Anomalous advection-diffusion in two-dimensional semiconductor systems with coexisting energetic and structural disorder is described in the framework of a generalized model of multiple trapping on a comb-like structure. The basic equations of the model contain fractional-order derivatives. To validate the model, we compare analytical solutions with results of a Monte Carlo simulation of phonon-assisted tunneling in two-dimensional patterns of a porous nanoparticle agglomerate and a phase-separated bulk heterojunction. To elucidate the role of directed percolation, we calculate transient current curves of the time-of-flight experiment and the evolution of the mean squared displacement averaged over medium realizations. The variations of the anomalous advection-diffusion parameters as functions of electric field intensity, levels of energetic, and structural disorder are presented.

The effect of anomalous diffusion of lithium on the discharge curves and impedance spectra of lithium-ion batteries (LIB) is studied within the fractional differential generalization of the single-particle model. The distribution of lithium ions in electrolyte and electrode particles is expressed through the Mittag–Leffler function and the Lévy stable density. Using the new model, we generalize the equivalent circuit of LIB. The slope of the low-frequency rectilinear part of the Nyquist diagram does not always unambiguously determine the subdiffusion index and can be either larger or smaller than the slope corresponding to normal diffusion. The new aspect of capacity degradation related to a change in the type of ion diffusion in LIB components is discussed.

The development of portable electronic devices has greatly stimulated the need for miniaturized power sources. Planar supercapacitors are micro-scale electrochemical energy storage devices that can be integrated with other microelectronic devices on a chip. In this paper, we study the behavior of microsupercapacitors with in-plane interdigital electrodes of carbon nanotube array under sinusoidal excitation, step voltage input and sawlike voltage input. Considering the anomalous diffusion of ions in the array and interelectrode space, we propose a fractional-order equivalent circuit model that successfully describes the measured impedance spectra. We demonstrate that the response of the investigated micro-supercapacitors is linear and the system is time-invariant. The numerical inversion of the Laplace transforms for electric current response in an equivalent circuit with a given impedance leads to results consistent with potentiostatic measurements and cyclic voltammograms. The use of electrodes based on an ordered array of nanotubes reduces the role of nonlinear effects in the behavior of a supercapacitor. The effect of the disordering of nanotubes with increasing array height on supercapacitor impedance is considered in the framework of a distributed-order subdiffusion model.

The kinetics of the subdiffusion-limited growth of spherical precipitates is studied. The process is described by the equation of anomalous diffusion with a fractional derivative with respect to time. It is shown that a decrease in the concentration of monomers is described by the law exp(−kt ^3α/2) at the initial stage and the power law t ^−α at large times, where 0 < α ≤ 1 is the dispersion parameter coinciding with the order of time derivative in the subdiffusion equation. The time dependence of the size of a spherical precipitate is obtained. The results generalize the Ham diffusion theory and are in agreement with the Monte Carlo simulation data.

Kinetic models of aggregation and dissolution of clusters in disordered heterogeneous materials based on subdiffusive equations containing fractional derivatives are studied. Using the generalized fractional Fick law and fractional Fokker–Planck equation for impurity diffusion with localization, we consider modifications of the classical models of Ham, Aaron–Kotler, and Lifshitz–Slezov for nucleation and decomposition of solid solutions. The asymptotic time dependencies of supersaturation degree, average cluster size, and other characteristics at the stages of subdiffusion-limited nucleation and coalescence are calculated and analyzed.