Hopping and diffusion models are extremely useful for describing processes occurring in extended systems, on time scales far longer than some underlying molecular time, such as a characteristic solvent vibrational time or (for crystals) an inverse Debye frequency. Most applications of hopping models to problems in chemistry and materials science assume the presence of two time scales, a residence time and hopping time, and that the time of residence is far longer than the time involved in hopping from one site to another. We describe a generalization of this model to deal with systems in which the underlying structure exhibits dynamical disorder - that is, in which in addition to the species undergoing hopping motions, the lattice itself is reorganizing in time. An important example is glass-forming liquids above their glass transition temperature, especially polymeric materials. In its simplest realization, this multiple time scale hopping model involves only two times - a hopping time and a renewal time τR characterizing the average relaxation time of the underlying lattice motions. One then is faced with the analysis of a problem involving motion on these two time scales, and with the application of that model to a number of systems. Experimentally, a model was originally developed to deal with polymer electrolyte materials, in which ionic diffusion occurs in polymer hosts well above their glass transition temperatures. In this case, the renewal time can be roughly correlated with the glass transition relaxation in the neat polymer host. The dynamic disorder hopping model, or dynamic bond percolation model, is closely related to other models used in solid-state theory, such as the continuous time random walk of Scher and Lax, or the stirred percolation model used in the study of microemulsions. It has a very simple chemical interpretation, since only two times are defined. We describe the nature of the dynamic disorder models, their solutions in particular cases, and their application to a number of physical systems. Particularly important results include formal proofs that, when dynamic disorder is present, percolation thresholds disappear and the system is always diffusive over times long compared to the renewal time. One can also derive generalized analytic continuation results relating the frequency-dependent diffusion in the dynamically renewing lattice to the frequency-dependent diffusion in the static lattice. While the model was originally developed to deal with ionic transport in polymer media, a number of interesting applications in other areas, including polymer viscosity and polymer dynamics, are also briefly discussed.
ASJC Scopus subject areas
- Physical and Theoretical Chemistry