Summary A wide range of physical systems can be formally mapped to a linear chain of sorted objects. Upon introduction of intrachain interactions, such a chain can “fold” to elaborate topological structures, analogous to folded linear polymer systems. Two distinct chain-topology theories, knot theory and circuit topology, have separately provided insight into the structure, dynamics, and evolution of folded linear polymers such as proteins and genomic DNA. Knot theory, however, ignores intrachain interactions (contacts), whereas chain crossings are ignored in circuit topology. Thus, there is a need for a universal approach that can provide topological description of any folded linear chain. Here, we generalize circuit topology in order to grasp particularities typically addressed by knot theory. We develop a generic approach that is simple, mathematically rigorous, and practically useful for structural classification, analysis of structural dynamics, and engineering applications.

Based on the Poisson-Nernst-Planck equations (PNP), the spatiotemporal charge, concentration profile, and the electric field in polyelectrolytes are analyzed. The system is subjected to a dc applied voltage. Different to recent papers we obtain an exact analytical solution of the PNP in the linear regime, which is characterized by an inevitable coupling between the spatial and the temporal behavior. In the long time limit the systems tends in a nonexponential manner to the steady state predicted by the Debye-Hueckel theory, where the time scale for the crossover into the steady state is determined by the Debye screening length and the initial concentration. The higher the initial concentration is the faster the system evolves into the stationary state. The Debye screening length characterizes not only the asymptotic behavior but also the spatiotemporal evolution of the system at finite times. Using experimental data the concentration profile and the electric field is shown to be on a master curve parametrized by the screening length.

The analytical solution of the Poisson-Nernst-Planck equations is found in the linear regime as response to a dc-voltage. In deriving the results a new approach is suggested, which allows to fulfill all initial and boundary conditions and guarantees the absence of Faradaic processes explicitly. We obtain the spatiotemporal distribution of the electric field and the concentration of the charge carriers valid in the whole time interval and for an arbitrary initial concentration of ions. A different behavior in the short- and the long-time regime is observed. The crossover between these regimes is estimated.

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