coupled to solvent motion. in terms of the near unity value of the ratio of conductivity relaxation3,9,10 time to structural relaxation time again reflects this coupling, as does the linear relation between the logarithm of D or a 11 (conductivity) and the shift factor of Williams, Landel and Ferry, that measures mechanical properties. 12
Phenomenologically the ion mobility p can be expressed by the form A(T)
= Ao exp{-B/R(T-To)}
,
13
(1)
where B is a so-called pseudo activation energy and To is an empiricallydetermined temperature often called the equilibrium glass transition temperature, and often about 50K below the thermal glass transition temperature T Most analysis of the thermal dependence of the conduction in polymer electrolytes starts with the VTF form (1) often rationalized in 14 9 10 ,13, terms of quasithermodynamic free-volume or excess entropy models. ,
Mat. Res. Soc. Symp. Proc. Vol. 210. 01991 Materials Research Society
110
In an attempt to formulate a microscopic model for ionic conduction in these materials, we have discussed a dynamic bond percola
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