| In this work, we examine pattern formation as a generic phenomenon as well as its occurrence in electrochemical systems. First, a global feedback control of pattern formation in a wide class of systems described by the Swift-Hohenberg (SH) equation is investigated theoretically, by means of stability analysis and numerical simulations. Two cases are considered: (i) feedback control of the competition between hexagon and roll patterns described by a supercritical SH equation, and (ii) the use of feedback control to suppress the blow-up in a system described by a subcritical SH equation. In case (i), it is shown that feedback control can change the hexagon and roll stability regions in the parameter space as well as cause a transition from up- to down-hexagons and stabilize a skewed (mixed mode) hexagonal pattern. In case (ii), it is demonstrated that feedback control can suppress blow-up and lead to the formation of spatially-localized patterns in the weakly nonlinear regime. The effects of a delayed feedback are also investigated for both cases, and it is shown that delay can induce temporal oscillations as well as blow-up.;Next, pattern formation resulting from the self-organization of porous nanostructures in anodic metal oxide is considered. Two possible mechanisms are proposed: (i) elastic stress caused by electrostrictive effects, and (ii) electrochemical transport of oxygen ions within the oxide layer. In each case, a mathematical model is developed which also incorporates the chemical reactions at the metal-oxide and oxide-electrolyte interfaces. It is then shown through linear stability analysis that a short-wave instability exists in certain parameter regimes for both cases which can lead to the formation of hexagonally ordered pores observed in anodized aluminum oxide. Numerical simulations of case (ii) validate these results.;Finally, we consider the self-organization of intercalating particles within crystals induced by spinodal decomposition and the anisotropy of coherency strains. In particular, we examine lithium intercalation in lithium iron phosphate crystalline cathodes during battery charging and discharging. A phase-field model is developed which incorporates the energetic contributions from the entropy, enthalpy and elastic properties of the host material. Bulk dynamics of the system is then analyzed using linear stability theory to obtain spinodal regions in the parameter space, and numerical simulations are used to predict long-term evolution and relaxational steady-states. |
