Stripes on finite domains: Why the zigzag instability is only a partial story

On the bandwidth of stable nonlinear stripe patterns in finite size systems

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers q, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for the Turing pattern in cells. In addition, we explain why below a certain system width, stripe patterns behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.

Bending and pinching of three-phase stripes: From secondary instabilities to morphological deformations in organic photovoltaics

Optimizing the properties of the mosaic nanoscale morphology of bulk heterojunction (BHJ) organic photovoltaics (OPV) is not only challenging technologically but also intriguing from the mechanistic point of view. Among the recent breakthroughs is the identification and utilization of a three-phase (donor-mixed-acceptor) BHJ, where the (intermediate) mixed phase can inhibit mesoscale morphological changes, such as phase separation. Using a mean-field approach, we reveal and distinguish between generic mechanisms that alter, through transverse instabilities, the evolution of stripes: the bending (zigzag mode) and the pinching (cross-roll mode) of the donor-acceptor domains. The results are summarized in a parameter plane spanned by the mixing energy and illumination, and show that donor-acceptor mixtures with higher mixing energy are more likely to develop pinching under charge-flux boundary conditions. The latter is notorious as it leads to the formation of disconnected domains and hence to loss of charge flux. We believe that these results provide a qualitative road map for BHJ optimization, using mixed-phase composition and, therefore, an essential step toward long-lasting OPV. More broadly, the results are also of relevance to study the coexistence of multiple-phase domains in material science, such as in ion-intercalated rechargeable batteries.